Well-posedness and properties of the flow for semilinear evolution equations

Abstract

We derive conditions for well-posedness of semilinear evolution equations with unbounded input operators. Based on this, we provide sufficient conditions for such properties of the flow map as Lipschitz continuity, bounded-implies-continuation property, boundedness of reachability sets, etc. These properties represent a basic toolbox for stability and robustness analysis of semilinear boundary control systems. We cover systems governed by general $C_0$-semigroups, and analytic semigroups that may have both boundary and distributed disturbances. We illustrate our findings on an example of a Burgers’ equation with nonlinear local dynamics and both distributed and boundary disturbances.

1 Introduction

1.1 Semilinear evolution equations

In this work, we analyze the well-posedness and properties of the flow for semilinear evolution equations of the form

$$\begin{aligned} \dot{x}(t)= & {} Ax(t) + B_2f(x(t),u(t)) + Bu(t),\quad t>0, \end{aligned}$$

(1a)

$$\begin{aligned} x(0)= & {} x_0. \end{aligned}$$

(1b)

Here, A generates a strongly continuous semigroup over a Banach space X, the operators B and $B_2$ are admissible with respect to some function space, and f is Lipschitz continuous in the first variable (see Assumption 3.3 for precise requirements on f). This class of systems is rather general:

• If B and $B_2$ are bounded operators, (1) corresponds to the classic semilinear evolution equations covering broad classes of semilinear PDEs with distributed inputs. If A is a bounded operator, such a theory was developed in [7]. In the case of unbounded generators A, we refer to [15, 43], [Chapter 11][3, 6], etc.

• If $B_2=0$, and B is an admissible operator, then (1) reduces to the class of general linear control systems that fully covers linear boundary control systems (see [6, 11, 23, 56, 57] for an overview). In particular, this class includes linear evolution PDEs with boundary inputs.

• Consider a linear system

  $$\begin{aligned} \dot{x} = Ax + Bv, \end{aligned}$$

  (2)

  with admissible B. Let us apply a feedback controller $v(x)=f(x,u_1)+u_2$ that is subject to additive actuator disturbance $u_2$ and further disturbance input $u_1$. Substituting this controller into (2), we arrive at systems (1), with $B_2=B$.

• In [17], it was shown that the class of systems (1) includes 2D Navier–Stokes equations (under certain boundary conditions) with in-domain inputs and disturbances. Furthermore, in [17] the authors have designed an error feedback controller that guarantees approximate local velocity output tracking for a class of reference outputs. Viscous Burgers’ equation with nonlinear local terms and boundary inputs of Dirichlet or Neumann type falls into the class (1) as well.

• In [51], it was shown that semilinear boundary control systems with linear boundary operators could be considered a special case of systems (1). In this case, it suffices to consider $B_2$ as the identity operator. Furthermore, in [51], the well-posedness and input-to-state stability of a class of analytic boundary control systems with nonlinear dynamics and a linear boundary operator were analyzed with the methods of operator theory.

1.2 ISS for infinite-dimensional systems

Our main motivation to analyze the systems (1) stems from the robust stability theory. During the last decade, we have witnessed tremendous progress in robust stability analysis of nonlinear infinite-dimensional systems subject to unknown unstructured disturbances. Input-to-state stability (ISS) framework admits a significant place in this development, striving to become a unifying paradigm for robust control and observation of PDEs and their interconnections, including ODE–PDE and PDE–PDE cascades [30, 38, 51].

Powerful techniques proposed to analyze the ISS property include: criteria of ISS and ISS-like properties in terms of weaker stability concepts [20, 40, 48], constructions of ISS Lyapunov functions for PDEs with in-domain and/or boundary controls [10, 44, 55, 60], efficient functional-analytic methods for the study of linear systems with unbounded input operators (including linear boundary control systems) [20, 22, 24, 29, 30, 33, 59], non-coercive ISS Lyapunov functions [19, 40], as well as small-gain stability analysis of finite [8, 26, 28, 35] and infinite networks, [9, 31, 32, 37], etc.

To make this powerful machinery work for any given system, one needs to verify its well-posedness, properties of reachability sets, and regularity of the flow induced by this system. Usually this is done for PDE systems in a case-by-case manner. In this paper, motivated by [51], we develop sufficient conditions that help to derive these crucial properties for systems (1), which cover many important PDE systems.

1.3 State of the art

The systems (1) have been studied (up to the assumptions on f, and the choice of the space of admissible inputs) in [41] under the requirement that its linearization is an exponentially stable regular linear system in the sense of [53, 56, 57]. Natarajan and Bentsman [41] ensure local well-posedness of regular nonlinear systems assuming the Lipschitz continuity of nonlinearity, and invoking regularity of the linearization. On this basis, the authors show in [41] that an error feedback controller designed for robust output regulation of a linearization of a regular nonlinear system achieves approximate local output regulation for the original regular nonlinear system.

Control of systems (1) has been studied recently in several papers. In particular, in [42], the exact controllability of a class of regular nonlinear systems was studied using back-and-forth iterations. A problem of robust observability was studied for a related class of systems in [25].

Stabilization of linear port-Hamiltonian systems by means of nonlinear boundary controllers was studied in [2, 46]. Bounded controls with saturations (a priori limitations of the input signal) have been employed for PDE control in [39, 45, 54]. Recently, several papers appeared that treat nonlinear boundary control systems within the input-to-state stability framework. Nonlinear boundary feedback was employed for the ISS stabilization of linear port-Hamiltonian systems in [50].

Several types of infinite-dimensional systems, distinct from (1), have been studied as well. One of such classes is time-variant infinite-dimensional semilinear systems that have been first studied (as far as the author is concerned) for systems without disturbances in [18]. Recently, in [49], sufficient conditions for well-posedness and uniform global stability have been obtained for scattering-passive semilinear systems (see [49, Theorem 3.8]).

Another important extension of (1) are semilinear systems with outputs. Such systems with globally Lipschitz nonlinearities have been analyzed in [57, Section 7], and it was shown that such systems are well-posed and forward complete provided that the Lipschitz constant is small enough. In [14] employing a counterexample, it was shown that a linear transport equation with a locally Lipschitz boundary feedback might fail to be well-posed. Well-posedness of incrementally scattering-passive nonlinear systems with outputs has been analyzed in [52] by applying Crandall–Pazy theorem [5] on generation of nonlinear contraction semigroups to a Lax–Phillips nonlinear semigroup representing the system together with its inputs and outputs.

1.4 Contribution

Our first main result is Theorem 3.7 guaranteeing (under proper conditions on f and the input operators) the local existence and uniqueness of solutions for the system (1) with a locally essentially bounded input u.

There are several existence and uniqueness theorems in the literature. For example, [41, Proposition 3.2] covers semilinear systems with $L^\infty$-inputs; [57, Theorem 7.6], [16, Lemma 2.8] treat the case of bilinear systems of various type, and [51] considers the case of systems with linearly bounded nonlinearities. In contrast to the usual formulations of such results (including a closely related result [41, Proposition 3.2]), we also provide a uniform existence time for solutions that controls the maximal deviation of the trajectory from the given set of initial conditions.

Next, we show in Theorems 3.17 and 3.18 that under natural conditions, the system (1) is a well-posed control system in the sense of [38]. Finally, we study the fundamental properties of the flow map, such as Lipschitz continuity with respect to initial states, boundedness of reachability sets, and boundedness-implies-continuation property. These properties are important in their own right. Moreover, they are key components for the robust stability analysis of systems (1) as we explained before.

The structure of semilinear evolution equations allows combining the “linear” methods of admissibility theory with “nonlinear” methods, such as fixed point theorems and Lyapunov methods. We consider the case of general $C_0$-semigroups and the special case of analytic semigroups, for which one can achieve stronger results. This synergy of tools is one of the novelties of this paper. For systems without inputs and without the presence of unbounded operators, the existence and uniqueness results as well as the properties of the flow are classical both for general and analytic case [15, 43]. To show the applicability of our methods, we analyze well-posedness of semilinear parabolic systems with Dirichlet boundary inputs (motivated by [15, p. 57]). Also, we reformulate semilinear boundary control systems in terms of evolution equations, which makes our results applicable to boundary control systems as well.

As argued at the previous pages, having developed conditions ensuring the well-posedness and “nice” properties of the flow map of systems (1), we can analyze the ISS of (1) via such powerful tools as coercive and non-coercive ISS Lyapunov functions [19], ISS superposition theorems [40], small-gain theorems for general systems [37], etc. We expect that this will help to prove many results available for particular PDE systems, in a more general fashion. For example, see [51] for an abstract version of the results obtained for particular classes of parabolic systems in [60]. To make the paper accessible for the researchers trained primarily in nonlinear control and nonlinear ISS theory, we spell out the proofs in great detail with a tutorial flavor.

1.5 Notation

By $\mathbb {N}$, $\mathbb {R}$, $\mathbb {R}_+$, we denote the sets of natural, real, and nonnegative real numbers, respectively. $\overline{S}$ denotes the closure of a set S (in a given topology).

By $t \rightarrow a \pm 0$, we denote the fact that t approaches a from the right/left.

Vector spaces considered in this paper are assumed to be real.

Let S be a normed vector space. The distance from $z\in S$ to the set $Z \subset S$ we denote by $\textrm{dist}\,(z,Z){:}{=}\inf \{\Vert z-y\Vert _S: y \in Z \}$. We denote an open ball of radius r around $Z \subset S$ by $B_{r,S}(Z){:}{=}\{y \in X:\textrm{dist}\,(y,Z)<r\}$, and we set also $B_{r,S}(x){:}{=}B_{r,S}(\{x\})$ for $x \in X$, and $B_{r,S}{:}{=}B_{r,S}(0)$. If $S =X$ (the state space of the system), we write for short $B_{r}(Z){:}{=}B_{r,X}(Z)$, $B_{r}(x){:}{=}B_{r,X}(x)$, etc.

Denote by $\mathcal {K}$ the class of continuous strictly increasing functions $\gamma :\mathbb {R}_+\rightarrow \mathbb {R}_+$, satisfying $\gamma (0)=0$. $\mathcal {K_\infty }$ denotes the set of unbounded functions from $\mathcal {K}$.

For normed vector spaces X, U, denote by L(X, U) the space of bounded linear operators from X to U. We endow L(X, U) with the standard operator norm $\Vert A\Vert {:}{=}\sup _{\Vert x\Vert _X=1}\Vert Ax\Vert _U$. We write for short $L(X){:}{=}L(X,X)$. By C(X, U), we denote the space of continuous maps from X to U. Similarly, by $C(\mathbb {R}_+,X)$ we understand the space of continuous maps from $\mathbb {R}_+$ to X. The domain of definition, kernel, and image of an operator A we denote by D(A), $\textrm{Ker}\,(A)$, and $\textrm{Im}(A)$, respectively. By $\sigma (A)$, we denote the spectrum of a closed operator $A:D(A)\subset X \rightarrow X$, and by $\rho (A)$ the resolvent set of A. We denote by $\omega _0(T)$ the growth bound of a $C_0$-semigroup T.

Let X be a Banach space, and let I be a closed subset of $\mathbb {R}$. We define for $p\in [1,\infty )$ the following spaces of vector-valued functions

$$\begin{aligned} M(I,X)&{:}{=}&\{f: I \rightarrow X: f \text { is strongly measurable} \},\\ L^p(I,X)&{:}{=}&\Big \{f \in M(I,X): \Vert f\Vert _{L^p(I,X)}{:}{=}\Big (\int _I\Vert f(s)\Vert ^p_X{\text {d}}s\Big )^{\frac{1}{p}}< \infty \Big \},\\ L^p_{{{\,\textrm{loc}\,}}}(\mathbb {R}_+,X)&{:}{=}&\{{f|_{[0,t]}} \in L^p([0,t],X)\quad \forall t>0\},\\ L^\infty (I,X)&{:}{=}&\{f \in M(I,X): \Vert f\Vert _{L^\infty (I,X)}{:}{=}\mathop {\mathrm {ess\,sup}}\limits _{s\in I}\Vert f(s)\Vert _X < \infty \},\\ L^\infty _{{{\,\textrm{loc}\,}}}(I,X)&{:}{=}&\{f \in L^\infty ([0,t],X)\quad \forall t>0\}. \end{aligned}$$

Denote also $L^p(a,b)$ ${:}{=}L^p([a,b],\mathbb {R})$, where $p\in [1,\infty ]$. The space $H^{k}(a,b)$, $k\in \mathbb {N}$, is a Sobolev space of functions $u \in L^2(a,b)$, such that for each natural $j\le k$, the weak derivative $u^{(j)}$ exists and belongs to $L^2(a,b)$. $H^{k}(a,b)$ is endowed with the norm $u\mapsto \Big (\sum _{j \le k}\int _a^b{\big | u^{(j)} (x)\big |^2 dx} \Big )^{\frac{1}{2}}$. $H^{k}_0(a,b)$ denotes the closure of smooth functions with compact support in (a, b) in the norm of $H^{k}(a,b)$,   $k\in \mathbb {N}$.

2 General class of systems

We start with a general definition of a control system that we adopt from [38].

Definition 2.1

Consider the triple $\Sigma =(X,\mathcal {U},\phi )$ consisting of

• (i) A normed vector space $(X,\Vert \cdot \Vert _X)$, called the state space, endowed with the norm $\Vert \cdot \Vert _X$.

• (ii) A normed vector space of inputs $\mathcal {U}\subset \{u:\mathbb {R}_+ \rightarrow U\}$ endowed with a norm $\Vert \cdot \Vert _{\mathcal {U}}$, where U is a normed vector space of input values. We assume that the following two axioms hold: The axiom of shift invariance: for all $u \in \mathcal {U}$ and all $\tau \ge 0$ the time shift $u(\cdot + \tau )$ belongs to $\mathcal {U}$ with $\Vert u\Vert _\mathcal {U}\ge \Vert u(\cdot + \tau )\Vert _\mathcal {U}$. The axiom of concatenation: for all $u_1,u_2 \in \mathcal {U}$ and for all $t>0$ the concatenation of $u_1$ and $u_2$ at time t, defined by

  $$\begin{aligned} {\big ({u_1\, \underset{t}{\lozenge }\,{u_2}}\big )(\tau ){:}{=}} {\left\{ \begin{array}{ll} u_1(\tau ), &{} \text { if } \tau \in [0,t], \\ u_2(\tau -t), &{} \text { otherwise}, \end{array}\right. } \end{aligned}$$

  (3)

  belongs to $\mathcal {U}$.

• (iii) A map $\phi :D_{\phi } \rightarrow X$, $D_{\phi }\subseteq \mathbb {R}_+ \times X \times \mathcal {U}$ (called transition map), such that for all $(x,u)\in X \times \mathcal {U}$ it holds that $D_{\phi } \cap \big (\mathbb {R}_+ \times \{(x,u)\}\big ) = [0,t_m)\times \{(x,u)\} \subset D_{\phi }$, for a certain $t_m=t_m(x,u)\in (0,+\infty ]$. The corresponding interval $[0,t_m)$ is called the maximal domain of definition of $t\mapsto \phi (t,x,u)$.

The triple $\Sigma$ is called a (control) system, if the following properties hold:

$(\sum 1)$:

The identity property: for every $(x,u) \in X \times \mathcal {U}$ it holds that $\phi (0, x,u)=x$.

$(\sum 2)$:

Causality: for every $(t,x,u) \in D_\phi$, for every $\tilde{u} \in \mathcal {U}$, such that $u(s) = \tilde{u}(s)$ for all $s \in [0,t]$ it holds that $[0,t]\times \{(x,\tilde{u})\} \subset D_\phi$ and $\phi (t,x,u) = \phi (t,x,\tilde{u})$.

$(\sum 3)$:

Continuity: for each $(x,u) \in X \times \mathcal {U}$ the map $t \mapsto \phi (t,x,u)$ is continuous on its maximal domain of definition.

$(\sum 4)$:

The cocycle property: for all $x \in X$, $u \in \mathcal {U}$, for all $t,h \ge 0$ so that $[0,t+h]\times \{(x,u)\} \subset D_{\phi }$, we have

$$\begin{aligned} \phi \big (h,\phi (t,x,u),u(t+\cdot )\big )=\phi (t+h,x,u). \end{aligned}$$

Definition 2.1 can be viewed as a direct generalization, and a unification of the concepts of strongly continuous nonlinear semigroups [4, 5] with abstract linear control systems [58].

This class of systems encompasses control systems generated by ordinary differential equations (ODEs), switched systems, time-delay systems, evolution partial differential equations (PDEs), abstract differential equations in Banach spaces and many others [27, Chapter 1].

Definition 2.2

We say that a control system $\Sigma =(X,\mathcal {U},\phi )$ is forward complete (FC), if $D_\phi = \mathbb {R}_+ \times X\times \mathcal {U}$, that is for every $(x,u) \in X \times \mathcal {U}$ and for all $t \ge 0$ the value $\phi (t,x,u) \in X$ is well defined.

Forward completeness alone does not imply, in general, the existence of any uniform bounds on the trajectories emanating from bounded balls that are subject to uniformly bounded inputs [40, Example 2, p. 1612]. Systems exhibiting such bounds deserve a special name.

Definition 2.3

We say that a control system $\Sigma =(X,\mathcal {U},\phi )$ has bounded reachability sets (BRS), if for any $C>0$ and any $\tau >0$ it holds that

$$\begin{aligned} \sup \big \{ \Vert \phi (t,x,u)\Vert _X: \Vert x\Vert _X\le C,\ \Vert u\Vert _{\mathcal {U}} \le C,\ t \in [0,\tau ]\big \} < \infty . \end{aligned}$$

For a wide class of control systems, the boundedness of a solution implies the possibility of prolonging it to a larger interval, see [27, Chapter 1]. Next, we formulate this property for abstract systems:

Definition 2.4

We say that a control system $\Sigma =(X,\mathcal {U},\phi )$ satisfies the boundedness-implies-continuation (BIC) property if for each $(x,u)\in X \times \mathcal {U}$ with $t_m(x,u)<\infty$ it holds that

$$\begin{aligned} \limsup _{t \rightarrow t_m(x,u)-0}\Vert \phi (t,x,u)\Vert _X = \infty . \end{aligned}$$

3 Semilinear evolution equations with unbounded input operators

Consider a Cauchy problem for infinite-dimensional evolution equations of the form

$$\begin{aligned} \dot{x}(t)= & {} Ax(t) + B_2f(x(t),u(t)) + Bu(t),\quad t>0, \end{aligned}$$

(4a)

$$\begin{aligned} x(0)= & {} x_0, \end{aligned}$$

(4b)

where $A: D(A)\subset X \rightarrow X$ generates a strongly continuous semigroup $T=(T(t))_{t\ge 0}$ of bounded linear operators on a Banach space X; U is a Banach space of input values, and $x_0\in X$ is a given initial condition. As the input space, we take $\mathcal {U}{:}{=}L^\infty (\mathbb {R}_+,U)$.

The map $f: X\times U \rightarrow V$ is defined on the whole $X \times U$ and maps to a Banach space V. Furthermore, $B\in L(U,X_{-1})$ and $B_2 \in L(V,X_{-1})$. Here, the extrapolation space $X_{-1}$ is the closure of X in the norm $x \mapsto \Vert (aI-A)^{-1}x\Vert _X$, $x \in X$, where $a \in \rho (A)$ (different choices of $a \in \rho (A)$ induce equivalent norms on X). Note that the operators B and $B_2$ are unbounded, if they are understood as operators that map to X.

3.1 Admissible input operators and mild solutions

First, consider the linear counterpart of the system (4).

$$\begin{aligned} \dot{x}(t)= & {} Ax(t) + Bu(t),\quad t>0, \end{aligned}$$

(5a)

$$\begin{aligned} x(0)= & {} x_0, \end{aligned}$$

(5b)

for the same A, B as above. As the image of B does not necessarily lie in X, one has to be careful when defining the concept of a solution for (5). Since $B \in L(U,X_{-1})$, it is natural to consider the system (5) on the space $X_{-1}$. Note that the semigroup $(T(t))_{t\ge 0}$ extends uniquely to a strongly continuous semigroup $(T_{-1}(t))_{t\ge 0}$ on $X_{-1}$ whose generator $A_{-1}$ acting in $X_{-1}$ is an extension of A with $D(A_{-1}) = X$, see, e.g., [12, Section II.5]. Recall the definitions of the spaces $L^p$, $L^p_{{{\,\textrm{loc}\,}}}$ from Sect. 1.

The mild solution of (5) for any $x \in X$ and $u \in L^1_{{{\,\textrm{loc}\,}}}(\mathbb {R}_+,U)$ is given by

$$\begin{aligned} \phi _L(t,x,u) = T(t)x + \int _0^t T_{-1}(t-s)Bu(s){\text {d}}s,\quad t\ge 0. \end{aligned}$$

The integral term here, however, belongs in general to $X_{-1}$.

Thus, the existence and uniqueness of a mild solution depend on whether $\int _0^t T_{-1}(t-s)Bu(s){\text {d}}s \in X$. This leads to the following concept:

Definition 3.1

Let $q\in [1,\infty ]$. The operator $B\in L(U,X_{-1})$ is called a q-admissible control operator for $(T(t))_{t\ge 0}$, if there is $t>0$ so that

$$\begin{aligned} u\in L^q_{{{\,\textrm{loc}\,}}}(\mathbb {R}_+,U) \quad \Rightarrow \quad \int _0^t T_{-1}(t-s)Bu(s){\text {d}}s\in X. \end{aligned}$$

(6)

Define for each $t\ge 0$ an operator $\Phi (t): L^q_{{{\,\textrm{loc}\,}}}(\mathbb {R}_+,U) \rightarrow X_{-1}$ by

$$\begin{aligned} \Phi (t)u {:}{=} \int _0^t T_{-1}(t-s)Bu(s){\text {d}}s. \end{aligned}$$

Note that as $B \in L(U,X_{-1})$, the operators $\Phi (t)$ are well defined as maps from $L^1_{{{\,\textrm{loc}\,}}}(\mathbb {R}_+,U)$ to $X_{-1}$ for all t. The next result (see [58, Proposition 4.2], [56, Proposition 4.2.2]) shows that q-admissibility of B ensures that the image of $\Phi (t)$ is in X for all $t \ge 0$ and $\Phi (t)\in L(L^q(\mathbb {R}_+,U),X)$ for all $t>0$.

Proposition 3.2

Let X, U be Banach spaces, and let $q \in [1,\infty ]$ be given. Then, $B \in L(U,X_{-1})$ is q-admissible if and only if for all $t>0$ there is $h_t>0$ so that for all $u \in L^{q}_{{{\,\textrm{loc}\,}}}(\mathbb {R}_+,U)$ it holds that $\Phi (t)u \in X$ and

$$\begin{aligned} \left\| \int _0^t T_{-1}(t-s)Bu(s)\,{\text {d}}s\right\| _X \le h_t \Vert u\Vert _{L^q([0,t],U)}. \end{aligned}$$

(7)

The function $t\mapsto h_t$ we assume wlog to be non-decreasing in t.

An important consequence of Proposition 3.2 is that well-posedness (and thus forward completeness) of the system (5) already implies the boundedness of reachability sets property for (5), with a bound given by (7).

As $t\mapsto h_t$ is non-decreasing in t, there is a limit $h_0{:}{=}\lim _{t\rightarrow +0}h_t \ge 0$, which is not necessarily zero. Operators for which $h_0=0$ deserve a special name.

Definition 3.3

Let $q \in [1,\infty ]$. A q-admissible operator $B\in L(U, X_{-1})$ is called zero-class q-admissible, if the constants $(h_t)_{t>0}$ can be chosen such that $h_0=0$.

All $B \in L(U,X)$ are zero-class 1-admissible. If X is reflexive, then 1-admissible operators are necessarily bounded. At the same time, there are unbounded zero-class admissible operators, see Proposition 4.2. Consider [21, Examples 3.8, 3.9] for unbounded admissible observation operators that are not zero-class admissible.

The above considerations motivate us to impose

Assumption 3.1

The operator $B\in L(U,X_{-1})$ is $\infty$-admissible, and the map $(t,u) \mapsto \Phi (t)u$ is continuous on $\mathbb {R}_+\times L^\infty (\mathbb {R}_+,U)$.

In particular, this assumption holds if B is a q-admissible operator with $q<\infty$, see [58, Proposition 2.3].

To define the concept of a mild solution, we also require the following:

Assumption 3.2

We assume that $B_2$ is zero-class $\infty$-admissible and for all $u\in L^\infty (\mathbb {R}_+,U)$ and any $x\in C(\mathbb {R}_+,X)$ the map $s\mapsto f\big (x(s),u(s)\big )$ is in $L^\infty _{{{\,\textrm{loc}\,}}}(\mathbb {R}_+,V)$.

Due to [20, Proposition 2.5], these conditions ensure that for above x, u the map

$$\begin{aligned} t \mapsto \int _0^t T_{-1}(t-s)B_2f\big (x(s),u(s)\big ){\text {d}}s \end{aligned}$$

(8)

is well defined and continuous on $\mathbb {R}_+$.

Remark 3.4

Assumption 3.2 holds, in particular, if $B_2 \in L(V,X)$, and

• (i) $f(x,u)=g(x) + Ru$, $x \in X$, $u\in U$, where $R\in L(U,V)$, and g is continuous on X. Indeed, for a continuous x, the map $s\mapsto g\big (x(s)\big )$ is continuous either, and thus Riemann integrable. The map $s\mapsto T(t-s) B_2Ru(s)$ is Bochner integrable for any $u \in L^1_{{{\,\textrm{loc}\,}}}(\mathbb {R}_+,U)$ by [1, Proposition 1.3.4], [23, Lemma 10.1.6]. This ensures that Assumption 3.2 holds.

• (ii) If f is continuous on $X \times U$, and u is piecewise right-continuous, then the map $s\mapsto f\big (x(s),u(s)\big )$ is also piecewise right-continuous, and thus, it is Riemann integrable.

• (iii) (ODE systems). Let $X=\mathbb {R}^n$, $U=\mathbb {R}^m$, $A=0$ (and thus $T(t)={\text {id}}$ for all t), $B_2={\text {id}}$, $B=0$, and f be continuous on $X \times U$. With these assumptions, the equations (4) take the form

  $$\begin{aligned} \dot{x} = f(x,u). \end{aligned}$$

  (9)

  Then for each $u\in L^\infty (\mathbb {R}_+,U)$ and each $x \in C(\mathbb {R}_+,X)$, the map $s\mapsto f(x(s),u(s))$ is Lebesgue integrable, and thus, Assumption 3.2 holds. Indeed, as x is a solution of (9) on $[0,\tau )$, x is continuous on $[0,\tau )$. By assumptions, u is measurable on $[0,\tau )$, and f is continuous on $\mathbb {R}^n \times \mathbb {R}^m$. Arguing similarly to [47, Proposition 7] (where it was shown that a composition of a continuous and measurable function defined on a measurable set E is measurable on E), we see that the map $q: [0,\tau )\rightarrow \mathbb {R}^n$, $q(s){:}{=} f(x(s),u(s))$, is a measurable map. As u is essentially bounded, and x and f map bounded sets into bounded sets, q is essentially bounded on $[0,\tau )$. Thus, $q \in L^{\infty }(\mathbb {R}_+,\mathbb {R}^n)$, and thus, q is integrable on $[0,\tau )$.

Next we define mild solutions of (4).

Definition 3.5

(Mild solutions) Let Assumptions 3.1 and 3.2 hold and $\tau >0$ be given. A function $x \in C([0,\tau ], X)$ is called a mild solution of (4) on $[0,\tau ]$ corresponding to certain $x_0\in X$ and $u \in L^\infty _{{{\,\textrm{loc}\,}}}(\mathbb {R}_+,U)$, if x solves the integral equation

$$\begin{aligned} x(t)=T(t) x_0 + \int _0^t T_{-1}(t-s) B_2f\big (x(s),u(s)\big ){\text {d}}s + \int _0^t T_{-1}(t-s) Bu(s){\text {d}}s. \end{aligned}$$

(10)

Here, the integrals are Bochner integrals of $X_{-1}$-valued maps.

We say that $x:\mathbb {R}_+\rightarrow X$ is a mild solution of (4) on $\mathbb {R}_+$ corresponding to certain $x_0\in X$ and $u \in L^\infty _{{{\,\textrm{loc}\,}}}(\mathbb {R}_+,U)$, if $x|_{[0,\tau ]}$ is a mild solution of (4) (with $x_0, u$) on $[0,\tau ]$ for all $\tau >0$.

3.2 Local existence and uniqueness

Assumptions 3.1, 3.2 guarantee that the integral terms in (10) are well defined. To ensure the existence and uniqueness of mild solutions, we impose further restrictions on f.

Recall the notation $B_{C,U} = \{v \in U:\Vert v\Vert _U < C\}$ and $B_{C} = \{v \in X:\Vert v\Vert _X < C\}$.

Definition 3.6

We call $f:X \times U \rightarrow V$

• (i) Lipschitz continuous (with respect to the first argument) on bounded subsets of X if for any $C>0$ there is $L(C)>0$, such that $\forall x,y \in B_C$, $\forall v \in B_{C,U}$ it holds that

  $$\begin{aligned} \Vert f(y,v)-f(x,v)\Vert _V \le L(C) \Vert y-x\Vert _X. \end{aligned}$$

  (11)

• (ii) uniformly globally Lipschitz continuous (with respect to the first argument) if (11) holds for all $x,y \in X$, and all $v \in U$ with a constant L that does not depend on x, y, v.

We omit the indication “with respect to the first argument” wherever this is clear from the context.

For the well-posedness analysis, we rely on the following assumption on the nonlinearity f in (4).

Assumption 3.3

The nonlinearity f satisfies the following properties:

• (i) $f: X \times U \rightarrow V$ is Lipschitz continuous on bounded subsets of X.

• (ii) $f(x,\cdot )$ is continuous for all $x \in X$.

• (iii) There exist $\sigma \in \mathcal {K_\infty }$ and $c>0$ so that for all $u \in U$ the following holds:

  $$\begin{aligned} \Vert f(0,u)\Vert _V \le \sigma (\Vert u\Vert _U) + c. \end{aligned}$$

  (12)

Recall the notation for the distances and balls in normed vector spaces, introduced in the end of Sect. 1. Finally, for a set $\mathcal {S}\subset U$, denote the set of inputs with essential image in $\mathcal {S}$ as $\mathcal {U}_\mathcal {S}$:

$$\begin{aligned} \mathcal {U}_\mathcal {S}{:}{=}\{u\in \mathcal {U}: u(t) \in \mathcal {S}, \text { for a.e. } t\in \mathbb {R}_+\}. \end{aligned}$$

(13)

We start with the following sufficient condition for the existence and uniqueness of solutions of a system (4) with inputs in $L^\infty (\mathbb {R}_+,U)$.

Recall the notation $h_0{:}{=}\lim _{t\rightarrow +0}h_t$, where $h_t$ is defined as in (7).

Theorem 3.7

(Picard–Lindelöf theorem) Let Assumptions 3.1, 3.2, and 3.3 hold.

Assume that $(T(t))_{t\ge 0}$ satisfies for certain $M \ge 1$, $\lambda >0$ the estimate

$$\begin{aligned} \Vert T(t)\Vert \le Me^{\lambda t},\quad t\ge 0. \end{aligned}$$

(14)

For any compact set $Q \subset X$, any $r>0$, any bounded set $\mathcal {S}\subset U$, and any $\delta >0$, there is a time $t_1 = t_1(Q,r,\mathcal {S},\delta )>0$, such that for any $w\in Q$, for any $x_0 \in W{:}{=}B_r(w)$, and for any $u\in \mathcal {U}_{\mathcal {S}}$ there is a unique mild solution of (4) on $[0,t_1]$, and $\phi ([0,t_1],x_0,u) \subset B_{Mr+h_0\Vert u\Vert _{L^\infty ([0,t_1],U)}+\delta }(w)$.

Proof

First, we show the claim for the case if Q is a single point in X, that is, $Q=\{w\}$, for some $\omega \in X$. Pick any $C>0$ such that $W {:}{=} B_r(w) \subset B_C$, and $\mathcal {U}_\mathcal {S}\subset B_{C,\mathcal {U}}$. Pick any $u \in \mathcal {U}_\mathcal {S}$. Also, take any $\delta >0$, and consider the following sets (depending on the parameter $t>0$):

$$\begin{aligned} \hspace{10mm}Y_{t}{:}{=} \big \{x \in C([0,t],X): \sup _{s \in [0,t]} \Vert x(s) - w\Vert _X\le Mr + h_0\Vert u\Vert _{L^\infty ([0,t],U)} + \delta \big \}, \nonumber \\ \end{aligned}$$

(15)

endowed with the metric $\rho _{t}(x,y){:}{=}\sup _{s \in [0,t]} \Vert x(s)-y(s)\Vert _X$. As the sets $Y_t$ are closed subsets of the Banach spaces C([0, t], X), for all $t>0$, the space $Y_t$ is a complete metric space.

Pick any $x_0 \in W$. We are going to prove that for small enough t, the spaces $Y_{t}$ are invariant under the operator $\Phi _u$, defined for any $x \in Y_{t}$ and all $\tau \in [0,t]$ by

$$\begin{aligned} \Phi _u(x)(\tau )&{{:}{=}} T(\tau )x_0 + \int _0^\tau T_{-1}(\tau -s)B_2f\big (x(s),u(s)\big ){\text {d}}s + \int _0^\tau T_{-1}(\tau -s) Bu(s){\text {d}}s. \end{aligned}$$

(16)

By Assumptions 3.1, 3.2, the function $\Phi _u(x)$ is continuous for any $x \in Y_t$.

Fix any $t>0$ and pick any $x \in Y_{t}$. As $x_0 \in W=B_r(w)$, there is $a \in B_r$ such that $x_0=w+a$.

Then for any $\tau <t$, it holds that

$$\begin{aligned}&\Vert \Phi _{u}(x)(\tau )-w\Vert _X \\&\quad \le \Big \Vert T(\tau )x_0 - w\Big \Vert _X + \Big \Vert \int _0^\tau T_{-1}(\tau -s) Bu(s){\text {d}}s\Big \Vert _X \\&\qquad + \Big \Vert \int _0^\tau T_{-1}(\tau -s) B_2f(x(s),u(s)){\text {d}}s\Big \Vert _X \\&\quad \le \Vert T(\tau )(w+a) - w \Vert _X + h_\tau \Vert u\Vert _{L^\infty ([0,\tau ],U)} \\&\qquad + c_\tau \Vert f(x(\cdot ),u(\cdot ))\Vert _{L^\infty ([0,\tau ],V)} \\&\quad \le \Vert T(\tau )w - w\Vert _X + \Vert T(\tau )a\Vert _X + h_\tau \Vert u\Vert _{L^\infty ([0,\tau ],U)}\\&\qquad + c_\tau \Vert f(x(\cdot ),u(\cdot )) - f(0,u(\cdot ))\Vert _{L^\infty ([0,\tau ],V)} + c_\tau \Vert f(0,u(\cdot ))\Vert _{L^\infty ([0,\tau ],V)}. \end{aligned}$$

Now, for all $s\in [0,t]$

$$\begin{aligned} \Vert x(s)\Vert _X&\le \Vert w\Vert _X+Mr+ h_0\Vert u\Vert _{L^\infty ([0,t],U)} +\delta \\&\le M(\Vert w\Vert _X+r) + h_0 C+\delta \le (M+h_0)C+\delta =:K. \end{aligned}$$

In view of Assumption 3.3(iii), it holds that

$$\begin{aligned} \Vert f(0,u(s))\Vert _V \le \sigma (\Vert u(s)\Vert _U) + c,\quad \text { for a.e. } s \in [0,t]. \end{aligned}$$

As $M\ge 1$, it holds that $K>C$, and the Lipschitz continuity of f on bounded balls ensures that there is $L(K)>0$, such that for all $\tau \in [0,t]$

$$\begin{aligned} \Vert \Phi _{u}(x)(\tau )-w\Vert _X&\le \Vert T(\tau )w - w\Vert _X + Me^{\lambda t} r + h_t \Vert u\Vert _{L^\infty ([0,t],U)} \\&\qquad \qquad + c_\tau \big ( L(K) \Vert x\Vert _{L^\infty ([0,t],X)} + \sigma (\Vert u\Vert _{L^\infty ([0,t],U)}) + c\big )\\&\le \Vert T(\tau )w - w\Vert _X + Me^{\lambda t} r + h_t \Vert u\Vert _{L^\infty ([0,t],U)} \\&\qquad \qquad + c_t \big ( L(K)K + \sigma (C) + c\big ). \end{aligned}$$

Since T is a strongly continuous semigroup, as $h_t\rightarrow h_0$ whenever $t\rightarrow +0$, and since $c_t\rightarrow 0$ as $t\rightarrow +0$, there exists $t_1$, such that

$$\begin{aligned} \Vert \Phi _u(x)(t)-w\Vert _X \le Mr+ h_0 \Vert u\Vert _{L^\infty ([0,t_1],U)} +\delta ,\quad \text { for all } t \in [0,t_1]. \end{aligned}$$

This means that $Y_{t}$ is invariant with respect to $\Phi _u$ for all $t \in (0,t_1]$, and $t_1$ does not depend on the choice of $x_0 \in W$.

Now, pick any $t>0$, $\tau \in [0, t]$, and any $x, y \in Y_{t}$. It holds that

$$\begin{aligned} \Vert \Phi _u(x)(\tau ) - \Phi _u(y)(\tau )\Vert _X&\le \Big \Vert \int _0^\tau T_{-1}(\tau -s) B_2\big (f(x(s),u(s)) - f(y(s),u(s))\big ){\text {d}}s\Big \Vert _X \\&\le c_\tau \Vert f(x(\cdot ),u(\cdot )) - f(y(\cdot ),u(\cdot ))\Vert _{L^\infty ([0,\tau ],V)} \\&\le c_t L(K)\rho _t(x,y) \\&\le \frac{1}{2} \rho _t(x,y), \end{aligned}$$

for $t \le t_2$, where $t_2>0$ is a small enough real number, that does not depend on the choice of $x_0 \in W$.

According to the Banach fixed point theorem, there exists a unique solution of $x(t)=\Phi _u(x)(t)$ on $[0,\min \{t_1,t_2\}]$, which is a mild solution of (4).

General compact Q Till now, we have shown that for any $w\in Q$, any $r>0$, any bounded set $\mathcal {S}\subset U$, and any $\delta >0$, there is a time $t_1 = t_1(w,r,\mathcal {S},\delta )>0$ (that we always take the maximal possible), such that for any $x_0 \in W{:}{=}B_r(w)$, and for any $u\in \mathcal {U}_{\mathcal {S}}$ there is a unique solution of (4) on $[0,t_1]$, and it lies in the ball $B_{Mr+h_0\Vert u\Vert _{L^\infty ([0,t_1],U)}+\delta }(w)$.

It remains to show that $t_1$ can be chosen uniformly in $w \in Q$, that is $\inf _{w \in Q} t_1(w,r,\mathcal {S},\delta )>0$. Let this not be so, that is, $\inf _{w \in Q} t_1(w,r,\mathcal {S},\delta )=0$. Then, there is a sequence $(w_k) \subset Q$, such that the corresponding times $\big (t_1(w_k,r,\mathcal {S},\delta )\big )_{k\in \mathbb {N}}$ monotonically decay to zero. As Q is compact, there is a converging subsequence of $(w_k)$, converging to some $w^* \in Q$. However, $t_1(w^*,r,\mathcal {S},\delta )>0$, which easily leads to a contradiction. $\square$

Remark 3.8

The technique of proving the Picard–Lindelöf theorem is quite classical. Note, however, that here we need to tackle the influence of unbounded input operators, and also we provide a uniform existence time for solutions that controls the maximal deviation of the trajectory from the given set of initial conditions, which is realized by the choice of the spaces $Y_t$ in (15). This leads to several changes in the proof of the invariance of $Y_t$ with respect to the operator $\Phi _u(x)$.

Corollary 3.9

(Picard–Lindelöf theorem for zero-class admissible B and quasi-contractive semigroups) Let Assumptions 3.1, 3.2, 3.3 hold. Let also B be zero-class admissible, and T be a quasi-contractive strongly continuous semigroup; that is, there is $\lambda >0$ such that

$$\begin{aligned} \Vert T(t)\Vert \le e^{\lambda t},\quad t\ge 0. \end{aligned}$$

(17)

For any bounded ball $W \subset X$ (with corresponding $w\in X$ and $r>0$: $W= B_r(w)$), any bounded set $\mathcal {S}\subset U$, and any $\delta >0$, there is a time $t_1 = t_1(W,\mathcal {S},\delta )>0$, such that for any $x_0 \in W$ and any $u\in \mathcal {U}_{\mathcal {S}}$ there is a unique solution of (4) on $[0,t_1]$, and it lies in the ball $B_{r+\delta }(w)$.

Proof

The claim follows directly from Theorem 3.7. $\square$

Remark 3.10

Without an assumption of quasi-contractivity, Corollary 3.9 does not hold. Consider the special case $f \equiv 0$ and $B\equiv 0$. Then, the system (4) is linear, and for a given $x_0\in X$ the solution of (4) exists globally and equals $t\mapsto T(t)x_0$. Now, take $w{:}{=}0$ and pick any $r>0$ and $t_1>0$. Then,

$$\begin{aligned} \sup _{\tau \in [0,t_1]}\sup _{\Vert x\Vert _X\le r}\Vert T(\tau )x\Vert _X= r\sup _{\tau \in [0,t_1]}\Vert T(\tau )\Vert . \end{aligned}$$

Since T is merely strongly continuous, the map $t\mapsto \Vert T(t)\Vert$ does not have to be continuous at $t=0$, and it may happen that $\lim _{t_1\rightarrow 0}\sup _{\tau \in [0,t_1]}\Vert T(\tau )\Vert >1$.

Hence, in general, it is not possible to prove that the solution starting at arbitrary $x_0 \in B_r(w)$ will stay in $B_{r+\delta }(w)$ during a sufficiently small and uniform in $x_0 \in B_r(w)$ time.

The following example shows that Theorem 3.7 does not hold in general if W is a bounded set (and not only a bounded ball over a compact set), even for linear systems governed by contraction semigroups on a Hilbert space.

Example 3.11

Let $X=\ell _2$, and consider a diagonal semigroup, defined by $T(t)x{:}{=}(e^{-kt}x_k)_k$, for all $x=(x_k)_k\in X$ and all $t\ge 0$. This semigroup is strongly continuous and contractive. Consider a bounded and closed set $W{:}{=}\{x \in \ell _2:\ \Vert x\Vert _X =1\}$. Yet $\Vert T(t)e_k\Vert _X = e^{-kt}$, and thus for each $\delta \in (0,1)$ and for each time $t_1>0$, we can find $k\in \mathbb {N}$, such that $\Vert T(t_1)e_k\Vert _X<1-\delta$, which means that $T(t_1)e_k \notin B_\delta (W)$.

At the same time, a stronger Picard–Lindelöf-type theorem can be shown for uniformly continuous semigroups (this encompasses, in particular, the case of infinite ODE systems, also called “ensembles”), which fully extends the corresponding result for ODE systems, see [36, Chapter 1].

Theorem 3.12

(Picard–Lindelöf theorem for uniformly continuous semigroups) Let Assumptions 3.1, 3.2, 3.3 hold. Let further T be a uniformly continuous semigroup (not necessarily quasi-contractive). For any bounded set $W \subset X$, any bounded set $\mathcal {S}\subset U$ and any $\delta >0$, there is a time $\tau = \tau (W,\mathcal {S},\delta )>0$, such that for any $x_0 \in W$, and $u\in \mathcal {U}_{\mathcal {S}}$ there is a unique solution of (4) on $[0,\tau ]$, and it lies in $B_\delta (W)$.

Proof

First note that since $A \in L(X)$, for any $a \in \rho (A)$ the norm $x \mapsto \Vert (aI-A)^{-1}x\Vert _X$, $x \in X$, is equivalent to the original norm on X. Thus, $X = X_{-1}$ up to the equivalence of norms. Hence, as $B \in L(U,X_{-1})$, then also $B \in L(U,X)$, and thus, in particular, B is zero-class $\infty$-admissible operator.

Pick any $C>0$ such that $W \subset B_C$, and $\mathcal {U}_\mathcal {S}\subset B_{C,\mathcal {U}}$. Take also any $\delta >0$, and consider the following sets (depending on a parameter $t>0$):

$$\begin{aligned} Y_{t}{:}{=} \big \{x \in C([0,t],X): \textrm{dist}\,(x(t),W) \le \delta \ \ \forall t \in [0,t]\big \}, \end{aligned}$$

(18)

endowed with the metric $\rho _{t}(x,y){:}{=}\sup _{s \in [0,t]} \Vert x(s)-y(s)\Vert _X$, making them complete metric spaces.

Pick any $x_0 \in W$ and any $u \in \mathcal {U}_\mathcal {S}$. We are going to prove that for small enough t, the spaces $Y_{t}$ are invariant under the operator $\Phi _u$, defined for any $x \in Y_{t}$ and all $\tau \in [0,t]$ by (16). By Assumptions 3.1, 3.2, the function $\Phi _u(x)$ is continuous.

Fix any $t>0$ and pick any $x \in Y_{t}$. Then for any $\tau <t$, it holds that

$$\begin{aligned}&\textrm{dist}\,(\Phi _u(x)(\tau ),W) \le \Vert \Phi _u(x)(\tau )-x_0\Vert _X \\&\quad \le \Big \Vert T(\tau )x_0 - x_0\Big \Vert _X + \Big \Vert \int _0^\tau T_{-1}(\tau -s) Bu(s){\text {d}}s\Big \Vert _X \\&\qquad + \Big \Vert \int _0^\tau T_{-1}(\tau -s) B_2f(x(s),u(s)){\text {d}}s\Big \Vert _X \\&\quad \le \Vert T(\tau )-I \Vert _X \Vert x_0\Vert _X + h_\tau \Vert u\Vert _{L^\infty ([0,\tau ],U)} \\&\qquad + c_\tau \Vert f(x(\cdot ),u(\cdot ))\Vert _{L^\infty ([0,\tau ],V)} \\&\quad \le C \Vert T(\tau )-I \Vert _X + h_\tau \Vert u\Vert _{L^\infty ([0,\tau ],U)}\\&\qquad + c_\tau \Vert f(x(\cdot ),u(\cdot )) - f(0,u(\cdot ))\Vert _{L^\infty ([0,\tau ],V)} \\&\qquad + c_\tau \Vert f(0,u(\cdot ))\Vert _{L^\infty ([0,\tau ],V)}. \end{aligned}$$

Now for all $s\in [0,t]$

$$\begin{aligned} \Vert x(s)\Vert _X&\le C +\delta =: K. \end{aligned}$$

In view of Assumption 3.3(iii), it holds that

$$\begin{aligned} \Vert f(0,u(s))\Vert _V \le \sigma (\Vert u(s)\Vert _U) + c,\quad \text { for a.e. } s \in [0,t]. \end{aligned}$$

Now, Lipschitz continuity of f on bounded balls ensures that there is $L(K)>0$, such that for all $\tau \in [0,t]$

$$\begin{aligned} \Vert \Phi _{t}(x)(\tau )-w\Vert _X&\le C \Vert T(\tau )-I \Vert _X + h_t \Vert u\Vert _{L^\infty ([0,t],U)} \\&\quad + c_\tau \big ( L(K) \Vert x\Vert _{L^\infty ([0,t],X)} + \sigma (\Vert u\Vert _{L^\infty ([0,t],U)}) + c\big )\\&\le C \Vert T(\tau )-I \Vert _X + h_t \Vert u\Vert _{L^\infty ([0,t],U)} \\&\quad + c_t \big ( L(K)K + \sigma (C) + c\big ). \end{aligned}$$

Since T is a uniformly continuous semigroup, $h_t\rightarrow 0$ as $t\rightarrow +0$, and $c_t\rightarrow 0$ as $t\rightarrow +0$, from this estimate it is clear that there exists $t_1>0$, depending solely on C and $\delta$, such that

$$\begin{aligned} \textrm{dist}\,(\Phi _u(x)(t),W)\le \delta ,\quad \text { for all } t \in [0,t_1]. \end{aligned}$$

This means that $Y_{t}$ is invariant with respect to $\Phi _u$ for all $t \in (0,t_1]$, and $t_1$ does not depend on the choice of $x_0 \in W$. The rest of the proof is analogous to the proof of Theorem 3.7. $\square$

3.3 Well-posedness

Our next aim is to study the prolongations of solutions and their asymptotic properties.

Definition 3.13

Let $x_1(\cdot )$, $x_2(\cdot )$ be mild solutions of (4) defined on the intervals $[0,t_1)$ and $[0,t_2)$, respectively, $t_1,t_2>0$. We call $x_2$ an extension of $x_1$ if $t_2>t_1$, and $x_2(t)=x_1(t)$ for all $t \in [0,t_1)$.

Lemma 3.14

Let Assumptions 3.1, 3.2, 3.3 hold. Take any $x_0 \in X$ and $u\in \mathcal {U}$. Any two solutions of (4) coincide in their common domain of existence.

The proof is similar to the ODE case [36, Lemma 1.13] as is omitted.

Definition 3.15

A solution $x(\cdot )$ of (4) is called

• (i) maximal if there is no solution of (4) that extends $x(\cdot )$,

• (ii) global if $x(\cdot )$ is defined on $\mathbb {R}_+$.

A central property of the system (4) is

Definition 3.16

We say that the system (4) is well-posed if for every initial value $x_0 \in X$ and every external input $u \in \mathcal {U}$, there exists a unique maximal solution $\phi (\cdot ,x_0,u):[0,t_m(x_0,u)) \rightarrow X$, where $0 < t_m(x_0,u) \le \infty$.

We call $t_m(x_0,u)$ the maximal existence time of a solution corresponding to $(x_0,u)$.

The map $\phi$, defined in Definition 3.16, and describing the evolution of the system (4), is called the flow map, or just flow. The domain of definition of the flow $\phi$ is

$$\begin{aligned} D_\phi {:}{=}\cup _{x_0\in X,\ u\in \mathcal {U}}[0,t_m(x_0,u))\times \{(x_0,u)\}. \end{aligned}$$

In the following pages, we will always deal with maximal solutions. We will usually denote the initial condition by $x \in X$.

Theorem 3.17

(Well-posedness) Let Assumptions 3.1, 3.2, 3.3 hold. Then, (4) is well-posed.

The proof is similar to the ODE case [36, Theorem 1.16] as is omitted.

Now, we show that well-posed systems (4) are a special case of general control systems, introduced in Definition 2.1.

Theorem 3.18

Let (4) be well-posed. Then, the triple $(X,\mathcal {U},\phi )$, where $\phi$ is a flow map of (4), constitutes a control system in the sense of Definition 2.1.

Proof

The continuity axiom holds by the definition of a mild solution. Let us check the cocycle property.

Take any initial condition $x \in X$, any input $u \in \mathcal {U}$, and any $t,\tau \ge 0$, such that $[0,t+\tau ]\times \{(x,u)\} \subset D_{\phi }$. Define an input v by $v(r)=u(r+\tau )$, $r \ge 0$.

Due to (10), we have:

$$\begin{aligned}&\phi (t + \tau ,x,u) = T(t+\tau )x + \int _0^{t+\tau } T_{-1}(t+\tau -s) B_2f(\phi (s,x,u),u(s)){\text {d}}s\\&\qquad \qquad \qquad \qquad \qquad + \int _0^{t+\tau } T_{-1}(t+\tau -s) Bu(s){\text {d}}s. \end{aligned}$$

As $T_{-1}(t)$ is a bounded operator, it can be taken out of the Bochner integral:

$$\begin{aligned}&\phi (t + \tau ,x,u) = T(t)T(\tau )x + T_{-1}(t)\int _0^{\tau } T_{-1}(\tau -s) B_2f(\phi (s,x,u),u(s)){\text {d}}s \\&\quad + T_{-1}(t)\int _0^{\tau } T_{-1}(\tau -s) Bu(s){\text {d}}s \\&\quad + \int _{\tau }^{t+\tau } T_{-1}(t+\tau -s) B_2f(\phi (s,x,u),u(s)){\text {d}}s + \int _{\tau }^{t+\tau } T_{-1}(t+\tau -s) Bu(s){\text {d}}s. \end{aligned}$$

As B is $\infty$-admissible, we have that $\int _0^{\tau } T_{-1}(\tau -s) Bu(s){\text {d}}s \in X$. Since $T_{-1}(\cdot )$ coincides with $T(\cdot )$ on X, we infer

$$\begin{aligned} T_{-1}(t)\int _0^{\tau } T_{-1}(\tau -s) Bu(s){\text {d}}s = T(t)\int _0^{\tau } T_{-1}(\tau -s) Bu(s){\text {d}}s. \end{aligned}$$

Finally,

$$\begin{aligned} \phi (t + \tau ,x,u)&= T(t) \phi (\tau ,x,u) + \int _{0}^{t} T_{-1}(t-s) B_2 f(\phi (s+\tau ,x,v),v(s)){\text {d}}s \\&\qquad \qquad \qquad \qquad \qquad + \int _{0}^{t} T_{-1}(t-s) Bv(s){\text {d}}s\\&= \phi (t,\phi (\tau ,x,u),v), \end{aligned}$$

and the cocycle property holds. The rest of the properties of control systems are fulfilled by construction. $\square$

We proceed with a proof of a boundedness-implies-continuation property.

Proposition 3.19

Let Assumptions 3.1, 3.2, 3.3 hold. Then, (4) has the BIC property.

Proof

Pick any $x \in X$, any $u \in \mathcal {U}$, and consider the corresponding maximal solution $\phi (\cdot ,x,u)$, defined on $[0,t_m(x,u))$. Assume that $t_m(x,u)<+\infty$, but at the same time $\mathop {\underline{\lim }}_{t\rightarrow t_m(x,u)-0}\Vert \phi (t,x,u)\Vert _X <\infty$. Then, there is a sequence $(t_k)$, such that $t_k \rightarrow t_m(x,u)$ as $k\rightarrow \infty$ and $\lim _{k\rightarrow \infty }\Vert \phi (t_k,x,u)\Vert _X <\infty$. Hence, also $\sup _{k\in \mathbb {N}}\Vert \phi (t_k,x,u)\Vert _X =:C<\infty$.

Let $\tau (C)>0$ be a uniform existence time for the solutions starting in the ball $\overline{B_C}$ subject to inputs of a magnitude not exceeding $\Vert u\Vert$, which exists and is positive in view of Theorem 3.7. Then, the solution of (4) starting in $\phi (t_k,x,u)$, corresponding to the input $u(\cdot + t_k)$, exists and is unique on $[0,\tau (C)]$ by Theorem 3.7, and by the cocycle property, $\phi (\cdot ,x,u)$ can be prolonged to $[0,t_k +\tau (C))$, which (since $t_k\rightarrow t_m(x,u)$ as $k\rightarrow \infty$) contradicts to the maximality of the solution corresponding to (x, u).

Hence, $\mathop {\underline{\lim }}_{t\rightarrow t_m(x,u)-0}\Vert \phi (t,x,u)\Vert _X =\infty$, which implies the claim. $\square$

3.4 Forward completeness and boundedness of reachability sets

Local Lipschitz continuity guarantees the local existence of solutions. To ensure the global existence of solutions, stronger requirements on nonlinearity are needed.

Proposition 3.20

Let Assumptions 3.1, 3.2, 3.3 hold. Let further f be uniformly globally Lipschitz. Then, (4) is forward complete and has BRS.

Proof

By Theorem 3.7, for any $x_0 \in X$ and any $u \in \mathcal {U}$ there exists a mild solution of (4), with a maximal existence time $t_m (x_0, u)$, which may be finite or infinite. Let $t_m (x_0, u)$ be finite.

Let $L>0$ be a uniform global Lipschitz constant for f. As $\Vert T(t)\Vert \le Me^{\lambda t}$ for some $M\ge 1$, $\lambda \ge 0$ and all $t\ge 0$, for any $t<t_m (x_0, u)$ we have according to the formula (10) the following estimates

$$\begin{aligned}&\Vert \phi (t,x_0,u)\Vert _X \le \Vert T(t)\Vert \Vert x_0\Vert _X + \Big \Vert \int _{0}^{t} T_{-1}(t-s) Bu(s){\text {d}}s\Big \Vert _X \\&\qquad + \Big \Vert \int _0^t T_{-1}(t-s)B_2 f(\phi (s,x_0,u),u(s)){\text {d}}s\Big \Vert _X \\&\quad \le Me^{\lambda t} \Vert x_0\Vert _X + h_t \Vert u\Vert _{\mathcal {U}} + c_t \Vert f(\phi (\cdot ,x_0,u),u(\cdot ))\Vert _{L^\infty ([0,t],V)}\\&\quad \le Me^{\lambda t} \Vert x_0\Vert _X + h_t \Vert u\Vert _{\mathcal {U}} + c_t \Vert f(\phi (\cdot ,x_0,u),u(\cdot )) - f(0, u(\cdot ))\Vert _{L^\infty ([0,t],V)}\\&\qquad + c_t \Vert f(0,u(\cdot ))\Vert _{L^\infty ([0,t],V)}\\&\quad \le Me^{\lambda t} \Vert x_0\Vert _X + h_t \Vert u\Vert _{\mathcal {U}} + c_t L \Vert \phi (\cdot ,x_0,u)\Vert _{L^\infty ([0,t],X)} + c_t (\sigma (\Vert u\Vert _\mathcal {U}) + c). \end{aligned}$$

Since $c_t \rightarrow 0$ whenever $t \rightarrow +0$, there is some $t_1 \in (0,t_m (x_0, u))$ such that $c_{t_1}L\le \frac{1}{2}$. Then, it holds that

$$\begin{aligned} \sup _{t \in [0,t_1]}\Vert \phi (t,x_0,u)\Vert _X \le 2\Big (Me^{\lambda t_1} \Vert x_0\Vert _X + h_{t_1} \Vert u\Vert _{\mathcal {U}} + c_{t_1} (\sigma (\Vert u\Vert _\mathcal {U}) + c)\Big ). \end{aligned}$$

(19)

Note that $t_1$ does not depend on $x_0$ and u. Hence, using cocycle property and with $\phi (t_1,x_0,u)$ instead of $x_0$, we obtain a uniform bound for $\phi (\cdot ,x_0,u)$ on $2t_1$, $3t_1$, and so on. Thus, $\phi (\cdot ,x_0,u)$ is uniformly bounded on $[0,t_m (x_0, u))$, and hence can be prolonged to a larger interval by the BIC property ensured by Proposition 3.19, a contradiction to the definition of $t_m (x_0, u)$. Overall, $\Sigma$ is forward complete, and the estimate (19) iterated as above to larger intervals shows that (4) has BRS. $\square$

3.5 Regularity of the flow map

We start this section with a basic result describing the exponential deviation between two trajectories.

Theorem 3.21

Let Assumptions 3.1, 3.2, 3.3 hold. Take $M \ge 1$, $\lambda \ge 0$ such that $\Vert T(t)\Vert \le Me^{\lambda t}$ for all $t\ge 0$. Pick any $x_1,x_2 \in X$, any $u\in \mathcal {U}$, and let $\phi (\cdot ,x_1,u)$ and $\phi (\cdot ,x_2,u)$ be defined on a certain common interval $[0,\tau ]$.

Then, there exists $R=R(x_1,x_2,\tau ,u)>0$, such that

$$\begin{aligned} \Vert \phi (t,x_1,u) - \phi (t,x_2,u)\Vert _X \le 2M\Vert x_1-x_2\Vert _X e^{Rt},\quad t\in [0,\tau ]. \end{aligned}$$

(20)

Proof

Pick any $x_1,x_2 \in X$, any $u \in \mathcal {U}$, and let $\phi _i(t){:}{=}\phi (t,x_i,u)$, $i=1,2$ be the corresponding (unique) maximal solutions of (4) (guaranteed by Theorem 3.17), defined on $[0,\tau ]$, for a certain $\tau >0$.

Set

$$\begin{aligned} K{:}{=}\max \big \{\sup _{0 \le t \le \tau }\Vert \phi _1(t)\Vert _X,\sup _{0 \le t \le \tau }\Vert \phi _2(t)\Vert _X,\Vert u\Vert _\mathcal {U}\big \}<\infty , \end{aligned}$$

where K is finite due to the continuity of trajectories.

Due to (10), and using Lipschitz continuity of f (see (11)), we have for any $t \in [0,\tau ]$:

$$\begin{aligned}&\Vert \phi _1(t) - \phi _2(t)\Vert _X \le \Vert T(t)\Vert \Vert x_1-x_2\Vert _X \\&\qquad + \Big \Vert \int _0^t T_{-1}(t-s) B_2 \Big (f(\phi _1(s),u(s)) - f(\phi _2(s),u(s))\Big ){\text {d}}s\Big \Vert _X \\&\quad \le Me^{\lambda t} \Vert x_1-x_2\Vert _X + c_t \Vert f(\phi _1(\cdot ),u) - f(\phi _2(\cdot ),u)\Vert _{L^\infty ([0,t],X)}\\&\quad \le Me^{\lambda t} \Vert x_1-x_2\Vert _X + c_t L(K) \Vert \phi _1(\cdot )-\phi _2(\cdot )\Vert _{L^\infty ([0,t],X)}. \end{aligned}$$

As $c_t \rightarrow 0$ as $t \rightarrow +0$, there is some $t_1 \in (0,\tau )$ such that $c_{t_1}L(K)\le \frac{1}{2}$. Note that $t_1$ depends on $\tau$ only (as K does).

Then, taking the supremum of the previous expression over [0, t], with $t<t_1$, we obtain that

$$\begin{aligned} \Vert \phi _1(t) - \phi _2(t)\Vert _X&\le 2Me^{\lambda t} \Vert x_1-x_2\Vert _X,\quad t\in [0,t_1]. \end{aligned}$$

Take $k\in \mathbb {N}$ such that $kt_1 <\tau$ and $(k+1)t_1>\tau$. Then, using the cocycle property, for any $l\in \mathbb {N}$, $l\le k$ and all $t\in [0,t_1]$ s.t. $lt_1+t<\tau$ we have

$$\begin{aligned} \Vert \phi _1(lt_1 + t) - \phi _2(lt_1+t)\Vert _X&\le (2M)^{l+1} e^{\lambda (lt_1 + t)} \Vert x_1-x_2\Vert _X\\&= 2M e^{l \ln (2M) + \lambda lt_1 + \lambda t} \Vert x_1-x_2\Vert _X\\ [R{:}{=}\lambda + \frac{1}{t_1}\ln (2M)>\lambda ]&= 2M e^{l Rt_1 + \lambda t} \Vert x_1-x_2\Vert _X\\&\le 2M e^{R (lt_1 + t)} \Vert x_1-x_2\Vert _X. \end{aligned}$$

This shows (20). $\square$

Definition 3.22

The flow of a forward complete control system $\Sigma =(X,\mathcal {U},\phi )$, is called Lipschitz continuous on compact intervals (for uniformly bounded inputs), if for any $\tau >0$ and any $C>0$ there exists $L>0$ so that for any $x_1,x_2 \in \overline{B_C}$, for all $u \in B_{C,\mathcal {U}}$, it holds that

$$\begin{aligned} \Vert \phi (t,x_1,u) - \phi (t,x_2,u) \Vert _X \le L \Vert x_1-x_2\Vert _X,\quad t \in [0,\tau ]. \end{aligned}$$

(21)

Theorem 3.21 estimates the deviation between two trajectories. To have a stronger result, showing the Lipschitz continuity of the flow map $\phi$, we additionally assume the BRS property of (4).

Theorem 3.23

Suppose that Assumptions 3.1, 3.2, 3.3 hold and (4) has BRS. Then, the flow of (4) is Lipschitz continuous on compact intervals for uniformly bounded inputs.

Proof

Take any $C>0$ and pick any $x_1,x_2 \in B_C$, and any $u \in \mathcal {U}$ with $\Vert u\Vert _{\mathcal {U}} \le C$. Let $\phi _i(\cdot ){:}{=}\phi (\cdot ,x_i,u)$, $i=1,2$ be the corresponding maximal solutions of (4). These solutions are global since we assume that (4) is forward complete.

As (4) is BRS, the following quantity is finite for any $\tau >0$:

$$\begin{aligned} K(\tau ){:}{=}\sup _{t\in [0,\tau ],\ x\in B_C,\ u\in B_{C,\mathcal {U}}}\Vert \phi (t,x,u)\Vert _X<\infty . \end{aligned}$$

Following the lines of the proof of Theorem 3.21, we obtain the claim. $\square$

Definition 3.24

Let $\Sigma =(X,\mathcal {U},\phi )$ be a forward complete control system. We say that the flow $\phi$ depends continuously on inputs and on initial states, if for all $x \in X$, $u \in \mathcal {U}$, $\tau >0$, and all $\varepsilon >0$ there exists $\delta >0$, such that $\forall x' \in X: \Vert x-x'\Vert _X< \delta$ and $\forall u' \in \mathcal {U}: \Vert u-u'\Vert _{\mathcal {U}}< \delta$ it holds that

$$\begin{aligned} \Vert \phi (t,x,u)-\phi (t,x',u')\Vert _X< \varepsilon , \quad t \in [0,\tau ]. \end{aligned}$$

To obtain the continuity of the flow map with respect to both states and inputs, which is important for the application of the density argument, we impose additional conditions on the nonlinearity f.

Theorem 3.25

Let Assumptions 3.1, 3.2, 3.3 hold. Let further there exists $q \in \mathcal {K_\infty }$ such that for all $C>0$ there is $L(C)>0$: for all $x_1,x_2 \in \overline{B_C}$ and all $v_1,v_2 \in \overline{B_{C,U}}$ it holds that

$$\begin{aligned} \Vert f(x_1,v_1)-f(x_2,v_2)\Vert _V \le L(C) \big ( \Vert x_1-x_2\Vert _X + q(\Vert v_1-v_2\Vert _U) \big ). \end{aligned}$$

(22)

If (4) has the BRS property, then the flow of (4) depends continuously on initial states and inputs.

Proof

Pick any time $\tau >0$. Take any $C>0$, any $x_1,x_2 \in \overline{B_C}$, and any $u_1,u_2 \in \overline{B_{C,\mathcal {U}}}$. Let $\phi _i(\cdot )=\phi (\cdot ,x_i,u_i)$, $i=1,2$ be the corresponding global solutions.

Due to (10), we have:

$$\begin{aligned} \Vert \phi _1(t) - \phi _2(t)\Vert _X&\le \Vert T(t)\Vert \Vert x_1-x_2\Vert _X +h_t \Vert u_1-u_2\Vert _{\mathcal {U}}\\&\qquad + c_t \sup _{r\in [0,t]}\big \Vert f(\phi _1(r),u_1(r))-f(\phi _2(r),u_2(r))\big \Vert _V. \end{aligned}$$

In view of the boundedness of reachability sets for the system (4), we have

$$\begin{aligned} K{:}{=}\sup _{\Vert z\Vert _X\le C,\ \Vert u\Vert _\mathcal {U}\le C,\ t \in [0,\tau ]}\Vert \phi (t,z,u)\Vert _X < \infty . \end{aligned}$$

As $\Vert T(t)\Vert \le Me^{\lambda t}$ for some $M,\lambda \ge 0$ and all $t\ge 0$, and due to the property (22) with $L{:}{=}L(K)$ (note that $K\ge C$), we can continue above estimates to obtain

$$\begin{aligned} \Vert \phi _1(t)&- \phi _2(t)\Vert _X \le Me^{\lambda t} \Vert x_1-x_2\Vert _X + h_t \Vert u_1-u_2\Vert _{\mathcal {U}} \\&\qquad + c_t L(K) \sup _{r\in [0,t]} \Big ( \Vert \phi _1(r)-\phi _2(r)\Vert _X + q\big (\Vert u_1(r)-u_2(r)\Vert _U\big )\Big ). \end{aligned}$$

Since B is zero-class admissible, there is $t_1>0$ such that $c_{t_1} L(K) = \frac{1}{2}$, and thus taking the supremum of both sides over $t\in [0,t_1]$, we have for all $t\in [0,t_1]$ that

$$\begin{aligned} \Vert \phi _1(t)&- \phi _2(t)\Vert _X \le 2Me^{\lambda t_1} \Vert x_1-x_2\Vert _X + 2h_{t_1} \Vert u_1-u_2\Vert _{\mathcal {U}} + q\big (\Vert u_1-u_2\Vert _\mathcal {U}\big ). \end{aligned}$$

Thus, for each $\varepsilon >0$ there is $\delta >0$ so that for all $x_2 \in B_\delta (x_1)$ and for all $u_2 \in B_{\delta ,\mathcal {U}}(u_1)$ it holds that

$$\begin{aligned} \Vert \phi _1(t)&- \phi _2(t)\Vert _X \le \varepsilon ,\quad t\in [0,t_1]. \end{aligned}$$

This establishes the continuity over the interval $[0,t_1]$. To obtain continuity over the interval $[0,\tau ]$, one can follow the strategy in the second part of the proof of Lemma 3.27 (and noting that at all the steps the parameter K does not change). $\square$

3.6 Continuity at trivial equilibrium

Without loss of generality, we restrict our analysis to fixed points of the form $(0,0) \in X \times \mathcal {U}$. Note that (0, 0) is in $X \times \mathcal {U}$ since both X and $\mathcal {U}$ are linear spaces.

To describe the behavior of solutions near the equilibrium, the following notion is of importance:

Definition 3.26

Consider a system $\Sigma =(X,\mathcal {U},\phi )$ with equilibrium point $0\in X$. We say that $\phi$ is continuous at the equilibrium if for every $\varepsilon >0$ and for any $h>0$ there exists a $\delta = \delta (\varepsilon ,h)>0$, so that $[0,h] \times B_\delta \times B_{\delta ,\mathcal {U}} \subset D_\phi$, and

$$\begin{aligned} t\in [0,h],\ \Vert x\Vert _X \le \delta ,\ \Vert u\Vert _{\mathcal {U}} \le \delta \quad \Rightarrow \quad \Vert \phi (t,x,u)\Vert _X \le \varepsilon . \end{aligned}$$

(23)

In this case, we will also say that $\Sigma$ has the CEP property.

CEP property is a “local in time version” of Lyapunov stability and is important, in particular, for the ISS superposition theorems [40] and for the applications of the non-coercive ISS Lyapunov theory [19].

Lemma 3.27

(Continuity at equilibrium for (4)) Let Assumptions 3.1, 3.2, 3.3 hold, and let $f(0,0)=0$. Then, $\phi$ is continuous at the equilibrium.

Proof

Consider the following auxiliary system

$$\begin{aligned} \dot{x}(t)= & {} Ax(t) + B_2\tilde{f}(x(t),u(t)) + Bu(t),\quad t>0, \end{aligned}$$

(24a)

$$\begin{aligned} x(0)= & {} x_0, \end{aligned}$$

(24b)

where

$$\begin{aligned} \tilde{f}(x,u){:}{=}f\big (sat (x),sat _2(u)\big ),\quad x\in X,\ u\in U, \end{aligned}$$

and the saturation function is given for the vectors z in X and in U, respectively, by

$$\begin{aligned} sat (z){:}{=} {\left\{ \begin{array}{ll} z, &{} \Vert z\Vert _X\le 1, \\ \frac{z}{\Vert z\Vert _X}, &{} \text {otherwise}, \end{array}\right. } \qquad sat _2(z){:}{=} {\left\{ \begin{array}{ll} z, &{} \Vert z\Vert _U\le 1, \\ \frac{z}{\Vert z\Vert _U}, &{} \text {otherwise}. \end{array}\right. } \end{aligned}$$

As f satisfies Assumption 3.3, one can show that $\tilde{f}$ is uniformly globally Lipschitz continuous. Hence, (24) is forward complete and has BRS property by Proposition 3.20.

We denote the flow of (24) by $\tilde{\phi } = \tilde{\phi }(t,x,u)$. As $f(x,u)=\tilde{f}(x,u)$ whenever $\Vert x\Vert _X\le 1$ and $\Vert u\Vert _U\le 1$, it holds also

$$\begin{aligned} \phi (t,x,u) = \tilde{\phi }(t,x,u), \end{aligned}$$

provided that $\Vert u\Vert _{\mathcal {U}}\le 1$, $\phi (\cdot ,x,u)$ exists on [0, t], and $\Vert \phi (s,x,u)\Vert _X\le 1$ for all $s\in [0,t]$.

Pick any $\varepsilon \in (0,1)$, $\tau \ge 0$, $\delta \in (0,\varepsilon )$, $x \in B_\delta$, and any $u\in B_{\delta ,\mathcal {U}}$. It holds that

$$\begin{aligned} \Vert \tilde{\phi }(t,x,u) \Vert _X\le & {} \Vert \tilde{\phi }(t,x,u) -\tilde{\phi }(t,0,u)\Vert _X + \Vert \tilde{\phi }(t,0,u)\Vert _X. \end{aligned}$$

Since (24) has BRS property, by Theorem 3.23, the flow of (24) is Lipschitz continuous on compact time intervals. Hence, there exists a $L(\tau ,\delta )>0$ so that for all $t\in [0,\tau ]$

$$\begin{aligned} \Vert \tilde{\phi }(t,x,u) -\tilde{\phi }(t,0,u)\Vert _X \le L(\tau ,\delta )\Vert x\Vert _X \le L(\tau ,\delta )\delta . \end{aligned}$$

(25)

Let us estimate $\Vert \tilde{\phi }(t,0,u)\Vert _X$. We have:

$$\begin{aligned} \Vert \tilde{\phi }(t,0,u)\Vert _X&\le \Big \Vert \int _0^t T_{-1}(t-s) B_2\tilde{f}\big (\tilde{\phi }(s,0,u),u(s)\big ){\text {d}}s\Big \Vert _X + \Big \Vert \int _0^t T_{-1}(t-s)Bu(s){\text {d}}s\Big \Vert _X\\&\le c_t\mathop {\mathrm {ess\,sup}}\limits _{s\in [0,t]}\big \Vert \tilde{f}\big (\tilde{\phi }(s,0,u),u(s)\big )\big \Vert _X + h_t\big \Vert u\big \Vert _{L^\infty ([0,t],U)}\\&\le c_t\mathop {\mathrm {ess\,sup}}\limits _{s\in [0,t]}\big \Vert \tilde{f}\big (\tilde{\phi }(s,0,u), u(s)\big ) - \tilde{f}\big (0, u(s)\big )\big \Vert _X + c_t\mathop {\mathrm {ess\,sup}}\limits _{s\in [0,t]}\big \Vert \tilde{f}\big (0, u(s)\big )\big \Vert _X \\&\qquad \qquad \qquad + h_t\big \Vert u\big \Vert _{L^\infty ([0,t],U)}. \end{aligned}$$

Since $\tilde{f}(0,\cdot )$ is continuous, for any $\varepsilon _2>0$ there exists $\delta _2<\delta$ so that $u(s) \in B_{\delta _2}$ implies that $\Vert \tilde{f}(0,u(s))-\tilde{f}(0,0)\Vert _X \le \varepsilon _2$. Since $\tilde{f}(0,0)=0$, for the above u we have $\Vert \tilde{f}(0,u(s))\Vert _X \le \varepsilon _2$.

As $\tilde{f}$ is uniformly globally Lipschitz, there is $L>0$ such that for the inputs satisfying $\Vert u\Vert _\mathcal {U}\le \delta _2$ we have

$$\begin{aligned} \Vert \tilde{\phi }(t,0,u)\Vert _X \le c_tL\mathop {\mathrm {ess\,sup}}\limits _{s\in [0,t]}\big \Vert \tilde{\phi }(s,0,u)\big \Vert _X + c_t \varepsilon _2 + h_t\delta _2. \end{aligned}$$

As $c_t\rightarrow 0$ for $t\rightarrow +0$, there is $t_1>0$, such that $c_{t_1}L\le \frac{1}{2}$.

Then, we have that

$$\begin{aligned} \Vert \tilde{\phi }(t,0,u)\Vert _X \le 2c_{t_1} \varepsilon _2 + 2h_{t_1}\delta _2,\quad t \le t_1. \end{aligned}$$

(26)

Combining (25) with (26), we see that whenever $\Vert x\Vert _X\le \delta _2$ and $\Vert u\Vert _\mathcal {U}\le \delta _2$, it holds that

$$\begin{aligned} \Vert \tilde{\phi }(t,x,u) \Vert _X \le L(\tau ,\delta _2)\delta _2 + 2c_{t_1} \varepsilon _2 + 2h_{t_1}\delta _2,\quad t \le t_1. \end{aligned}$$

Now, for any $\varepsilon <1$ we can find $\delta _2<\varepsilon$, such that

$$\begin{aligned} \Vert \tilde{\phi }(t,x,u) \Vert _X \le \varepsilon ,\quad t \le t_1,\quad \Vert x\Vert _X\le \delta _2,\quad \Vert u\Vert _\mathcal {U}\le \delta _2. \end{aligned}$$

As $\tilde{\phi }(t,x,u) = \phi (t,x,u)$ whenever $\Vert \tilde{\phi }(t,x,u)\Vert _X<1$, we obtain that

$$\begin{aligned} \Vert \phi (t,x,u) \Vert _X \le \varepsilon ,\quad t \le t_1,\quad \Vert x\Vert _X\le \delta _2,\quad \Vert u\Vert _\mathcal {U}\le \delta _2. \end{aligned}$$

Note that $t_1$ depends on L only and does not depend on $\delta _2$. Thus, one can find $\delta _3<\delta _2$, such that

$$\begin{aligned} \Vert \phi (t,x,u) \Vert _X \le \delta _2,\quad t \le t_1,\ \Vert x\Vert _X\le \delta _3,\ \Vert u\Vert _\mathcal {U}\le \delta _3. \end{aligned}$$

By the cocycle property, we obtain that

$$\begin{aligned} \Vert \phi (t,x,u) \Vert _X \le \varepsilon ,\quad t \le 2t_1,\ \Vert x\Vert _X\le \delta _3,\ \Vert u\Vert _\mathcal {U}\le \delta _3. \end{aligned}$$

Iterating this procedure, we obtain that there is some $\omega >0$, such that

$$\begin{aligned} \Vert \phi (t,x,u) \Vert _X \le \varepsilon ,\quad t \in [0,\tau ],\ \Vert x\Vert _X\le \omega ,\ \Vert u\Vert _\mathcal {U}\le \omega . \end{aligned}$$

This shows the CEP property. $\square$

4 Semilinear analytic systems

4.1 Preliminaries for analytic semigroups and admissibility

Recall that $\omega _0(T)$ denotes the growth bound of the semigroup T. Pick any $\omega >\omega _0(T)$ and define the space $X_\alpha$ and the norm in it as

$$\begin{aligned} X_\alpha {:}{=}D((\omega I-A)^\alpha ),\quad \Vert x\Vert _{X_\alpha }{:}{=}\Vert (\omega I-A)^\alpha x\Vert _X,\quad x \in X_\alpha . \end{aligned}$$

(27)

Furthermore, define the spaces $X_{-\alpha }$ as the completion of X with respect to the norm $x \mapsto \Vert (\omega I-A)^{-\alpha }x\Vert _X$. For the theory of fractional powers of operators and fractional spaces, see [13] and [15, Section 1.4] and, for a very brief description of the essentials required here, [51].

The following well-known property holds:

Proposition 4.1

Let T be an analytic semigroup on a Banach space X with the generator A. Then for each $\omega ,\kappa >\omega _0(T)$, each $\alpha \in [0,1)$, and each $t>0$ we have $\textrm{Im}(T(t)) \subset X_\alpha$, and there is $C_\alpha >0$ such that

$$\begin{aligned} \Vert (\omega I-A)^\alpha T(t)\Vert \le \frac{C_\alpha }{t^\alpha }e^{\kappa t},\quad t>0. \end{aligned}$$

(28)

Furthermore, the map $t\mapsto (\omega I-A)^\alpha T(t)$ is continuous on $(0,+\infty )$ in the uniform operator topology.

Next, we formulate a sufficient condition for the zero-class admissibility of input operators for analytic systems. Part (ii) of the following proposition is (up to the zero-class statement) contained in [51, Proposition 2.13]. We, however, provide a short proof based on the statement (i) to be self-contained.

Proposition 4.2

Assume that A generates an analytic semigroup T and $B\in L(U,X_{-1+\alpha })$ for some $\alpha \in (0,1)$. Then:

(i):

For any $\omega >\omega _0(T)$, any $d \in [0,1)$ the operator $(\omega I-A_{-1})^{d}$ is zero-class p-admissible for any $p\in (\frac{1}{1-d},+\infty ]$. In particular, for any $g \in L^p_{{{\,\textrm{loc}\,}}}(\mathbb {R}_+,X)$, the following map

$$\begin{aligned} {\xi :t \mapsto \int _0^t (\omega I-A)^d T(t-s)g(s){\text {d}}s = \int _0^t T_{-1}(t-s)(\omega I-A)^d g(s){\text {d}}s} \nonumber \\ \end{aligned}$$

(29)

is well defined and continuous on $\mathbb {R}_+$. Furthermore, for any $\kappa >\omega _0(T)$ there is $R=R(\kappa ,d)$ such that for any $g \in L^\infty _{{{\,\textrm{loc}\,}}}(\mathbb {R}_+,X)$ the following holds:

$$\begin{aligned} \int _0^t \big \Vert (\omega I-A)^d T(t-s) g(s)\big \Vert _X {\text {d}}s \le R t^{1-d} e^{\kappa t} \Vert g\Vert _{L^\infty ([0,t],X)}. \end{aligned}$$

(30)

(ii):

B is zero-class q-admissible for $q\in (\frac{1}{\alpha },+\infty ]$.

(iii):

For any $\omega >\omega _0(T)$, any $d \in [0,\alpha )$ the operator $(\omega I-A)^{d}B$ is zero-class $\infty$-admissible.

Proof

(i). Since T is an analytic semigroup, T(t) maps X to D(A) for any $t>0$. As $D(A)\subset X_d$ for all $d\in [0,1]$, the integrand in (29) is in X for a.e. $s\in [0,t)$. Let us show the Bochner integrability of X-valued map $s \mapsto (\omega I-A)^d T(t-s)g(s)$ on [0, t].

As $g \in L^{1}_{{{\,\textrm{loc}\,}}}(\mathbb {R}_+,X)$, by the criterion of Bochner integrability, g is strongly measurable and $\int _I \Vert g(s)\Vert _X {\text {d}}s < \infty$ for any bounded interval $I \subset \mathbb {R}_+$.

Denote by $\chi _{\Omega }$ the characteristic function of the set $\Omega \subset \mathbb {R}_+$. Recall that the map $t\mapsto (\omega I-A)^d T(t)$ is continuous outside of $t=0$ in view of Proposition 4.1.

If $g(s) = \chi _{\Omega }(s)x$ for some measurable $\Omega \subset \mathbb {R}_+$ and $x \in X$, then the function

$$\begin{aligned} s\mapsto (\omega I-A)^d T(t-s)g(s) = (\omega I-A)^d T(t-s)\chi _{\Omega }(s)x \end{aligned}$$

is measurable as a product of a measurable scalar function and a continuous (and thus measurable) vector-valued function. By linearity, $s\mapsto (\omega I-A)^d T(t-s)g(s)$ is strongly measurable if g is a simple function (see [1, Section 1.1] for definitions).

As g is strongly measurable, there is a sequence of simple functions $(g_n)_{n\in \mathbb {N}}$, converging pointwise to g almost everywhere. Consider a sequence

$$\begin{aligned} \big (s\mapsto (\omega I-A)^d T(t-s)g_n(s)\big )_{n\in \mathbb {N}} \end{aligned}$$

(31)

and take any $s \in [0,t)$ such that $g_n(s) \rightarrow g(s)$ as $n\rightarrow \infty$. We have that

$$\begin{aligned} \big \Vert (\omega I-A)^d T(t-s)&g_n(s)- (\omega I-A)^d T(t-s)g(s)\big \Vert _X \\&\le \Vert (\omega I-A)^d T(t-s)\Vert \Vert g_n(s)-g(s)\Vert _X \rightarrow 0,\quad n\rightarrow \infty . \end{aligned}$$

Hence, a sequence of strongly measurable functions (31) converges a.e. to $s\mapsto (\omega I-A)^d T(t-s)g(s)$, and thus, $s\mapsto (\omega I-A)^d T(t-s)g(s)$ is strongly measurable by [1, Corollary 1.1.2].

Furthermore, for any $t >0$, using Proposition 4.1, we have that for any $\kappa >\omega _0(T)$ there is $C_d>0$ such that

$$\begin{aligned} \int _0^t \big \Vert (\omega I-A)^d T(t-s) g(s)\big \Vert _X {\text {d}}s&\le \int _0^t \Vert (\omega I-A)^d T(t-s)\Vert \Vert g(s)\Vert _X {\text {d}}s \nonumber \\&\le \int _0^t \frac{C_d}{(t-s)^d}e^{\kappa (t-s)} \Vert g(s)\Vert _X {\text {d}}s \nonumber \\&\le C_d e^{\kappa t}\int _0^t \frac{1}{(t-s)^d} \Vert g(s)\Vert _X {\text {d}}s. \end{aligned}$$

(32)

Using Hölder’s inequality with a finite $p>\frac{1}{1-d}$, we obtain

$$\begin{aligned} \int _0^t \Vert (\omega I-A)^d T(t-s) g(s)\Vert _X {\text {d}}s&\le C_d e^{\kappa t} \Big (\int _0^t \Big (\frac{1}{(t-s)^d}\Big )^b {\text {d}}s\Big )^{\frac{1}{b}} \Big ( \int _0^t \Vert g(s)\Vert ^p_X {\text {d}}s\Big )^{\frac{1}{p}}\nonumber \\&\le \frac{C_d}{(1-d b)^{1/b}} e^{\kappa t} t^{\frac{1-d b}{b}} \Big ( \int _0^t \Vert g(s)\Vert ^p_X {\text {d}}s\Big )^{\frac{1}{p}}, \end{aligned}$$

(33)

where $\frac{1}{b} + \frac{1}{p} = 1$, and thus, b satisfies $b<\frac{1}{d}$.

Finally, by [1, Theorem 1.1.4], the map $s \mapsto (\omega I-A)^d T(t-s)g(s)$ is Bochner integrable on each $[0,t]\subset \mathbb {R}_+$. This shows p-admissibility of $(\omega I-A)^d$, and if $p<+\infty$, (33) implies zero-class p-admissibility of $(\omega I-A)^d$. Continuity of the map $\xi$ follows from [58, Proposition 2.3].

For the last claim of item (i), we take $g\in L^\infty _{{{\,\textrm{loc}\,}}}(\mathbb {R}_+,X)$ and continue the estimates in (32) as follows:

$$\begin{aligned} \int _0^t \big \Vert (\omega I-A)^d T(t-s) g(s)\big \Vert _X {\text {d}}s&\le C_d e^{\kappa t} \int _0^t \frac{1}{(t-s)^d} {\text {d}}s \Vert g\Vert _{L^\infty ([0,t],X)}, \end{aligned}$$

and (30) holds with $R=\frac{C_d}{1-d}$. This implies zero-class $\infty$-admissibility of $(\omega I-A)^d$.

(ii). Take any $\omega >\omega _0(T)$, and consider the corresponding norm on $X_{-1+\alpha }$:

$$\begin{aligned} \Vert B\Vert _{L(U,X_{-1+\alpha })}&= \sup _{u\in U:\Vert u\Vert _U=1}\Vert Bu\Vert _{X_{-1+\alpha }}\\&= \sup _{u\in U:\Vert u\Vert _U=1}\Vert (\omega I-A)^{-1+\alpha }Bu\Vert _{X} = \Vert (\omega I-A)^{-1+\alpha }B\Vert _{L(U,X)}. \end{aligned}$$

Thus, the condition $B\in L(U,X_{-1+\alpha })$ is equivalent to $(\omega I-A)^{-1+\alpha }B \in L(U,X)$. With this in mind, we have

$$\begin{aligned} T_{-1}(t)B= & {} T_{-1}(t)(\omega I-A)^{1-\alpha }(\omega I-A)^{-1+\alpha }B. \end{aligned}$$

(34)

Due to [43, Theorem 2.6.13, p. 74], on $X_{1-\alpha } = D((\omega I-A)^{1-\alpha })$ it holds that

$$\begin{aligned} T_{-1}(t)(\omega I-A)^{1-\alpha } = (\omega I-A)^{1-\alpha }T_{-1}(t). \end{aligned}$$

Now, take any $f \in L^q_{{{\,\textrm{loc}\,}}}(\mathbb {R}_+,U)$ with $q>\frac{1}{\alpha }$. Representing

$$\begin{aligned} \int _0^t T_{-1}(t-s) Bf(s) {\text {d}}s = \int _0^t (\omega I-A)^{1-\alpha } T_{-1}(t-s) (\omega I-A)^{\alpha -1} Bf(s) {\text {d}}s \end{aligned}$$

and applying item (i) of this proposition and in particular the estimate (33) with $d{:}{=}1-\alpha$, $p{:}{=}q$, and with $g{:}{=}(\omega I-A)^{-1+\alpha }B f$ we see that B is zero-class q-admissible for $q \in (\frac{1}{\alpha },+\infty )$.

(iii). It holds that $\Vert (\omega I-A)^{d} B\Vert _{L(U,X_{-1+\alpha -d})} = \Vert (\omega I-A)^{-1+\alpha }B\Vert _{L(U,X)}$, and item (i) implies the claim. $\square$

Proposition 4.3

Assume that A generates an analytic semigroup T and $B\in L(U,X_{-1+\alpha })$ for some $\alpha \in (0,1)$.

For any $\omega >\omega _0(T)$, any $d \in [0,\alpha )$, and any $\kappa >\omega _0(T)$, there is $R>0$ such that for any $g \in L^\infty _{{{\,\textrm{loc}\,}}}(\mathbb {R}_+,X)$ the map

$$\begin{aligned} t\mapsto \int _0^t (\omega I-A)^d T(t-s) Bg(s) {\text {d}}s \end{aligned}$$

(35)

is continuous in X-norm, and the following holds:

$$\begin{aligned} \int _0^t \big \Vert (\omega I-A)^d T(t-s) Bg(s)\big \Vert _X {\text {d}}s \le R t^{\alpha -d} e^{\kappa t} \Vert g\Vert _{L^\infty ([0,t],X)}. \end{aligned}$$

(36)

Proof

For $d<\alpha$ and $g \in L^\infty _{{{\,\textrm{loc}\,}}}(\mathbb {R}_+,U)$, consider the map

$$\begin{aligned} s \mapsto (\omega I-A)^d T_{-1}(t-s)Bg(s)= & {} (\omega I-A)^{1-\alpha +d}T_{-1}(t-s)(\omega I-A)^{-1+\alpha }Bg(s). \end{aligned}$$

By item (i) of Proposition 4.2, this map is Bochner integrable and in view of (30) with $1-\alpha +d$ instead of d and $(\omega I-A)^{-1+\alpha }Bg$ instead of g, we see that the map (35) is continuous and (36) holds. $\square$

4.2 Semilinear analytic systems and their mild solutions

Consider again the system (4) with $B_2={\text {id}}$ that we restate next:

$$\begin{aligned} \dot{x}(t)= & {} Ax(t) + f(x(t),u(t)) + Bu(t),\quad t>0, \end{aligned}$$

(37a)

$$\begin{aligned} x(0)= & {} x_0, \end{aligned}$$

(37b)

In Sect. 3, we have assumed that f is a well-defined map from $X \times U$ to X. Although it sounds natural, it is, in fact, a quite restrictive assumption, as already basic nonlinearities, such as pointwise polynomial maps, do not satisfy it. Indeed, if $f(x) = x^2$, where $x \in X{:}{=}L^2(0,1)$, then f maps X to the space $L^1(0,1)$. However, as A generates an analytic semigroup, the requirements on f can be considerably relaxed. Namely, we assume in this section that there is $\alpha \in [0,1]$ such that f is a well-defined map from $X_\alpha \times U$ to X.

We note that systems (37) without inputs ($u=0$) have been analyzed several decades ago, see the classical monographs [15, 43]. The main difference to these works is the presence of unbounded input operators.

Next, we define mild solutions of (37). Note that the nonlinearity f is defined on $X_\alpha \times U$, and thus, we must require that the mild solution lies in $X_\alpha$ for all positive times. We cannot expect such a nice behavior for general semigroups, but thanks to the smoothing effect of analytic semigroups, this is what we can expect in the analytic case.

Definition 4.4

Let $\tau >0$ and $\alpha \in [0,1]$ be given. A function $x \in C([0,\tau ], X)$ is called a mild solution of (37) on $[0,\tau ]$ corresponding to certain $x_0\in X$ and $u \in L^1_{{{\,\textrm{loc}\,}}}(\mathbb {R}_+,U)$, if $x(s) \in X_\alpha$ for $s\in (0,\tau ]$, and x solves the integral equation

$$\begin{aligned} x(t)=T(t) x_0 + \int _0^t T(t-s) f\big (x(s),u(s)\big ){\text {d}}s + \int _0^t T_{-1}(t-s) Bu(s){\text {d}}s. \end{aligned}$$

(38)

We say that $x:\mathbb {R}_+\rightarrow X$ is a mild solution of (37) on $\mathbb {R}_+$ corresponding to certain $x_0\in X$ and $u \in L^1_{{{\,\textrm{loc}\,}}}(\mathbb {R}_+,U)$, if its restriction to $[0,\tau ]$ is a mild solution of (37) (with $x_0, u$) on $[0,\tau ]$ for all $\tau >0$.

Remark 4.5

Note that if $\alpha = 0$, then $X_\alpha = X_0 = X$, and the concept of a mild solution introduced for general and analytic semigroups coincide.

Assumption 4.1

Let the following hold:

• (i) $\alpha \in (0,1)$.

• (ii) $B\in L(U,X_{-1+\alpha +\varepsilon })$ for sufficiently small $\varepsilon >0$.

• (iii) $f \in C(X_\alpha \times U, X)$, and f is Lipschitz continuous in the first argument in the following sense: For each $r>0$, there is $L=L(r)>0$ such that for each $x_1,x_2 \in B_{r,X_\alpha }$ and all $u \in B_{r,U}$ it holds that

  $$\begin{aligned} \Vert f(x_1,u) - f(x_2,u)\Vert _X \le L\Vert x_1-x_2\Vert _{X_\alpha }. \end{aligned}$$

  (39)

• (iv) For all $u\in L^\infty (\mathbb {R}_+,U)$ and any $x\in C(\mathbb {R}_+,X)$ with $x((0,+\infty )) \subset X_\alpha$, the map $s\mapsto f\big (x(s),u(s)\big )$ is in $L^p_{{{\,\textrm{loc}\,}}}(\mathbb {R}_+,X)$ with a certain $p>\frac{1}{1-\alpha }$.

• (v) There is $\sigma \in \mathcal {K_\infty }$ such that

  $$\begin{aligned} \Vert f(0,u)\Vert _X \le \sigma (\Vert u\Vert _U) + c,\quad u\in U. \end{aligned}$$

4.3 Local existence and uniqueness

By Proposition 4.2, the condition $B\in L(U,X_{-1+\alpha +\varepsilon })$ with $\alpha ,\varepsilon >0$ implies that B is zero-class q-admissible for any $q\in (\frac{1}{\alpha +\varepsilon },+\infty ]$. This in turn implies that for such q the map $t\mapsto \int _0^t T_{-1}(t-s) Bu(s){\text {d}}s$ is continuous for any $u \in L^q(\mathbb {R}_+,U)$, by [58, Proposition 2.3].

By Assumption 4.1(iv), we see that for any $u\in L^\infty (\mathbb {R}_+,U)$ the map

$$\begin{aligned} t \mapsto \int _0^t T(t-s)f\big (x(s),u(s)\big ){\text {d}}s \end{aligned}$$

is well defined and continuous.

Hence, if $x\in C(\mathbb {R}_+,X)$ with $x((0,+\infty )) \subset X_\alpha$, then for any $u\in L^\infty (\mathbb {R}_+,U)$ the right-hand side of (38) is a continuous function of time.

Our next result is the local existence and uniqueness theorem for analytic systems with initial states in $X_\alpha$ and the inputs in $\mathcal {U}{:}{=}L^\infty _{{{\,\textrm{loc}\,}}}(\mathbb {R}_+,U)$. Recall the notation $\mathcal {U}_\mathcal {S}$ from (13).

Theorem 4.6

(Picard–Lindelöf theorem for analytic systems) Let Assumption 4.1 hold. Assume that T is an analytic semigroup, satisfying for certain $M \ge 1$, $\lambda >0$ the estimate

$$\begin{aligned} \Vert T(t)\Vert \le Me^{\lambda t},\quad t\ge 0. \end{aligned}$$

For any compact set $Q \subset X_\alpha$, any $r>0$, any bounded set $\mathcal {S}\subset U$, and any $\delta >0$, there is a time $t_1 = t_1(Q,r,\mathcal {S},\delta )>0$, such that for any $w \in Q$, any $x_0 \in W{:}{=}B_{r,X_\alpha }(w)$, and any $u\in \mathcal {U}_{\mathcal {S}}$ there is a unique mild solution of (4) on $[0,t_1]$, and it lies in the ball $B_{Mr+\delta ,X_\alpha }(w)$.

Proof

First, we show the claim for the case if $Q=\{w\}$ is a single point in $X_\alpha$.

(i) Take any $\omega >\omega _0(T)$, and consider the corresponding space $X_\alpha$. Pick any $r>0$ and any $C>0$ such that $W {:}{=} B_{r,X_\alpha }(w) \subset B_{C,X_\alpha }$, and $\mathcal {U}_\mathcal {S}\subset B_{C,\mathcal {U}}$. Pick any $u \in \mathcal {U}_\mathcal {S}$. Take also any $\delta >0$, and consider the following sets (depending on a parameter $t>0$):

$$\begin{aligned} Y_{t}{:}{=} \big \{y \in C([0,t],X): \Vert y(s) - (\omega I-A)^\alpha w\Vert _X\le Mr + \delta \ \ \forall s \in [0,t]\big \}, \end{aligned}$$

endowed with the metric $\rho _{t}(y_1,y_2){:}{=}\sup _{s \in [0,t]} \Vert y_1(s)-y_2(s)\Vert _X$, which makes $Y_t$ complete metric spaces for all $t>0$.

(ii) Pick any $x_0 \in W$. We are going to prove that C([0, t], X) is invariant under the operator $\Phi _u$, defined for any $y \in Y_{t}$ and all $\tau \in [0,t]$ by

$$\begin{aligned} \Phi _u(y)(\tau )&= (\omega I-A)^\alpha T(\tau )x_0 + \int _0^\tau (\omega I-A)^\alpha T_{-1}(\tau -s) Bu(s){\text {d}}s \nonumber \\&\qquad \qquad \qquad + \int _0^\tau (\omega I-A)^\alpha T(\tau -s)f\big ((\omega I-A)^{-\alpha } y(s),u(s)\big ){\text {d}}s. \end{aligned}$$

(40)

Since $y\in C([0,t],X)$, the map $s \mapsto (\omega I-A)^{-\alpha } y(s)$ is in $C([0,t],X_\alpha )$, as for any $s_1,s_2 \in [0,\tau ]$ we have that

$$\begin{aligned} \big \Vert (\omega I-A)^{-\alpha } y(s_1) - (\omega I-A)^{-\alpha } y(s_2)\big \Vert _{X_\alpha } = \Vert y(s_1) - y(s_2)\Vert _X. \end{aligned}$$

By Assumption 4.1, the map $s\mapsto f\big ((\omega I-A)^{-\alpha } y(s),u(s)\big )$ is in $L^p_{{{\,\textrm{loc}\,}}}(\mathbb {R}_+,X)$, with a certain $p>\frac{1}{1-\alpha }$. Proposition 4.2 ensures that the map

$$\begin{aligned} \tau \mapsto \int _0^\tau (\omega I-A)^\alpha T(\tau -s)f\big ((\omega I-A)^{-\alpha } y(s),u(s)\big ){\text {d}}s \end{aligned}$$

is continuous.

Since $B\in L(U,X_{-1+\alpha +\varepsilon })$, Proposition 4.2(ii) implies that

$$\begin{aligned} \tau \mapsto \int _0^\tau (\omega I-A)^\alpha T_{-1}(\tau -s) Bu(s){\text {d}}s \end{aligned}$$

belongs to $C([0,\tau ],X)$.

Overall, the function $\Phi _u(y)$ is continuous, and thus, $\Phi _u$ maps C([0, t], X) to C([0, t], X).

(iii) Now, we prove that for small enough t the spaces $Y_{t}$ are invariant under the operator $\Phi _u$.

Fix any $t>0$ and pick any $y \in Y_{t}$. As $x_0 \in W=B_{r,X_\alpha }(w)$, there is $a \in X_\alpha$: $\Vert a\Vert _{X_\alpha } <r$ such that $x_0=w+a$.

Then for any $\tau <t$, we obtain that

$$\begin{aligned} \Vert&\Phi _{t}(y)(\tau )- (\omega I-A)^\alpha w\Vert _X \nonumber \\&\le \Big \Vert (\omega I-A)^\alpha T(\tau )x_0 - (\omega I-A)^\alpha w\Big \Vert _X + \Big \Vert \int _0^\tau (\omega I-A)^\alpha T_{-1}(\tau -s) Bu(s){\text {d}}s\Big \Vert _X \nonumber \\&\qquad \qquad \qquad + \int _0^\tau \big \Vert (\omega I-A)^\alpha T(\tau -s)\big \Vert \big \Vert f((\omega I-A)^{-\alpha }y(s),u(s))\big \Vert _X{\text {d}}s. \nonumber \end{aligned}$$

We substitute $x_0{:}{=}w+a$ into the first term on the right-hand side of the above inequality. The last term we estimate using (28). To estimate the second term, we use that $(\omega I-A)^{\alpha } B \in L(U,X_{-1+\varepsilon })$. By Proposition 4.2, $(\omega I-A)^{\alpha } B$ is zero-class $\infty$-admissible, and thus, there is an increasing continuous function $t \mapsto h_t$ satisfying $h_0=0$, such that:

$$\begin{aligned}&\big \Vert \Phi _{t}(y)(\tau )- (\omega I-A)^\alpha w\big \Vert _X \le \big \Vert (\omega I-A)^\alpha T(\tau )w - (\omega I-A)^\alpha w\big \Vert _X \nonumber \\&\qquad + \Vert (\omega I-A)^\alpha T(\tau )a\Vert _X + h_{\tau } \Vert u\Vert _{L^\infty ([0,\tau ],U)} \nonumber \\&\qquad + \int _0^\tau \frac{C_\alpha e^{\lambda (\tau -s)}}{(\tau -s)^\alpha } \big (\Vert f(0,u(s))\Vert _X \nonumber \\&\qquad + \big \Vert f((\omega I-A)^{-\alpha }y(s),u(s))-f(0,u(s))\big \Vert _X\big ){\text {d}}s. \end{aligned}$$

(41)

To estimate the latter expression, note that

• $\Vert (\omega I-A)^\alpha a\Vert _X = \Vert a\Vert _{X_\alpha } < r$.

• For all $s\in [0,t]$, we have

  $$\begin{aligned} \Vert (\omega I-A)^{-\alpha }y(s)-0\Vert _{X_\alpha } = \Vert y(s)\Vert _X&\le \Vert (\omega I-A)^\alpha w\Vert _X+Mr+\delta \\&\le M(\Vert w\Vert _{X_\alpha }+r) +\delta \le MC+\delta {:}{=}K. \end{aligned}$$

• In view of Assumption 4.1, it holds that

  $$\begin{aligned} \Vert f(0,u(s))\Vert _X \le \sigma (\Vert u(s)\Vert _U) + c,\quad \text { for a.e. } s \in [0,t]. \end{aligned}$$

• h is a monotonically increasing continuous function.

As $M\ge 1$, it holds that $K>C$, and Lipschitz continuity of f on bounded balls ensures that there is $L(K)>0$, such that for all $\tau \in [0,t]$

$$\begin{aligned}&\Vert \Phi _{t}(y)(\tau )-(\omega I-A)^\alpha w\Vert _X \\&\quad \le \Vert (\omega I-A)^\alpha T(\tau )w - (\omega I-A)^\alpha w\Vert _X + \Vert T(\tau )(\omega I-A)^\alpha a\Vert _X + h_{\tau } \Vert u\Vert _{L^\infty ([0,\tau ],U)}\\&\qquad + \int _0^\tau \frac{C_\alpha }{(\tau -s)^\alpha } e^{\lambda (\tau -s)} \big (\sigma (\Vert u(s)\Vert _U) + c + L(K)\Vert (\omega I-A)^{-\alpha }y(s)\Vert _{X_\alpha }\big ){\text {d}}s\\&\quad \le \sup _{\tau \in [0,t]}\Vert T(\tau )(\omega I-A)^\alpha w - (\omega I-A)^\alpha w\Vert _X + Me^{\lambda t}r + h_{t} \Vert u\Vert _{L^\infty ([0,t],U)}\\&\qquad + C_\alpha e^{\lambda t} \big (\sigma (C) + c + L(K)K\big ) \int _0^t \frac{1}{s^\alpha } {\text {d}}s. \end{aligned}$$

Since T is a strongly continuous semigroup, and $h_t\rightarrow 0$ as $t\rightarrow +0$, from this estimate, it is clear that there exists $t_1$, such that

$$\begin{aligned} \Vert \Phi _u(y)(t)-w\Vert _X \le Mr +\delta ,\quad \text { for all } t \in [0,t_1]. \end{aligned}$$

This means, that $Y_{t}$ is invariant with respect to $\Phi _u$ for all $t \in (0,t_1]$, and $t_1$ does not depend on the choice of $x_0 \in W$.

(iv) Now, pick any $t>0$, $\tau \in [0, t]$, and any $y_1, y_2 \in Y_{t}$. Then, it holds that

$$\begin{aligned}&\Vert \Phi _u(y_1)(\tau ) - \Phi _u(y_2)(\tau )\Vert _X \\&\quad \le \int _0^\tau \Vert (\omega I-A)^\alpha T(\tau -s)\Vert \\&\quad \cdot \big \Vert f((\omega I-A)^{-\alpha }y_1(s),u(s)) - f((\omega I-A)^{-\alpha }y_2(s),u(s))\big \Vert _X{\text {d}}s \\&\quad \le \int _0^t L(K)\frac{C_\alpha }{(\tau -s)^\alpha } e^{\lambda (\tau -s)} \Vert y_1(s)-y_2(s)\Vert _X {\text {d}}s \\&\quad \le L(K) C_\alpha e^{\lambda t} \int _0^t s^{-\alpha } {\text {d}}s \rho _t(y_1,y_2) \\&\quad \le L(K) C_\alpha e^{\lambda t} \frac{t^{1-\alpha }}{1-\alpha } \rho _t(y_1,y_2) \\&\quad \le \frac{1}{2} \rho _t(y_1,y_2), \end{aligned}$$

for $t \le t_2$, where $t_2>0$ is a small enough real number that does not depend on the choice of $x_0 \in W$.

According to Banach fixed point theorem, there exists a unique $y \in Y_t$ that is a fixed point of $\Phi _u$, that is,

$$\begin{aligned} y(\tau )&= (\omega I-A)^\alpha T(\tau )x_0 + \int _0^\tau (\omega I-A)^\alpha T_{-1}(\tau -s) Bu(s){\text {d}}s \nonumber \\&\qquad \qquad \qquad + \int _0^\tau (\omega I-A)^\alpha T(\tau -s)f\big ((\omega I-A)^{-\alpha } y(s),u(s)\big ){\text {d}}s. \end{aligned}$$

(42)

on $[0,\min \{t_1,t_2\}]$.

As $(\omega I-A)^{\alpha }$ is invertible with a bounded inverse, y solves (42) if and only if y solves

$$\begin{aligned} (\omega I-A)^{-\alpha } y(\tau )&= T(\tau )x_0 + \int _0^\tau T_{-1}(\tau -s) Bu(s){\text {d}}s \nonumber \\&\qquad \qquad \qquad + \int _0^\tau T(\tau -s)f\big ((\omega I-A)^{-\alpha } y(s),u(s)\big ){\text {d}}s. \end{aligned}$$

(43)

As $y \in C([0, \min \{t_1,t_2\}],X)$, the map $x{:}{=}(\omega I-A)^{-\alpha } y$ is in $C([0, \min \{t_1,t_2\}],X_\alpha )$, and is the unique mild solution of (37).

(v). General compact Q. Similar to the corresponding part of the proof of Theorem 3.7. $\square$

Remark 4.7

For systems without inputs, Theorem 4.6 was shown (in a somewhat different formulation without bounds on the growth of the solution) in [43, Theorem 3.1]. We have proved our local existence result for initial conditions that are in $X_\alpha$. To ensure local existence and uniqueness for the initial states outside of $X_\alpha$, stronger requirements on f have to be imposed, see [34, Theorems 7.1.5, 7.1.6].

Introducing the concepts of maximal solutions and of well-posedness and arguing similar to Sects. 3.2, 3.3, we obtain the following well-posedness theorem.

Theorem 4.8

Let A generate an analytic semigroup, Assumption 4.1 hold, and let $\mathcal {U}{:}{=}L^\infty (\mathbb {R}_+,U)$. Then:

• (i) For each $x\in X_\alpha$ and each $u \in \mathcal {U}$, there is a unique maximal solution of (37), defined over the certain maximal time-interval $[0,t_m(x,u))$. We denote this solution as $\phi (\cdot ,x,u)$.

• (ii) The triple $\Sigma {:}{=}(X_\alpha ,\mathcal {U},\phi )$ is a well-defined control system in the sense of Definition 2.1.

• (iii) $\Sigma$ satisfies the BIC property; that is, if for a certain $x \in X_\alpha$ and $u\in \mathcal {U}$ we have $t_m(x,u)<\infty$, then $\Vert \phi (t,x,u)\Vert _{X_\alpha } \rightarrow \infty$ as $t \rightarrow t_m(x,u)-0$.

4.4 Global existence

Motivated by [43, Section 6.3, Theorem 3.3], we have the following result guaranteeing the forward completeness and BRS property for semilinear analytic systems.

Theorem 4.9

Let A generate an analytic semigroup, Assumption 4.1 hold, and let $\mathcal {U}{:}{=}L^\infty (\mathbb {R}_+,U)$. Assume further that there are $L,c>0$ and $\sigma \in \mathcal {K_\infty }$ such that

$$\begin{aligned} \Vert f(x,u)\Vert _X \le L\Vert x\Vert _{X_\alpha } + \sigma (\Vert u\Vert _U) + c,\quad x\in X_\alpha ,\quad u\in U. \end{aligned}$$

(44)

Then, $\Sigma {:}{=}(X_\alpha ,\mathcal {U},\phi )$ is a forward complete control system.

Proof

Take any positive $\omega >\omega _0(T)$ and define $X_\alpha$ as in (27).

We argue by a contradiction. Let $\Sigma$ be not forward complete. Then there are $(x_0,u)\in X_\alpha \times \mathcal {U}$ such that $t_m(x_0,u)<\infty$. By Theorem 4.8, we have that $\Vert \phi (t,x_0,u)\Vert _{X_\alpha } \rightarrow \infty$ as $t \rightarrow t_m(x_0,u)-0$.

For $t<t_m(x_0,u)$ denote $x(t){:}{=}\phi (t,x_0,u)$. As $x(\cdot ) \subset X_\alpha$, we can apply $(\omega I-A)^\alpha$ along the trajectory $x(\cdot )$ to obtain

$$\begin{aligned} (\omega I-A)^\alpha x(t)&= (\omega I-A)^\alpha T(t)x_0 + \int _0^\tau (\omega I-A)^\alpha T_{-1}(\tau -s) Bu(s){\text {d}}s \\&\qquad + \int _{0}^t (\omega I-A)^\alpha T(t-s)f(x(s),u(s)){\text {d}}s. \end{aligned}$$

We obtain

$$\begin{aligned} \Vert x(t)\Vert _\alpha&= \Vert (\omega I-A)^\alpha x(t)\Vert _X \\&\le \Vert (\omega I-A)^\alpha T(t)x_0\Vert _X + \Big \Vert \int _0^\tau (\omega I-A)^\alpha T_{-1}(\tau -s) Bu(s){\text {d}}s\Big \Vert _X \\&\qquad \qquad + \int _{0}^t \big \Vert (\omega I-A)^\alpha T(t-s)\big \Vert \big \Vert f(x(s),u(s))\big \Vert _X{\text {d}}s. \end{aligned}$$

We now estimate the second term as in (41), where h is a continuous increasing function with $h_0=0$. The last term we estimate using (28). Overall:

$$\begin{aligned} \Vert x(t)\Vert _\alpha&\le Me^{\omega t} \Vert (\omega I-A)^\alpha x_0\Vert _X + h_{\tau } \Vert u\Vert _{L^\infty ([0,t],U)}\\&\qquad \qquad + \int _{0}^t \frac{C_\alpha }{(t-s)^\alpha }e^{\omega (t-s)} \Big (L\Vert x(s)\Vert _{X_\alpha } + \sigma (\Vert u(s)\Vert _U) + c \Big ){\text {d}}s. \end{aligned}$$

Defining $z(t){:}{=}x(t)e^{-\omega t}$, we obtain from the previous estimate that

$$\begin{aligned} \Vert z(t)\Vert _\alpha&\le M\Vert (\omega I-A)^\alpha x_0\Vert _X + \int _{0}^t \frac{C_\alpha }{s^\alpha }{\text {d}}s \Big (\sigma (\Vert u\Vert _{L^\infty (\mathbb {R}_+,U)}) + c \Big ) + h_{\tau } \Vert u\Vert _{L^\infty (\mathbb {R}_+,U)}\\&\qquad + \int _{0}^t \frac{L C_\alpha }{(t-s)^\alpha }\Vert z(s)\Vert _{X_\alpha }{\text {d}}s. \end{aligned}$$

An analytic version of Gronwall inequality [15, p. 6] shows that z, and hence, x is uniformly bounded on $[0,t_m(x_0,u))$, and BIC property (Theorem 4.8(iii)) shows that $t_m(x_0,u)$ is not the finite maximal existence time. A contradiction. $\square$

4.5 Example: well-posedness of a Burgers’ equation with a distributed input

We consider the following semilinear reaction–diffusion equation of Burgers’ type on a domain $[0,\pi ]$, with distributed input u, boundary input d at $z=0$, and homogeneous Dirichlet boundary condition at $\pi$.

$$\begin{aligned} x_t&= x_{zz} - xx_z + f(z,x(z,t))+ u(z,t),\quad z\in (0,\pi ),\quad t>0, \end{aligned}$$

(45a)

$$\begin{aligned} x&(0,t)=d(t),\quad t>0,\end{aligned}$$

(45b)

$$\begin{aligned} x&(\pi ,t)=0. \end{aligned}$$

(45c)

Here, $f: [0,\pi ]\times \mathbb {R}\rightarrow \mathbb {R}$ is measurable in z, locally Lipschitz continuous in x uniformly in z, and

$$\begin{aligned} |f(z,y)| \le h(z)g(|y|),\quad \text {for a.e. } z \in [0,\pi ], \text { and all } y \in \mathbb {R}, \end{aligned}$$

(46)

where $h\in L^2(0,\pi )$, and g is continuous, increasing, and both h and g are positive.

This system with $u=0$ and $d=0$ was investigated in [15, p. 57]. Here, we give a detailed analysis of this system with distributed and boundary inputs.

We denote $X{:}{=}L^2(0,\pi )$. The operator $A{:}{=} \frac{d^2}{{\text {d}}z^2}$ with the domain $D(A) = H^2(0,\pi )\cap H^1_0(0,\pi )$ generates an analytic semigroup on X.

We assume that the distributed input u belongs to the space $\mathcal {U}=L^\infty (\mathbb {R}_+,U)$, with $U{:}{=}L^2(0,\pi )$, and the boundary input d belongs to $\mathcal {D}{:}{=}L^\infty (\mathbb {R}_+,\mathbb {R})$.

The system (45) can be reformulated as a semilinear evolution equation

$$\begin{aligned} x_t = Ax + F(x)+u + Bd, \end{aligned}$$

(47)

where we slightly abuse the notation and use x as an argument of the evolution equation.

The condition (ii) in Assumption 4.1 characterizing the admissibility properties of the boundary input operator B holds in view of [51, Example 2.16].

The space $X_{\frac{1}{2}}$ corresponding to the operator A is given by (see [56, Proposition 3.6.1])

$$\begin{aligned} X_{\frac{1}{2}} = H^1_0(0,\pi ), \end{aligned}$$

which is a Banach space with the norm

$$\begin{aligned} \Vert x\Vert _{\frac{1}{2}}{:}{=}\Big |\int _0^\pi |x'(z)|^2 {\text {d}}z\Big |^{\frac{1}{2}}, \quad x \in X_{\frac{1}{2}}. \end{aligned}$$

The nonlinearity $F: X_{\frac{1}{2}} \rightarrow X$ in (47) is given by

$$\begin{aligned} F(x)(z) = - x(z)x'(z) + f(z,x(z)). \end{aligned}$$

Proposition 4.10

For each $x_0 \in X_{\frac{1}{2}}$, each $u \in \mathcal {U}=L^\infty _{{{\,\textrm{loc}\,}}}(\mathbb {R}_+,U)$, and each boundary input $d \in \mathcal {D}=L^\infty _{{{\,\textrm{loc}\,}}}(\mathbb {R}_+,\mathbb {R})$, the system (45) possesses a unique maximal mild solution $\phi (\cdot ,x_0,(u,d))$. The system $\Sigma = (X_{\frac{1}{2}},\mathcal {U}\times \mathcal {D},\phi )$ is a control system satisfying the BIC property.

Proof

We proceed in 3 steps:

Step 1: F maps bounded sets of $X_{\frac{1}{2}}$ to bounded sets of X. Since the elements of $X_{\frac{1}{2}}=H^1_0(0,\pi )$ are absolutely continuous functions, using the Cauchy–Schwarz inequality, we obtain that for any $x \in X_{\frac{1}{2}}$ it holds that

$$\begin{aligned} \sup _{z\in (0,\pi )}|x(z)|= & {} \sup _{z\in (0,\pi )}\Big |\int _0^z x'(z) {\text {d}}z\Big | \le \sup _{z\in (0,\pi )}\int _0^z |x'(z)| {\text {d}}z = \int _0^\pi |x'(z) |{\text {d}}z \nonumber \\\le & {} \Big |\int _0^\pi 1 {\text {d}}z\Big |^{\frac{1}{2}} \Big |\int _0^\pi |x'(z)|^2 {\text {d}}z\Big |^{\frac{1}{2}} = \sqrt{\pi } \Vert x\Vert _{\frac{1}{2}}. \end{aligned}$$

(48)

For any $x \in X_{\frac{1}{2}}$, consider

$$\begin{aligned} \Vert F(x)\Vert _X^2 = \int _0^\pi |F(x)(z)|^2 {\text {d}}z= & {} \int _0^\pi |x(z)x'(z) + f(z,x(z))|^2 {\text {d}}z\\\le & {} \int _0^\pi 2|x(z)x'(z)|^2 + 2|f(z,x(z))|^2 {\text {d}}z. \end{aligned}$$

Using (48) and (46), we continue the estimates as follows:

$$\begin{aligned} \Vert F(x)\Vert _X^2\le & {} \int _0^\pi 2|x'(z)|^2 \pi \Vert x\Vert ^2_{\frac{1}{2}} + 2 |h(z)|^2 |g(|x(z)|)|^2 {\text {d}}z\\\le & {} 2\pi \Vert x\Vert ^4_{\frac{1}{2}} + 2 \int _0^\pi |h(z)|^2 |g(\sqrt{\pi } \Vert x\Vert _{\frac{1}{2}})|^2 {\text {d}}z\\= & {} 2\pi \Vert x\Vert ^4_{\frac{1}{2}} + 2 \Vert h\Vert ^2_X |g(\sqrt{\pi } \Vert x\Vert _{\frac{1}{2}})|^2. \end{aligned}$$

Taking the square root and using that $\sqrt{a+b}\le \sqrt{a} + \sqrt{b}$, for all $a,b\ge 0$, we finally obtain

$$\begin{aligned} \Vert F(x)\Vert _X\le & {} \sqrt{2\pi } \Vert x\Vert ^2_{\frac{1}{2}} + \sqrt{2} \Vert h\Vert _X |g(\sqrt{\pi } \Vert x\Vert _{\frac{1}{2}})|. \end{aligned}$$

(49)

This shows that F is well defined as a map from $X_{\frac{1}{2}}$ to X and F maps bounded sets of $X_{\frac{1}{2}}$ to bounded sets of X.

Step 2: F is Lipschitz continuous in x. For $x_0 \in X_{\frac{1}{2}}$, there is a neighborhood V of a compact set $\{(z,x_0(z)):z\in [0,\pi ]\}$ in $[0,\pi ]\times \mathbb {R}$ and positive constants $L,\theta$ so that for $(z,x_1) \in V$, $(z,x_2) \in V$ it holds that

$$\begin{aligned} |f(z,x_1) - f(z,x_2)| \le L |x_1-x_2|. \end{aligned}$$

Thus, there is a neighborhood U of $x_0$ in $X_{\frac{1}{2}}$ such that $x\in U$ implies that $(z,x(z))\in V$ for a.e. $z\in [0,\pi ]$ and for $x_1,x_2 \in U$ it holds that

$$\begin{aligned} \big \Vert f(\cdot ,x_1(\cdot )) - f(\cdot ,x_2(\cdot ))\big \Vert _{X}^2&= \int _0^\pi \big |f(z,x_1(z)) - f(z,x_2(z))\big |^2{\text {d}}z \\&\le L^2 \int _0^\pi |x_1(z)-x_2(z)|^2 {\text {d}}z, \end{aligned}$$

and using (48) we have that

$$\begin{aligned} \big \Vert f(\cdot ,x_1(\cdot )) - f(\cdot ,x_2(\cdot ))\big \Vert _{X}^2\le & {} L^2 \pi ^2 \Vert x_1-x_2\Vert ^2_{\frac{1}{2}}. \end{aligned}$$

Taking the square root, we have that

$$\begin{aligned} \big \Vert f(\cdot ,x_1(\cdot )) - f(\cdot ,x_2(\cdot ))\big \Vert _{X} \le \pi L \Vert x_1-x_2\Vert _{\frac{1}{2}}. \end{aligned}$$

(50)

Finally, for any $x_1,x_2\in X_{\frac{1}{2}}$ it holds that

$$\begin{aligned} \Vert x_1x_1' - x_2x_2'\Vert _X\le & {} \Vert x_1(x_1'-x_2')\Vert _X + \Vert (x_1-x_2)x_2'\Vert _X, \end{aligned}$$

and again using (48), we proceed to

$$\begin{aligned} \Vert x_1x_1' - x_2x_2'\Vert _X\le & {} \sqrt{\pi } \Vert x_1\Vert _{\frac{1}{2}} \Vert x_1-x_2\Vert _{\frac{1}{2}} + \sqrt{\pi } \Vert x_1-x_2\Vert _{\frac{1}{2}} \Vert x_2\Vert _{\frac{1}{2}}\nonumber \\= & {} \sqrt{\pi } ( \Vert x_1\Vert _{\frac{1}{2}} + \Vert x_2\Vert _{\frac{1}{2}})\Vert x_1-x_2\Vert _{\frac{1}{2}}. \end{aligned}$$

(51)

Combining (50) and (51), we obtain the required Lipschitz property for the function F.

Step 3: Application of general well-posedness theorems. Finally, Theorem 4.6 shows that the system (45) possesses a unique mild solution for each $x_0 \in X_{\frac{1}{2}}$, each $u \in \mathcal {U}=L^\infty _{{{\,\textrm{loc}\,}}}(\mathbb {R}_+,U)$, and each boundary input $d \in \mathcal {D}=L^\infty _{{{\,\textrm{loc}\,}}}(\mathbb {R}_+,\mathbb {R})$. Theorem 4.8 shows that $\Sigma$ is a control system satisfying the BIC property. $\square$

5 Boundary control systems

Control systems governed by partial differential equations are defined by PDEs describing the dynamics inside of the spatial domain and boundary conditions, describing the dynamics of the system at the boundary of the domain. Such systems look (at first glance) quite differently from the evolution equations in Banach spaces, studied in Sect. 3. This motivated the development of a theory of abstract boundary control systems that allows for a more straightforward interpretation of PDEs in the language of semigroup theory.

5.1 Linear boundary control systems

Let X and U be Banach spaces. Consider a system

$$\begin{aligned} \dot{x}(t)&= {\hat{A}} x(t), \qquad x(0) = x_0, \end{aligned}$$

(52a)

$$\begin{aligned} {\hat{R}}x(t)&= u(t), \end{aligned}$$

(52b)

where the formal system operator $\hat{A}: D( \hat{A}) \subset X \rightarrow X$ is a linear operator, the control function u takes values in U, and the boundary operator $\hat{R}: D( \hat{R}) \subset X \rightarrow U$ is linear and satisfies $D(\hat{A}) \subset D(\hat{R})$.

Definition 5.1

The system (52) is called a linear boundary control system (linear BCS) if the following conditions hold:

• (i) The operator $A: D(A) \rightarrow X$ with $D(A) = D({\hat{A}} ) \cap \textrm{Ker}\,({\hat{R}})$ defined by

  $$\begin{aligned} Ax = {\hat{A}}x \qquad \text {for} \quad x\in D(A) \end{aligned}$$

  (53)

  is the infinitesimal generator of a $C_0$-semigroup $(T(t))_{t\ge 0}$ on X.

• (ii) There is an operator $R \in \mathcal {L}(U,X)$ such that for all $u \in U$ we have $Ru \in D({\hat{A}})$, ${\hat{A}}R \in \mathcal {L}(U,X)$ and

  $$\begin{aligned} {\hat{R}}Ru = u, \qquad u\in U. \end{aligned}$$

  (54)

The operator R in this definition is sometimes called a lifting operator. Note that R is not uniquely defined by the properties in the item (ii).

Item (i) of Definition 5.1 shows that for $u\equiv 0$ the equations (52) are well-posed. In particular, as A is the generator of a certain strongly continuous semigroup $T(\cdot )$, for any $x\in D(A)$, it holds that $T(t)x \in D(A)$ and thus $T(t)x \in \textrm{Ker}\,({\hat{R}})$ for all $t\ge 0$, which means that (52b) is satisfied.

Item (ii) of the definition implies, in particular, that the range of the operator ${\hat{R}}$ equals U, and thus, the values of inputs are not restricted.

5.2 Semilinear boundary control systems

Let $(\hat{A},\hat{R})$ be a linear BCS. We consider $D(\hat{A}) \subset X$ as a linear space equipped with the graph norm

$$\begin{aligned} \Vert \cdot \Vert _{D(\hat{A})}{:}{=}\Vert \cdot \Vert _{X}+\big \Vert \hat{A}\cdot \big \Vert _{X}. \end{aligned}$$

Motivated by [51], we consider the following class of semilinear boundary control systems.

Definition 5.2

Consider a linear BCS $(\hat{A},\hat{R})$. Consider the following system

$$\begin{aligned} \dot{x}(t)&= \hat{A}x(t)+f(x(t),w(t)), \ t>0, \end{aligned}$$

(55a)

$$\begin{aligned} {\hat{R}}x(t)&= u(t),\ t>0, \end{aligned}$$

(55b)

$$\begin{aligned} x(0)&= x_0, \end{aligned}$$

(55c)

with a nonlinearity $f:X\times W\rightarrow X$, where W is a Banach space.

The system (55) we call a semilinear boundary control system (semilinear BCS).

Following [51], we define classical solutions to the semilinear BCS (55).

Definition 5.3

Let $x_{0}\in D(\hat{A})$, $\tau >0$ and $u\in C([0,\tau ],U)$. A function

$$\begin{aligned} x\in C([0,T],D(\hat{A}))\cap C^{1}([0,T],X) \end{aligned}$$

is called a classical solution to the semilinear BCS (55) on $[0,\tau ]$ if $x(t)\in X$ for all $t>0$ and the equations (55) are satisfied pointwise for $t\in (0,\tau ]$.

A function $x:[0,\infty )\rightarrow X$ is called (global) classical solution to the semilinear BCS (55), if $x|_{[0,\tau ]}$ is a classical solution on $[0,\tau ]$ for every $\tau >0$.

If $x\in C([0,\tau ],D(\hat{A}))\cap C^{1}((0,\tau ],X)$ and $x(t)\in X$ for all $t>0$ and the equations (55) are satisfied pointwise for $t\in (0,\tau ]$, then we say that x is a classical solution on $(0,\tau ]$.

The next theorem gives a representation for the (unique) solutions of (55) for smooth enough inputs.

Theorem 5.4

Consider the boundary control system (52) with $f \in C(X\times W,X)$. Let $u \in C^2([0,\tau ],U)$, and $w\in C([0,\tau ],W)$ for some $\tau >0$, and let $x_0\in X$ be such that $x_0 - Ru(0) \in D(A)$. Assume that the classical solution of semilinear BCS $\phi (\cdot ,x_0,u)$ exists on $[0,\tau ]$. Then, it can be represented as

$$\begin{aligned} \phi (t,x_0,u)&= T(t)\big (x_0-Ru(0)\big ) \nonumber \\&\qquad + \int _0^t T(t-r)\Big (f(x(r),w(r)) + {\hat{A}}Ru(r)-R\dot{u}(r)\Big ) {\text {d}}r + Ru(t) \end{aligned}$$

(56a)

$$\begin{aligned}&= T(t)x_0 + \int _0^t T(t-r)\Big (f(x(r),w(r))+{\hat{A}}Ru(r)\Big ){\text {d}}r\nonumber \\&\qquad - A\int _0^t T(t-r)Ru(r) {\text {d}}r \end{aligned}$$

(56b)

$$\begin{aligned}&= T(t)x_0 + \int _0^t T_{-1}(t-r)\Big (f\big (x(r),w(r)\big )+({\hat{A}}R - A_{-1}R)u(r)\Big ){\text {d}}r , \end{aligned}$$

(56c)

where $A_{-1}$ and $T_{-1}$ are the extensions of the infinitesimal generator A and of the semigroup T to the extrapolation space $X_{-1}$. Furthermore, $A_{-1}R \in L(U,X_{-1})$ (and thus ${\hat{A}}R - A_{-1}R \in L(U,X_{-1})$).

The proof of the linear case (with $f=0$) should be well known; see, e.g., [38, Theorem 4.4], [51, pp. 93–94] for the proofs of this fact, and [23, Theorem 11.1.2] for a partial result. The nonlinear result can be obtained in a similar manner. Hence we omit the proof.

An advantage of the representation formula (56a) is in the boundedness of the operators R and $\hat{A}R$ involved in the expression. Its disadvantage is that the derivative of u is employed. Still, the expression in the right-hand side of (56a) makes sense for any $x \in X$ and for any $u \in H^1([0,\tau ],U)$ and can be called a mild solution of BCS (52), as is done, e.g., in [23, p. 146].

The formula (56c) does not involve any derivatives of inputs and again is given in terms of a bounded operator ${\hat{A}}R - A_{-1}R \in L(U,X_{-1})$. Moreover, if we consider the expression in the right-hand side of (56c) in the extrapolation spaces $X_{-1}$, then it makes sense for all $x \in X$ and all $u\in L^{1}_{{{\,\textrm{loc}\,}}}(\mathbb {R}_+,U)$ and constitutes a mild solution of

$$\begin{aligned} \dot{x}(t) = Ax(t) + f(x(t),w(t)) + Bu(t), \end{aligned}$$

(57)

with

$$\begin{aligned} B{:}{=}{\hat{A}}R-A_{-1}R. \end{aligned}$$

(58)

This motivates us to define the mild solutions of semilinear BCS by means of the formula (56c), as was proposed in [51].

Definition 5.5

Let $(\hat{A},\hat{R},f)$ be a semilinear boundary control system with corresponding A, R. Let $x_{0}\in X$, $\tau >0$, $w\in L_{{{\,\textrm{loc}\,}}}^{1}([0,\tau ],W)$, and $u\in L_{{{\,\textrm{loc}\,}}}^{1}([0,\tau ],U)$. A continuous function $x:[0,\tau ]\rightarrow X$ is called mild solution to the semilinear BCS (55) on $[0,\tau ]$ if $x(t)\in X$ for all $t>0$ and x solves

$$\begin{aligned} x(t)= T(t)x_{0}+\int _{0}^{t}T_{-1}(t-s)\big (f(x(s),w(s))+Bu(s)\big ){\text {d}}s, \end{aligned}$$

for all $t\in [0,\tau ]$ and where $B=\hat{A}B_{0}-A_{-1}B_{0}$. A function $x:\mathbb {R}_+ \rightarrow X$ is called a global mild solution if $x|_{[0,\tau ]}$ is a mild solution on $[0,\tau ]$ for all $\tau >0$.

In other words, x is a mild solution of a semilinear BCS (5.2), if x is a mild solution of (57) with $B={\hat{A}}R - A_{-1}R$.

Thus, semilinear boundary control systems are a special case of semilinear evolution equations studied in Sect. 3, and we can use our well-posedness theory for semilinear evolution equations to analyze semilinear BCS.