1 Introduction

Convexity properties are particularly significant for many optimization problems, and when they are fully present, they lead to an efficient type of duality. Our goals in this paper are twofold. On the one hand, to formulate sufficient conditions of optimality for hyperbolic partial differential inclusions for convex Darboux problems and prove optimality conditions for polyhedral problem with state constraints. On the other hand, to construct the dual problem to optimal control problem given by Darboux differential inclusions with state constraints and prove duality relations.

One can use the differential calculus of set-valued mappings for looking for global set-valued solutions to hyperbolic of both partial differential equations, and inclusions [1, 4, 14,15,16,17, 20, 22, 25, 27]. The paper [5] studies an optimal control problem given by hyperbolic differential inclusions with boundary conditions and endpoint constraints. It is suggested a method for second-order optimality conditions. In the article [12], Jackson discusses how certain types of chemical plants’ optimum start-up and control problems can be expressed mathematically as variational problems involving two independent variables with hyperbolic partial differential equations acting as side-conditions. The author also investigates how the first-order variational theory might be expanded to conclude similar to Pontryagin’s maximal principle.

The existence of the solutions set of hyperbolic differential equations have received much attention during the last three decades. Using the mixed generalized Lipschitz and Caratheodory’s criteria, the work [10] gives sufficient conditions for the existence of solutions to the initial value problem for perturbed hyperbolic differential inclusions. The paper [2] deals with the existence of solutions to the Darboux problem with hyperbolic differential inclusion. The authors’ approach is based on the existence of upper and lower solutions and on a fixed point theorem for condensing set-valued mappings.

In the classical calculus of variations, convexity and duality first enter the picture in the correspondence between Lagrangian and Hamiltonian functions and in the way this is connected with necessary conditions and the existence of solutions. This correspondence is a result of the Legendre transformation, which converts Lagrangian functions into Hamiltonians and vice versa. Expressed in terms of the Hamiltonian, the optimality conditions for an arc x pair it with an “adjoint” arc \(x^*\). The pairing carries over to problems of optimal control via the maximum principle. Duality theory in this context aims at uncovering and analyzing cases where \(x^*\) happens to solve a dual problem for which x is in turn the adjoint arc. But although this is the principal motivation, some side issues have to be explored along the way, and these suggest new approaches even to problems where duality is not at stake [26].

In this paper, the duality theorems proved to allow one to conclude that a sufficient condition for an extremum is an extremal relation for the primal and dual problems. The latter means that if some pair of admissible solutions satisfy this relation, then each of them is a solution to the corresponding (primal and dual) problem. We remark that a significant part of the investigations of Ekeland and Temam [11] for simple variational problems is connected with such problems, and there are similar results for ordinary differential inclusions [7,8,9]. Some duality relations and optimality conditions for an extremum of different control problems with partial differential inclusions can be found in Mahmudov [18]. In particular, the paper [19] establish the duality theorems for Goursat-Darboux-type problems with state constraints.

Due to the significance of its applications, duality theory is one of the main approaches to convex optimality problems, and it is understood differently for various concrete cases [3]. The work [24] develops a duality theory that is used to derive the variational principle for the relevant situation. The author derives the variational principle for minimizing sequences from duality, which allows for numerical characterization of minimizing sequences and approximation of its infimum. The study [23] investigates optimum control problems for partial differential equations with control functions in the Dirichlet boundary conditions under point-wise control constraints. The author changes all notions of dynamic programming to dual space, then develops a dual dynamic approach with a dual Hamilton-Jacobi equation. Moreover, the author formulates sufficient optimality conditions for the concerned problem and defines optimal dual feedback control. The study of economic models, where dual variables can be thought of as prices, is one theoretical application of this duality. It is sometimes helpful in computation methods. The study of duality frequently advances a subject by providing opposing viewpoints, even though it may be pertinent to a particular subclass of problems.

Duality at its fullest, depending essentially on convexity, leads to an enrichment of all aspects of the analysis of optimization problems. In this paper, we construct the dual problem to the convex problem for differential inclusion of considered hyperbolic type, formulate duality results, and investigate the conditions under which such duality relations connect primal and dual problems. The nature of the problems presented and their optimality conditions are different from those that had been previously encountered. The structure of this paper is as follows:

The set-valued analysis notations and fundamental knowledge that will be used throughout this research are introduced in Sect. 2. We provide a brief definition of the set-valued mappings and cones of tangent directions that are necessary for the paper; the definitions and concepts are available in the books [13] and [21].

In Sect. 3, as discrete steps converge to zero when employing the discrete approximation method, the formulation of sufficient conditions is realized by passing the formal limit in the discrete-approximation problem. To avoid any numerical issues, we exclude the discrete approximation method calculations of the original problem (P) in this work. Therefore, we give sufficient conditions of optimality for convex hyperbolic differential inclusions with state constraints and prove the optimality conditions in the Theorem 3.1. To support our main findings for the fundamental convex problem with state constraint, we present optimality conditions for the considered problem in the Corollary 3.2. Finally, we take into account differential inclusion polyhedral problems of the Darboux type and provide the optimality conditions in the Theorem 3.3.

Section 4 is devoted to the duality in the convex problem with the Darboux differential inclusions and state constraints. It is proven that the Euler-Lagrange type inclusion is a dual relation and the optimal values in the primal convex and dual concave problems are equal in Theorems 4.1 and 4.2. Further, it is shown that the duality theorem for the polyhedral problem is proved in Corollary 4.4. Finally, the dual problem to the polyhedral Darboux problem is established, and an example is given to illustrate the key constructs of our approach. Here we get dual results thanks to the dual operations of addition and infimal convolution of convex functions. However, it takes a lot of time and effort to comprehend the computational components of building the duality problem with the help of discrete and discrete approximation problems. In order to avoid deviating from the central theory, we have excluded this.

2 Preliminary problem statements

In this section, first, we recall some basic definitions and properties of set-valued mappings that will be useful in the sequel; then, we state and prove our main results. Books [13] and [21] can provide the reader with further information and debates on significant developments in this area. Let (x, v) be a pair of \(x,v \in {\mathbb {R}}^n\), which denotes the n-dimensional Euclidean space with the norm \(||\cdot ||\) induced by the inner product \(\langle x, v\rangle\).

Assume that \(F:{\mathbb {R}}^{n} \rightrightarrows {\mathbb {R}}^n\) is a set-valued mapping from \({\mathbb {R}}^{n}\) into the set of subsets of \({\mathbb {R}}^{n}\). If the graph of a set-valued mapping F, \(gph F = \{(x,v): v \in F(x)\}\) is a convex subset of \({\mathbb {R}}^{2n}\), then the mapping is said to be convex. If the gphF of a set-valued mapping F is a convex-closed set in \({\mathbb {R}}^{2n}\), then the mapping is convex-closed. If F(x) is a convex set for each \(x \in dom F\), where \(dom F = \{x: F(x)\ne \emptyset \}\), it is convex-valued. For a set-valued mapping F, the Hamiltonian function and argmaximum set are defined as follows:

$$\begin{aligned} H_F(x,v^*)= & {} \sup \limits _{v}\Big \{\langle v,v^*\rangle :v\in F(x)\Big \}, \\ F_A(x;v^*)= & {} \Big \{v\in F(x):\langle v,v^*\rangle =H_F(x,v^*)\Big \} \end{aligned}$$

\(v^*\in {\mathbb {R}}^n\), respectively. If F is convex and \(F(x)=\emptyset\), we set \(H_F(x,v^*)=-\infty\). Clearly, the Hamiltonian function is concave in x and convex in \(v^*\) for the convex set-valued mapping F.

The concepts of the derivative and gradient functions are useful for describing the local properties of differentiable functions. We use subgradients and subdifferentials for convex functions rather than the gradient because they are effective tools for examining the local features of convex functions. The subgradient of a convex function f at \(x_0\), denoted \(x^*\), is defined by the system of inequalities \(f(x)-f(x_0)\ge \langle x^*,x-x_0\rangle \;\forall \;x\). The set of all subgradients at \(x_0\) denoted \(\partial f(x_0)\) is referred to as the subdifferential of f at \(x_0\). If the subdifferential at \(x_0\) consists of only one element, it is equal to the gradient of f at \(x_0\). A function \(\psi =\psi (x,y)\) is called a proper function if it does not assume the value \(-\infty\) and is not identically equal to \(+\infty\). Clearly, \(\psi\) is proper if and only is \(dom\psi \ne \emptyset\) and \(\psi (x,y)\) is finite for \((x,y)\in dom \psi =\{(x,y):\psi (x,y)<+\infty \}\). The support function \(W_A(x^*)\) of a nonempty set A in \({\mathbb {R}}^{n}\) is defined by \(W_A(x^*)=\sup _x\{\langle x,x^* \rangle : x\in A\}, x^*\in {\mathbb {R}}^{n}\). The function \(\psi ^*(\eta ^*)=\sup \limits _{\eta } \{\langle \eta ,\eta ^*\rangle -\psi (\eta )\}\) is called the conjugate of \(\psi\). The conjugate of a function is closed and convex. The operation of infimal convolution \(\oplus\) of proper convex functions \(f_1,f_2\) is defined as follows:

$$\begin{aligned} (f_1\oplus f_2)(u)=\inf \Big \{f_1(u_1)+f_2(u_2):\; u_1,u_2\in {\mathbb {R}}^{n},\; u_1+u_2=u\Big \}. \end{aligned}$$

The infimal convolution \((f_1\oplus f_2)\) is said to be exact provided the infimum above is attained for every \(u\in {\mathbb {R}}^{n}\). One has dom\((f_1\oplus f_2)=dom f_1+dom f_2\).

As is well known, the Moreau-Rockafellar-Robinson internal point qualification condition is sufficient to ensure that the infimal convolution of the conjugates of two extended-real-valued convex lower semi-continuous functions defined on a locally convex space is exact and that the subdifferential of the sum of these functions is the sum of their subdifferentials. The following theorem yields a simple characterization in the finite-dimensional setting of the subdifferential of the sum of these functions.

Theorem 2.1

[13](Moreau-Rockafellar) Let \(f_1\), \(f_2\) be a proper convex function and \(f=f_1+f_2\), \(x_0\in dom f_1 \cap dom f_2\). Suppose that either (1) there is a point \(x_1\in dom f_1 \cap dom f_2\) where \(f_1\) is continuous or (2) \(ri dom f_1 \cap ridom f_2\ne \emptyset\). Then

$$\begin{aligned} \partial f(x_0)=\partial f_1(x_0)+\partial f_2(x_0). \end{aligned}$$

One of the key ideas in convex analysis and extremal theory is the convex cone. The calculation of the dual cone is related to the examination of its properties. Our terminology and notation typically follow Mahmudov’s usage. The definitions of the dual cone and the cone of tangent directions is directly extracted from his book [13]. The set of vectors \(x^*\in {\mathbb {R}}^{n}\) for which \(\langle x,x^*\rangle \ge 0\) holds for all \(x\in K\) is called the dual cone to the cone K and is denoted by \(K^*\). The convex cone \(K_A(z_0)\) is called the cone of tangent directions at a point \(z_0\in A\) to the set A if from \(\overline{z}\in K_A(z_0)\) it follows that \(\overline{z}\) is a tangent vector to the set A at point \(z_0\in A\), i.e., there exists such function \(\gamma (\lambda )\in {\mathbb {R}}^{n}\) such that \(z_0+\lambda \overline{z}+\gamma (\lambda )\in A\) for sufficiently small \(\lambda >0\) and \(\lambda ^{-1}\gamma (\lambda )\rightarrow 0\) as \(\lambda \downarrow 0\).

For a convex mapping F at a point \(w_0=(x_0,v_0)\in gph F\) setting \(\gamma (\lambda )\equiv 0\), we have \(K_{gphF}(w_0)=cone\big [gph F-(w_0)\big ]=\big \{(\overline{x},\overline{v}):\;\overline{x}=\lambda (x-x_0),\overline{v}=\lambda (v-v_0),\) \(\big \},\;\forall \;(x,v)\in gph F.\) For a convex mapping F a set-valued function defined by

$$\begin{aligned} F^*\big (v^*;w_0\big ):=\big \{x^*:(x^*,-v^*)\in K^*_{gphF}(w_0)\big \} \end{aligned}$$

is a locally adjoint set-valued mapping (LAM) to F at a point \(w_0\in gphF\), where \(K^*_{gphF} (w_0)\) is the dual to the cone of tangent vectors \(K_{gph F}(w_0)\).

Theorem 2.2

[13](Theorem 2.1.) Let \(F:{\mathbb {R}}^{n} \rightrightarrows {\mathbb {R}}^n\) be a convex set-valued mapping. Then

$$\begin{aligned} F^*(v^*;(x,v))= & {} {\left\{ \begin{array}{ll} -\partial _x \big (-H_F(x,v^*)\big ), &{} v\in F_A(x;v^*),\\ \emptyset , &{} v\not \in F_A(x;v^*). \end{array}\right. } \end{aligned}$$

By Theorem 2.2, all sets \(F^*(v^*;(x,v))\) for every \(v\in F_A(x;v^*)\) are the same and equal to \(-\partial _x \big (-H_F(x,v^*)\big )\), where \(H_F\) is Hamiltonian function and \(F_A\) is an argmaximum set.

A polyhedral convex set in \({\mathbb {R}}^n\) is a set that can be described as the intersection of a finite family of closed half-spaces, that is, as the set of solutions to a finite system of inequalities of the form \(\langle x,x_k^*\rangle \le \beta _k,\;k=1,\ldots ,l\). In particular, if the finite system of inequalities is homogeneous, the set of solutions to this finite system of inequalities is called the polyhedral cone. A polyhedral mapping is defined, the graph of which is the following polyhedral set in \({\mathbb {R}}^{2n}\):

$$\begin{aligned} gph F=\Big \{(u,v): Pu-Qv\le d\Big \},\;F(u)=\Big \{v: Pu-Qv \le d\Big \}, \end{aligned}$$

where P and Q, \(m\times n\) dimensional matrices, d is a m-dimensional column-vector. The Hamiltonian function \(H_F(\cdot ,v^*)\) for a polyhedral mapping is closed and its LAM is the step function in the argument (u, v):

$$\begin{aligned} F^*\big (v^*;(u,v)\big ):=\Big \{-P^*\lambda :v^*=-Q^*\lambda ,\;\lambda \ge 0,\;\langle Pu-Qv-d,\;\lambda \rangle =0\Big \}. \end{aligned}$$

Let us denote the following:

$$\begin{aligned} M_F(x^*,y^*):=\inf _{x,y}\Big \{\langle x,x^*\rangle -\langle y,y^*\rangle :\; (x,y)\in gphF\Big \}. \end{aligned}$$

It is clear that for every \(x\in {\mathbb {R}}^{n}\)

$$\begin{aligned} M_F(x^*,y^*) \le \langle x,x^*\rangle -H_F(x,y^*). \end{aligned}$$

Here the Hamiltonian function \(H_F(x,y^*)=\sup \limits _y\{\langle y,y^*\rangle : y\in F(x)\}, y^*\in {\mathbb {R}}^{n}\) and for convex set-valued mapping in the case \(F(x)=\emptyset\), we set \(H_F(x,y^*)=-\infty\). Moreover it is easy to see that the function

$$\begin{aligned} M_F(x^*,y^*)=\inf _{x}\{\langle x,x^*\rangle -H_F(x,y^*)\} \end{aligned}$$

is a support function of the set gphF taken with a minus sign. It follows that for a fixed \(y^*\) \(M_F(x^*,y^*)=-\big (H_F(\cdot ,y^*)\big )^*(x^*)\).

Let us propose our so-called Darboux differential inclusion problem (P), which involves hyperbolic differential inclusion (2) and state constraints (3). We give our first problem as follows:

$$\begin{aligned} \text{ minimize }\;&J[u(\cdot ,\cdot )]= \iint \limits _{D}f(u(x,t),x,t)dx dt, \end{aligned}$$
(1)
$$\begin{aligned} (P)\qquad u_{xt}(x,t)&\in F(u(x,t),x,t), \; (x,t)\in D:=[0,1]\times [0,1], \end{aligned}$$
(2)
$$\begin{aligned}&u_t(0,t)\in F_1(u(0,t),t), \nonumber \\&u_x(x,0)\in F_2(u(x,0),x), \nonumber \\&u(x,t)\in \Psi (x,t). \end{aligned}$$
(3)

Here \(F(\cdot ,x,t):{\mathbb {R}}^{n}\rightrightarrows {\mathbb {R}}^n\), \(F_1(\cdot ,t):{\mathbb {R}}^{n}\rightrightarrows {\mathbb {R}}^n\) and \(F_2(\cdot ,x):{\mathbb {R}}^{n}\rightrightarrows {\mathbb {R}}^n\) are set-valued mappings for all fixed \((x,t)\in D\), \(t\in [0,1]\) and \(x\in [0,1]\), respectively, and \(f(\cdot ,x,t):{\mathbb {R}}^{n}\rightarrow {\mathbb {R}}^n\) is continuous function and \(\Psi :D\rightarrow {\mathbb {R}}^{n}\) is a convex-valued mapping.

The problem is to find a solution \(\widetilde{u}(x,t)\) of the problem (P) satisfying (2) almost everywhere on D and the boundary conditions and state constraints (3) that minimizes \(J[u(\cdot ,\cdot )]\). Here, an admissible solution is understood to be an absolutely continuous function defined on D with an integrable mixed derivative \(u_{xt}(\cdot ,\cdot )\) satisfying (2) almost everywhere on D and the boundary conditions in (3) on [0, 1]. Note that a function \(u(\cdot ,\cdot )\) is said to be absolutely continuous on D if there exist \(g:D\rightarrow {\mathbb {R}}^n\) such that

$$\begin{aligned} u(x,t)=\int \limits _{0}^t\int \limits _0^xg(\xi ,\eta )d\xi d\eta ,\quad (x,t)\in D. \end{aligned}$$

It is known that the space B of absolutely continuous functions \(u:D\rightarrow {\mathbb {R}}^n\) is a Banach space endowed with the norm

$$\begin{aligned} ||u(\cdot ,\cdot )||_B=\iint \limits _{D}||u_{xt}(x,t)||dx dt. \end{aligned}$$

3 Optimality conditions for the Darboux differential problem

We employ difference approximations of partial derivatives and grid functions on a uniform grid by approximating the problem (P) with the hyperbolic differential inclusions to construct sufficient conditions for optimality. In other words, we consider the discrete approximation method to obtain the necessary and sufficient conditions for the optimality of the discrete approximation problem associated with the differential problem. The approach in question necessitates some specific LAM equivalence theorems that arise in discrete and discrete approximation problems and the acquired equivalence theorems allow us to transition between discrete and differential problems. We use the tools of the locally adjoint mappings for convex problems to obtain sufficient optimality conditions for hyperbolic differential inclusions. The derivation of sufficient conditions is carried out by reaching the formal limit as the discrete steps approach zero. To avoid numerical complications, we skip the discrete approximation method calculations of the original problem (P). The discrete approximation method has been applied successfully to a variety of optimal control problems with partial differential inclusions, and the reader can find it in detail in the paper [15].

As indicated in Sect. 1, we need to formulate sufficient conditions of optimality for the problem (P) to construct duality relations. The optimality conditions for our first Darboux-type differential inclusion problem (P) can be stated by the following theorem.

Theorem 3.1

Assume that f is continuous and convex function on the first argument u, and that set-valued mappings \(F, F_1\), and \(F_2\) are also convex. Then for the optimality of a feasible solution \(\widetilde{u}(x,t)\), it is sufficient that there exists an absolutely continuous function \(u^*(x,t)\) on D with integrable mixed derivative \(u_{xt}^*(x,t)\) such that conditions \((i)-(iv)\) hold almost everywhere:

$$\begin{aligned} (i){} & {} u_{xt}^*(x,t)\in F^*\Big (u^*(x,t),(\tilde{u}(x,t),\tilde{u}_{xt}(x,t)),x,t\Big )-\partial f(\tilde{u}(x,t),x,t),\\ (ii){} & {} u_x^*(x,1)\in -K^*_{\Psi (x,1)}(\widetilde{u}(x,1))\\{} & {} u_t^*(1,t)\in -K^*_{\Psi (1,t)}(\widetilde{u}(1,t))\\{} & {} u^*(0,0)=u^*(1,1)=0 \\ (iii){} & {} \widetilde{u}_{xt}(x,t)\in F_A\Big (\widetilde{u}(x,t);u^*(x,t),x,t\Big ) \\{} & {} \widetilde{u}_{t}(0,t)\in F_{1A}\Big (\widetilde{u}(0,t);u^*(0,t),t\Big ) \\{} & {} \widetilde{u}_{x}(x,0)\in F_{2A}\Big (\widetilde{u}(x,0);u^*(x,0),x\Big ) \\ (iv){} & {} H_{F_1}\Big (\widetilde{u}(0,t);u^*(0,t)\Big )=\Big \langle \widetilde{u}_t(0,t),\;u^*(0,t)\Big \rangle \\{} & {} H_{F_2}\Big (\widetilde{u}(x,0);u^*(x,0)\Big )=\Big \langle \widetilde{u}_x(x,0),\;u^*(x,0)\Big \rangle \\ \end{aligned}$$

Proof

Since all sets \(F^*\Big (u^*(x,t),(\tilde{u}(x,t),\tilde{u}_{xt}(x,t)),x,t\Big )\) for every \(\widetilde{u}_{xt}(x,t)\in\) \(F_A\Big (\widetilde{u}(x,t);u^*(x,t),x,t\Big )\) are the same and equal to \(-\partial _u \Big ( -H_F \big (\tilde{u}(x,t),u^*(x,t),x,t\big )\Big )\) in the Theorem 2.2, and by Theorem 2.1, the condition (i) of the theorem implies that

$$\begin{aligned} -u_{xt}^*(x,t)\in \partial _u \Big (-H_F\big (\tilde{u}(x,t),u^*(x,t),x,t\big )+f(\tilde{u}(x,t),x,t)\Big ). \end{aligned}$$

By definition of subdifferential, we have for any feasible solution u(x, t)

$$\begin{aligned}{} & {} H_F\big (u(x,t),u^*(x,t),x,t\big )-H_F\big (\widetilde{u}(x,t),u^*(x,t),x,t\big )\\{} & {} \quad -f(u(x,t),x,t)+f(\tilde{u}(x,t),x,t)\\{} & {} \quad \le \Big \langle u_{xt}^*(x,t),\;u(x,t)-\widetilde{u}(x,t)\Big \rangle . \end{aligned}$$

Further, using the definition of the Hamiltonian function, we obtain that

$$\begin{aligned}{} & {} f(u(x,t),x,t)-f(\tilde{u}(x,t),x,t)\\{} & {} \quad \ge \Big \langle u_{xt}(x,t),u^*(x,t)\Big \rangle -\Big \langle \widetilde{u}_{xt}(x,t), u^*(x,t)\Big \rangle -\Big \langle u_{xt}^*(x,t),\;u(x,t)-\widetilde{u}(x,t)\Big \rangle \end{aligned}$$

By integrating this inequality over D, we have

$$\begin{aligned}{} & {} \displaystyle \iint \limits _D \Big [f(u(x,t),x,t)-f(\tilde{u}(x,t),x,t)\Big ]dxdt\nonumber \\{} & {} \quad \ge \displaystyle \iint \limits _D\Big \langle u_{xt}(x,t)-\widetilde{u}_{xt}(x,t),u^*(x,t)\Big \rangle dxdt\nonumber \\{} & {} \quad -\displaystyle \iint \limits _D \Big \langle u_{xt}^*(x,t),\;u(x,t)-\widetilde{u}(x,t)\Big \rangle dxdt. \end{aligned}$$
(4)

Let us rewrite double integrals on the right-hand side as follows:

$$\begin{aligned}{} & {} \displaystyle \iint \limits _D\Big \langle u_{xt}(x,t)-\widetilde{u}_{xt}(x,t),u^*(x,t)\Big \rangle dxdt\nonumber \\{} & {} \quad =\displaystyle \int \limits _0^1\Big \langle u_{t}(1,t)-\widetilde{u}_{t}(1,t),u^*(1,t)\Big \rangle dt-\displaystyle \int \limits _0^1\Big \langle u_{t}(0,t)-\widetilde{u}_{t}(0,t),u^*(0,t)\Big \rangle dt\nonumber \\{} & {} \quad -\displaystyle \iint \limits _D\Big \langle u_{t}(x,t)-\widetilde{u}_{t}(x,t),\;u_x^*(x,t)\Big \rangle dxdt \end{aligned}$$
(5)

and

$$\begin{aligned}{} & {} \displaystyle \iint \limits _D\Big \langle u_{xt}^*(x,t),\;u(x,t)-\widetilde{u}(x,t)\Big \rangle dxdt\nonumber \\{} & {} \quad =\displaystyle \int \limits _0^1\Big \langle u_{x}^*(x,1),\;u(x,1)-\widetilde{u}(x,1)\Big \rangle dx-\displaystyle \int \limits _0^1\Big \langle u_{x}^*(x,0),\;u(x,0)-\widetilde{u}(x,0)\Big \rangle dx\nonumber \\{} & {} \quad -\displaystyle \iint \limits _D\Big \langle u_x^*(x,t),\;u_{t}(x,t)-\widetilde{u}_{t}(x,t)\Big \rangle dxdt \end{aligned}$$
(6)

Then, taking into account formulas (5) and (6) in inequality (4), we find that

$$\begin{aligned}{} & {} \displaystyle \iint \limits _D \Big [f(u(x,t),x,t)-f(\tilde{u}(x,t),x,t)\Big ]dxdt\nonumber \\{} & {} \quad \ge \displaystyle \int \limits _0^1\Big \langle u_{t}(1,t)-\widetilde{u}_{t}(1,t),u^*(1,t)\Big \rangle dt-\displaystyle \int \limits _0^1\Big \langle u_{t}(0,t)-\widetilde{u}_{t}(0,t),u^*(0,t)\Big \rangle dt\nonumber \\{} & {} \quad -\displaystyle \int \limits _0^1\Big \langle u_{x}^*(x,1),\;u(x,1)-\widetilde{u}(x,1)\Big \rangle dx+\displaystyle \int \limits _0^1\Big \langle u_{x}^*(x,0),\;u(x,0)-\widetilde{u}(x,0)\Big \rangle dx. \end{aligned}$$
(7)

Moreover, by condition (iii) of the theorem, we have \(\Big \langle u_{t}(0,t)-\widetilde{u}_{t}(0,t),u^*(0,t)\Big \rangle \le 0\) and\(\Big \langle u_{x}(x,0)-\widetilde{u}_{x}(x,0),u^*(x,0)\Big \rangle \le 0\). Then we write in more convenient form (7) as follows

$$\begin{aligned}{} & {} \displaystyle \iint \limits _D \Big [f(u(x,t),x,t)-f(\tilde{u}(x,t),x,t)\Big ]dxdt\nonumber \\{} & {} \quad \ge \displaystyle \int \limits _0^1\Big \langle u_{t}(1,t)-\widetilde{u}_{t}(1,t),u^*(1,t)\Big \rangle dt-\displaystyle \int \limits _0^1\Big \langle u_{x}^*(x,1),\;u(x,1)-\widetilde{u}(x,1)\Big \rangle dx\nonumber \\{} & {} \quad +\displaystyle \int \limits _0^1\Big \langle u_{x}^*(x,0),\;u(x,0)-\widetilde{u}(x,0)\Big \rangle dx+\displaystyle \int \limits _0^1\Big \langle u_{x}(x,0)-\widetilde{u}_{x}(x,0),u^*(x,0)\Big \rangle dx. \end{aligned}$$
(8)

Furthermore, by condition (ii) of the theorem, we have \(\Big \langle -u_x^*(x,1)\;,\;u(x,1)-\widetilde{u}(x,1)\Big \rangle \ge 0\) and \(\Big \langle -u_t^*(1,t)\;,\;u(1,t)-\widetilde{u}(1,t)\Big \rangle \ge 0\). Then it is clear that

$$\begin{aligned} \displaystyle \int \limits _0^1 \Big \langle -u_x^*(x,1),\;u(x,1)-\widetilde{u}(x,1)\Big \rangle dx+\displaystyle \int \limits _0^1\Big \langle -u_t^*(1,t),\;u(1,t)-\widetilde{u}(1,t)\Big \rangle \ge 0 \end{aligned}$$
(9)

By summing up the inequalities in (8) and (9), we deduce that

$$\begin{aligned}{} & {} \displaystyle \iint \limits _D \Big [f(u(x,t),x,t)-f(\tilde{u}(x,t),x,t)\Big ]dxdt+\displaystyle \int \limits _0^1 \Big \langle -u_x^*(x,1),\;u(x,1)-\widetilde{u}(x,1)\Big \rangle dx\\{} & {} \quad +\displaystyle \int \limits _0^1\Big \langle -u_t^*(1,t),\;u(1,t)-\widetilde{u}(1,t)\Big \rangle \ge \displaystyle \int \limits _0^1\Big \langle u_{t}(1,t)-\widetilde{u}_{t}(1,t),u^*(1,t)\Big \rangle dt\\{} & {} \quad -\displaystyle \int \limits _0^1\Big \langle u_{x}^*(x,1),\;u(x,1)-\widetilde{u}(x,1)\Big \rangle dx\\{} & {} \quad +\displaystyle \int \limits _0^1\Big \langle u_{x}^*(x,0),\;u(x,0)-\widetilde{u}(x,0)\Big \rangle dx+\displaystyle \int \limits _0^1\Big \langle u_{x}(x,0)-\widetilde{u}_{x}(x,0),u^*(x,0)\Big \rangle dx. \end{aligned}$$

or, equivalently

$$\begin{aligned}{} & {} \displaystyle \iint \limits _D \Big [f(u(x,t),x,t)-f(\tilde{u}(x,t),x,t)\Big ]dxdt\nonumber \\{} & {} \quad \ge \displaystyle \int \limits _0^1\Big \langle u_t^*(1,t),\;u(1,t)-\widetilde{u}(1,t)\Big \rangle +\displaystyle \int \limits _0^1\Big \langle u_{t}(1,t)-\widetilde{u}_{t}(1,t),u^*(1,t)\Big \rangle dt\nonumber \\{} & {} \quad +\displaystyle \int \limits _0^1\Big \langle u_{x}^*(x,0),\;u(x,0)-\widetilde{u}(x,0)\Big \rangle dx+\displaystyle \int \limits _0^1\Big \langle u_{x}(x,0)-\widetilde{u}_{x}(x,0),u^*(x,0)\Big \rangle dx. \end{aligned}$$
(10)

Then, we rewrite the inequality (10) in the form

$$\begin{aligned}{} & {} \displaystyle \iint \limits _D \Big [f(u(x,t),x,t)-f(\tilde{u}(x,t),x,t)\Big ]dxdt\\{} & {} \quad \ge \displaystyle \int \limits _0^1d_t\Big \langle u^*(1,t),\;u(1,t)-\widetilde{u}(1,t)\Big \rangle +\displaystyle \int \limits _0^1d_x\Big \langle u^*(x,0),\;u(x,0)-\widetilde{u}(x,0)\Big \rangle \end{aligned}$$

and it is not hard to see that

$$\begin{aligned}{} & {} \displaystyle \iint \limits _D \Big [f(u(x,t),x,t)-f(\tilde{u}(x,t),x,t)\Big ]dxdt\nonumber \\{} & {} \quad \ge \Big \langle u^*(1,1),\;u(1,1)-\widetilde{u}(1,1)\Big \rangle -\Big \langle u^*(1,0),\;u(1,0)-\widetilde{u}(1,0)\Big \rangle \nonumber \\{} & {} \quad +\Big \langle u^*(1,0),\;u(1,0)-\widetilde{u}(1,0)\Big \rangle -\Big \langle u^*(0,0),\;u(0,0)-\widetilde{u}(0,0)\Big \rangle . \end{aligned}$$
(11)

Since \(u^*(0,0)=u^*(1,1)=0\), we have

$$\begin{aligned} \displaystyle \iint \limits _D \Big [f(u(x,t),x,t)-f(\tilde{u}(x,t),x,t)\Big ]dxdt\ge 0. \end{aligned}$$
(12)

Finally, we obtain that \(J[u(\cdot ,\cdot )]\ge J[\widetilde{u}(\cdot ,\cdot )]\), i.e., \(\widetilde{u}(\cdot ,\cdot )\) is optimal. The proof of the theorem is completed. \(\square\)

Let’s consider the following problem

$$\begin{aligned} \text{ minimize }\;&J[u(\cdot ,\cdot )]= \iint \limits _{D}f(u(x,t),x,t)dx dt,\nonumber \\ u_{xt}(x,t)&\in F(u(x,t),x,t), \; (x,t)\in D, \nonumber \\&u(x,t)\in \Psi (x,t), \nonumber \\&u(x,0)=0\;,\;u(0,t)=0\;,\; x\in [0,1],\; t\in [0,1] \end{aligned}$$
(13)

where \(F(\cdot ,x,t):{\mathbb {R}}^{n}\rightrightarrows {\mathbb {R}}^n\) is set-valued mapping for all fixed \((x,t)\in D\) and \(f(\cdot ,x,t):{\mathbb {R}}^{n}\rightarrow {\mathbb {R}}^n\) is continuous function and \(\Psi :D\rightarrow {\mathbb {R}}^{n}\) is a convex-valued mapping. We note that, in contrast to problem (P), in the problem (13), the set-valued mappings \(F_1(\cdot ,t) \equiv {\mathbb {R}}^{n}\), \(F_2(\cdot ,x) \equiv {\mathbb {R}}^{n}\), and the boundary conditions are \(u(x,0)=0\;,\;u(0,t)=0\;,\; x\in [0,1],\; t\in [0,1]\). In what follows, these subtle changes significantly impact the construction of a dual problem.

Corollary 3.2

Let f be continuous and convex function on the first argument u and suppose that set-valued mapping F is convex. Then for the optimality of a feasible solution \(\widetilde{u}(x,t)\), it is sufficient that there exists an absolutely continuous function \(u^*(x,t)\) on D with integrable mixed derivative \(u_{xt}^*(x,t)\) such that conditions \((a)-(c)\) hold almost everywhere:

$$\begin{aligned} (a){} & {} u_{xt}^*(x,t)\in F^*\Big (u^*(x,t),(\tilde{u}(x,t),\tilde{u}_{xt}(x,t)),x,t\Big )-\partial f(\tilde{u}(x,t),x,t),\\ (b){} & {} u_x^*(x,1)\in -K^*_{\Psi (x,1)}(\widetilde{u}(x,1)),\\{} & {} u_t^*(1,t)\in -K^*_{\Psi (1,t)}(\widetilde{u}(1,t)),\\{} & {} u^*(0,0)=u^*(1,1)=0 , \\ (c){} & {} \widetilde{u}_{xt}(x,t)\in F_A\Big (\widetilde{u}(x,t);u^*(x,t),x,t\Big ). \end{aligned}$$

Proof

The proof of the corollary is similiar to the one for Theorem 3.1. \(\square\)

Now let’s consider the problem (13) in polyhedral case. Thus, F is a set-valued polyhedral mapping defined as \(F(u)=\{v:\;Pu-Qv\le d\}\) where P and Q, \(m\times n\) dimensional matrices, d is a m-dimensional column-vector.

Theorem 3.3

Assume that f is a proper polyhedral function and that set-valued mapping F is polyhedral. Then, for the optimality of a feasible solution \(\widetilde{u}(x,t)\), it is sufficient that there exists \(\lambda (x,t)\) with integrable mixed derivative \(\lambda _{xt}(x,t)\) such that conditions \((i)-(iii)\) hold almost everywhere:

$$\begin{aligned} (i){} & {} Q^*\lambda _{xt}(x,t)-P^*\lambda (x,t)\in \partial f(\tilde{u}(x,t),x,t),\;(x,t)\in D,\\ (ii){} & {} Q^*\lambda _x(x,1)\in K^*_{\Psi (x,1)}(\widetilde{u}(x,1)),\\{} & {} Q^*\lambda _t(1,t)\in K^*_{\Psi (1,t)}(\widetilde{u}(1,t)),\\{} & {} Q^*\lambda (0,0)=Q^*\lambda (1,1)=0 ,\\ (iii){} & {} \big \langle P\widetilde{u}(x,t)-Q\widetilde{u}_{xt}(x,t)-d,\;\lambda (x,t)\big \rangle =0,\; \lambda (x,t)\ge 0. \end{aligned}$$

Proof

For all feasible solutions u(x, t), it follows from the definition of subdifferential that

$$\begin{aligned}{} & {} f(u(x,t),x,t)-f(\widetilde{u}(x,t),x,t)\nonumber \\{} & {} \quad \ge \Big \langle Q^*\lambda _{xt}(x,t)-P^*\lambda (x,t),u(x,t)-\widetilde{u}(x,t)\Big \rangle ,\;(x,t)\in D. \end{aligned}$$
(14)

Then taking into account the structure of F in polyhedral case, it is easy to see that

$$\begin{aligned}{} & {} \Big \langle Pu(x,t)-Q u_{xt}(x,t)-d,\;\lambda (x,t)\Big \rangle \\{} & {} \quad \le \Big \langle P\widetilde{u}(x,t)-Q\widetilde{u}_{xt}(x,t)-d,\;\lambda (x,t)\Big \rangle ,\; \lambda (x,t)\ge 0. \end{aligned}$$

Hence the inequality (14) can be written as follows

$$\begin{aligned}{} & {} f(u(x,t),x,t)-f(\widetilde{u}(x,t),x,t)\ge \Big \langle Q^*\lambda _{xt}(x,t),\;u(x,t)-\widetilde{u}(x,t)\Big \rangle \\{} & {} \quad -\Big \langle Q^*\lambda (x,t),u_{xt}(x,t)-\widetilde{u}_{xt}(x,t)\Big \rangle . \end{aligned}$$

Integrating that inequality over the domain D, we have

$$\begin{aligned}{} & {} \displaystyle \iint \limits _D \Big ( f(u(x,t),x,t)-f(\widetilde{u}(x,t),x,t)\Big ) dxdt\nonumber \\{} & {} \quad \ge \displaystyle \iint \limits _D\Big \langle Q^*\lambda _{xt}(x,t),\;u(x,t)-\widetilde{u}(x,t)\Big \rangle dxdt\nonumber \\{} & {} \quad -\displaystyle \iint \limits _D\Big \langle Q^*\lambda (x,t),u_{xt}(x,t)-\widetilde{u}_{xt}(x,t)\Big \rangle dxdt. \end{aligned}$$
(15)

Here by necessary simplification, it is clear that

$$\begin{aligned}{} & {} \displaystyle \iint \limits _D\Big \langle Q^*\lambda _{xt}(x,t),\;u(x,t)-\widetilde{u}(x,t)\Big \rangle dxdt\nonumber \\{} & {} \quad =\displaystyle \iint \limits _D\frac{\partial }{\partial t}\Big \langle Q^*\lambda _{x}(x,t),\;u(x,t)-\widetilde{u}(x,t)\Big \rangle dxdt\nonumber \\{} & {} \quad -\displaystyle \iint \limits _D\Big \langle Q^*\lambda _{x}(x,t),\;u_t(x,t)-\widetilde{u}_t(x,t)\Big \rangle dxdt \end{aligned}$$
(16)

and

$$\begin{aligned}{} & {} \displaystyle \iint \limits _D\Big \langle Q^*\lambda (x,t),\;u_{xt}(x,t)-\widetilde{u}_{xt}(x,t)\Big \rangle dxdt\nonumber \\{} & {} \quad =\displaystyle \iint \limits _D\frac{\partial }{\partial x}\Big \langle Q^*\lambda (x,t),\;u_t(x,t)-\widetilde{u}_t(x,t)\Big \rangle dxdt\nonumber \\{} & {} \quad -\displaystyle \iint \limits _D\Big \langle Q^*\lambda _{x}(x,t),\;u_t(x,t)-\widetilde{u}_t(x,t)\Big \rangle dxdt. \end{aligned}$$
(17)

Then in view of the inequalities (16) and (17) in the formula (15), we derive that

$$\begin{aligned}{} & {} \displaystyle \iint \limits _D \Big ( f(u(x,t),x,t)-f(\widetilde{u}(x,t),x,t)\Big ) dxdt \\{} & {} \quad \ge \displaystyle \iint \limits _D\frac{\partial }{\partial t}\Big \langle Q^*\lambda _{x}(x,t),\;u(x,t)-\widetilde{u}(x,t)\Big \rangle dxdt\\{} & {} \quad -\displaystyle \iint \limits _D\frac{\partial }{\partial x}\Big \langle Q^*\lambda (x,t),\;u_t(x,t)-\widetilde{u}_t(x,t)\Big \rangle dxdt \end{aligned}$$

or, equivalently

$$\begin{aligned}{} & {} \displaystyle \iint \limits _D \Big ( f(u(x,t),x,t)-f(\widetilde{u}(x,t),x,t)\Big ) dxdt \nonumber \\{} & {} \quad \ge \displaystyle \int \limits _0^1\Big \langle Q^*\lambda _{x}(x,1),\;u(x,1)-\widetilde{u}(x,1)\Big \rangle dx-\displaystyle \int \limits _0^1\Big \langle Q^*\lambda _{x}(x,0),\;u(x,0)-\widetilde{u}(x,0)\Big \rangle dx\nonumber \\{} & {} \quad -\displaystyle \int \limits _0^1\Big \langle Q^*\lambda (1,t),\;u_t(1,t)-\widetilde{u}_t(1,t)\Big \rangle dt+\displaystyle \int \limits _0^1\Big \langle Q^*\lambda (0,t),\;u_t(0,t)-\widetilde{u}_t(0,t)\Big \rangle dt. \end{aligned}$$
(18)

By condition (ii) of the Theorem 3.3, we can write the following

$$\begin{aligned}{} & {} \Big \langle Q^*\lambda _x(x,1),\;u(x,1)-\widetilde{u}(x,1)\Big \rangle \ge 0,\\{} & {} \Big \langle Q^*\lambda _t(1,t),\;u(1,t)-\widetilde{u}(1,t)\Big \rangle \ge 0. \end{aligned}$$

Then, the inequality (18) can be converted to the relation as follows

$$\begin{aligned}{} & {} \displaystyle \iint \limits _D \Big ( f(u(x,t),x,t)-f(\widetilde{u}(x,t),x,t)\Big ) dxdt \ge -\displaystyle \int \limits _0^1\Big \langle Q^*\lambda _{x}(x,0),\;u(x,0)-\widetilde{u}(x,0)\Big \rangle dx\\{} & {} \quad +\displaystyle \int \limits _0^1\Big \langle Q^*\lambda (0,t),\;u_t(0,t)-\widetilde{u}_t(0,t)\Big \rangle dt-\displaystyle \int \limits _0^1\frac{\partial }{\partial t}\Big \langle Q^*\lambda (1,t),\;u(1,t)-\widetilde{u}(1,t)\Big \rangle dt. \end{aligned}$$

or more convenient form

$$\begin{aligned}{} & {} \displaystyle \iint \limits _D \Big ( f(u(x,t),x,t)-f(\widetilde{u}(x,t),x,t)\Big ) dxdt \ge -\displaystyle \int \limits _0^1\Big \langle Q^*\lambda _{x}(x,0),\;u(x,0)-\widetilde{u}(x,0)\Big \rangle dx\\{} & {} \quad +\displaystyle \int \limits _0^1\Big \langle Q^*\lambda (0,t),\;u_t(0,t)-\widetilde{u}_t(0,t)\Big \rangle dt-\Big \langle Q^*\lambda (1,1),\;u(1,1)-\widetilde{u}(1,1)\Big \rangle \\{} & {} \quad +\Big \langle Q^*\lambda (1,0),\;u(1,0)-\widetilde{u}(1,0)\Big \rangle . \end{aligned}$$

Since \(Q^*\lambda (0,0)=Q^*\lambda (1,1)=0\) and for all feasible solutions \(u(x,0)=0,\;u(0,t)=0\), we conclude that

$$\begin{aligned} \displaystyle \iint \limits _D \Big ( f(u(x,t),x,t)-f(\widetilde{u}(x,t),x,t)\Big ) dxdt \ge 0. \end{aligned}$$
(19)

Thus, we have proved the desired result. \(\square\)

Now we handle the following hyperbolic Darboux type differential inclusion without the state conditions in the problem (13)

$$\begin{aligned} \text{ minimize }\;&J[u(\cdot ,\cdot )]= \iint \limits _{D}f(u(x,t),x,t)dx dt,\nonumber \\ u_{xt}(x,t)&\in F(u(x,t),x,t), \; (x,t)\in D, \nonumber \\&u(x,0)=0\;,\;u(0,t)=0\;,\; x\in [0,1],\; t\in [0,1]. \end{aligned}$$
(20)

It is clear that, for the optimality of a feasible solution \(\widetilde{u}(x,t)\) in the problem (20), it is sufficient that there exists an absolutely continuous function \(u^*(x,t)\) on D with integrable mixed derivative \(u_{xt}^*(x,t)\) such that conditions \((a')-(c')\) hold almost everywhere:

$$\begin{aligned} (a'){} & {} u_{xt}^*(x,t)\in F^*\Big (u^*(x,t),(\tilde{u}(x,t),\tilde{u}_{xt}(x,t)),x,t\Big )-\partial f(\tilde{u}(x,t),x,t),\\ (b'){} & {} u^*(0,0)=u^*(1,1)=0 , \\ (c'){} & {} \widetilde{u}_{xt}(x,t)\in F_A\Big (\widetilde{u}(x,t);u^*(x,t),x,t\Big ). \end{aligned}$$

Now, consider the problem (20) with polyhedral Darboux differential inclusion where \(F(u)=\{v:\;Pu-Qv\le d\}\) where P and Q, \(m\times n\) dimensional matrices, d is a m-dimensional column-vector. We can formulate the optimality conditions for the problem (20) in the polyhedral case.

Corollary 3.4

Suppose that f is a proper polyhedral function and that set-valued mapping F is polyhedral. Then, for the optimality of a feasible solution \(\widetilde{u}(x,t)\), it is sufficient that there exists \(\lambda (x,t)\) with integrable mixed derivative \(\lambda _{xt}(x,t)\) such that conditions \((i)-(iii)\) hold almost everywhere:

$$\begin{aligned} (i){} & {} Q^*\lambda _{xt}(x,t)-P^*\lambda (x,t)\in \partial f(\tilde{u}(x,t),x,t),\;(x,t)\in D,\\ (ii){} & {} Q^*\lambda (0,0)=Q^*\lambda (1,1)=0 , \\ (iii){} & {} \big \langle P\widetilde{u}(x,t)-Q\widetilde{u}_{xt}(x,t)-d,\;\lambda (x,t)\big \rangle =0,\; \lambda (x,t)\ge 0. \end{aligned}$$

Proof

The proof is a simple modification of the proof of Theorem 3.3 and so is omitted. \(\square\)

To illustrate the main constructions of our approach, we give a simple example of the problem (20).

Example 3.5

Let us consider \(Q=[1]\), \(P=[-1]\), \(d=[0]\) and \(f(u)=u+1\) continuously differentiable function according to the problem (20) in polyhedral case. Then, by Corollary 3.4 for the optimality \(\widetilde{u}(x,t)\) to given example, it is sufficient that there exists \(\lambda (x,t)\) such that

$$\begin{aligned}{} & {} \lambda _{xt}(x,t)+\lambda (x,t)\in \partial f(\tilde{u}(x,t),x,t),\;(x,t)\in D,\nonumber \\{} & {} \lambda (0,0)=\lambda (1,1)=0 ,\nonumber \\{} & {} \Big (\widetilde{u}(x,t)+\widetilde{u}_{xt}(x,t)\Big )\lambda (x,t)=0,\; \lambda (x,t)\ge 0. \end{aligned}$$
(21)

Here it can be easily seen that the subdifferential of f is the gradient vector, that is \(\partial _uf(\widetilde{u}(x,t),x,t)=\{1\}\) and in view of relations of (21) we have a boundary value problem for partial differential equation

$$\begin{aligned}{} & {} \lambda _{xt}(x,t)+\lambda (x,t)=1,\;(x,t)\in D,\nonumber \\{} & {} \lambda (0,0)=\lambda (1,1)=0. \end{aligned}$$
(22)

The general solution of the boundary value problem (22) is

$$\begin{aligned} \lambda (x,t)=G(x)H(t)+1 \end{aligned}$$

where \(\dfrac{d}{dx} G(x)=CG(x)\) and \(\dfrac{d}{d t} H(t)=-\dfrac{H(t)}{C}\) and C is a non-zero arbitrary constant. By computing these equations we have \(G(x)=C_1e^{Cx}\) and \(H(t)=C_2e^{\frac{-t}{C}}\) where \(C_1,C_2\) are arbitrary constant. Therefore, we have

$$\begin{aligned} \lambda (x,t)=C_1C_2e^{Cx-\frac{t}{C}}+1. \end{aligned}$$

By using the boundary conditions, we find that \(\lambda (x,t)=1-e^{x-t}\). Now, since \(\lambda (x,t)\ne 0\), \((x,t)\in D,x\ne t\) from the third formula of (21), we deduce that \(\widetilde{u}(x,t)+\widetilde{u}_{xt}(x,t)=0\). By similar way, using boundary conditions \(\widetilde{u}(x,0)=\widetilde{u}(0,t)=0\), we have the solution of the stated problem in (20) with polyhedral case \(\widetilde{u}(x,t)=0\). Then its value is

$$\begin{aligned} \iint \limits _{D}f(\widetilde{u}(x,t),x,t)dx dt=\iint \limits _{D}\Big (\widetilde{u}(x,t)+1 \big )dx dt=1. \end{aligned}$$

4 On duality in the Darboux differential problem with state constraints

Our approach to establishing duality results for differential problems (13) and (20) with state constraints is based on the passage to the formal limit in the dual problem for the discrete approximate problem. To formulate the dual problems for the differential problems, we use the duality theorems of operations of addition and infimal convolution of convex functions. However, we skip it and just present dual problems for continuous problems (13) and (20) to prevent laborious and tiresome calculations. Moreover, it is demonstrated how the duality relations relate primal and dual problems to one another. An extremal relation for the primal and dual problems is a sufficient condition for an extremum according to the established duality theorems. It means that if some pair of feasible solutions \(u(\cdot )\) and \(u^*(\cdot )\) satisfy duality relations, then \(u(\cdot )\) and \(u^*(\cdot )\) are solutions of the primal and dual problem, respectively.

The maximization problem will be the dual problem to the continuous convex problem (13):

$$\begin{aligned} \sup \limits _{\begin{array}{c} u^*,\varphi ^*,\phi ^* \\ u^*(0,0)=u^*(1,1) \end{array}} J_*\Big ( u^*(x,t),\varphi ^*(x,t),\phi ^*(x,t)\Big ) \end{aligned}$$
(23)

where

$$\begin{aligned}{} & {} J_*\Big ( u^*(x,t),\varphi ^*(x,t),\phi ^*(x,t)\Big )=\displaystyle \iint \limits _D\Big [M_F\Big (u_{xt}^*(x,t) -\varphi ^*(x,t)+\phi ^*(x,t),\;u^*(x,t)\Big )\\{} & {} \quad -W_{\Psi (x,t)}(-\varphi ^*(x,t))-f^*(\phi ^*(x,t),x,t)\Big ]dxdt\\{} & {} \quad -\displaystyle \int _0^1W_{\Psi (1,t)}(u^*_t(1,t))dt-\displaystyle \int _0^1W_{\Psi (x,1)}(u^*_x(x,1))dx. \end{aligned}$$

Here, the problem (23) is called the dual problem to the primal convex problem (13). It is assumed that \(\varphi ^*,\phi ^*\) are absolutely continuous function on D and \(u^*\) is an absolutely continuous function on D having an integrable mixed partial derivative \(u^*_{xt}(\cdot ,\cdot )\).

Theorem 4.1

The inequality \(J[u(x,t)]\ge J_*\Big ( u^*(x,t),\varphi ^*(x,t),\phi ^*(x,t)\Big )\) holds for all feasible solutions u(x, t) and \(\{u^*(x,t),\varphi ^*(x,t),\phi ^*(x,t)\}\) of the primal problem and dual problem.

Proof

It is clear from the definition of function \(M_F\) that

$$\begin{aligned}{} & {} M_F\Big (u_{xt}^*(x,t)-\varphi ^*(x,t)+\phi ^*(x,t),\;u^*(x,t)\Big )\nonumber \\{} & {} \quad \le \Big \langle u(x,t),\;u_{xt}^*(x,t)-\varphi ^*(x,t)+\phi ^*(x,t)\Big \rangle \nonumber \\{} & {} \quad -\Big \langle u_{xt}(x,t),\;u^*(x,t)\Big \rangle . \end{aligned}$$
(24)

Moreover, by definitions of support function and conjugate function, we deduce that

$$\begin{aligned}{} & {} W_{\Psi (x,t)}(-\varphi ^*(x,t))\ge \Big \langle u(x,t),\;-\varphi ^*(x,t)\Big \rangle ,\nonumber \\{} & {} W_{\Psi (1,t)}(u_t^*(1,t))\ge \Big \langle u_t^*(1,t),\;u(1,t)\Big \rangle ,\nonumber \\{} & {} W_{\Psi (x,1)}(u_x^*(x,1))\ge \Big \langle u_x^*(x,1),\;u(x,1)\Big \rangle , \end{aligned}$$
(25)

and

$$\begin{aligned} f^*(\phi ^*(x,t),x,t)\ge \Big \langle \phi ^*(x,t),\; u(x,t)\Big \rangle -f(u(x,t),x,t). \end{aligned}$$
(26)

Then, from the relation (24)-(26) we derive that

$$\begin{aligned}{} & {} M_F\Big (u_{xt}^*(x,t)-\varphi ^*(x,t)+\phi ^*(x,t),\;u^*(x,t)\Big )-W_{\Psi (x,t)}(-\varphi ^*(x,t)) -f^*(\phi ^*(x,t),x,t)\nonumber \\{} & {} \quad \le \Big \langle u(x,t),\;u_{xt}^*(x,t)-\varphi ^*(x,t)+\phi ^*(x,t)\Big \rangle -\Big \langle u_{xt}(x,t),\;u^*(x,t)\Big \rangle \nonumber \\{} & {} \quad +\Big \langle u(x,t),\;\varphi ^*(x,t)\Big \rangle - \Big \langle \phi ^*(x,t), \; u(x,t)\Big \rangle +f(u(x,t),x,t). \end{aligned}$$
(27)

and

$$\begin{aligned} -W_{\Psi (1,t)}(u_t^*(1,t))-W_{\Psi (x,1)}(u_x^*(x,1))\le -\Big \langle u_t^*(1,t), \;u(1,t)\Big \rangle -\Big \langle u_x^*(x,1),\;u(x,1)\Big \rangle \end{aligned}$$
(28)

We rewrite the relation (27) in the form

$$\begin{aligned}{} & {} M_F\Big (u_{xt}^*(x,t)-\varphi ^*(x,t)+\phi ^*(x,t),\;u^*(x,t)\Big )-W_{\Psi (x,t)}(-\varphi ^*(x,t)) -f^*(\phi ^*(x,t),x,t)\nonumber \\{} & {} \quad \le \Big \langle u(x,t),\;u_{xt}^*(x,t)\Big \rangle -\Big \langle u_{xt}(x,t),\;u^*(x,t) \Big \rangle +f(u(x,t),x,t) \end{aligned}$$
(29)

Integrating the inequality (29) over the domain D and the inequality (28) over the interval [0, 1], and adding them, we have

$$\begin{aligned}{} & {} J_*\Big ( u^*(x,t),\varphi ^*(x,t),\phi ^*(x,t)\Big )\le \displaystyle \iint \limits _D \Big [\Big \langle u(x,t),\;u_{xt}^*(x,t)\Big \rangle \nonumber \\{} & {} \quad -\Big \langle u_{xt}(x,t),\;u^*(x,t) \Big \rangle \Big ]dxdt+J[u(x,t)]\nonumber \\{} & {} \quad -\displaystyle \int \limits _0^1\Big \langle u_t^*(1,t),\;u(1,t)\Big \rangle dt-\displaystyle \int \limits _0^1\Big \langle u_x^*(x,1),\;u(x,1)\Big \rangle dx. \end{aligned}$$
(30)

On the other hand it is easy to see that

$$\begin{aligned}{} & {} \Big \langle u(x,t),\;u_{xt}^*(x,t)\Big \rangle =\dfrac{d}{dt}\Big [\Big \langle u(x,t),\;u_{x}^*(x,t)\Big \rangle \Big ]-\Big \langle u_t(x,t),\;u_{x}^*(x,t)\Big \rangle \nonumber \\{} & {} \Big \langle u_{xt}(x,t),\;u^*(x,t)\Big \rangle =\dfrac{d}{dx}\Big [\Big \langle u_t(x,t),\;u^*(x,t)\Big \rangle \Big ]-\Big \langle u_t(x,t),\;u_{x}^*(x,t)\Big \rangle , \end{aligned}$$

and so,

$$\begin{aligned}{} & {} \displaystyle \iint \limits _D\Big [\Big \langle u(x,t),\;u_{xt}^*(x,t)\Big \rangle -\Big \langle u_{xt}(x,t),\;u^*(x,t)\Big \rangle \Big ]dxdt\nonumber \\{} & {} \quad =\displaystyle \int \limits _0^1\Big \langle u(x,1),\;u_{x}^*(x,1)\Big \rangle dx-\displaystyle \int \limits _0^1\Big \langle u(x,0),\;u_{x}^*(x,0)\Big \rangle dx\nonumber \\{} & {} \quad -\displaystyle \int \limits _0^1\Big \langle u_t(1,t),\;u^*(1,t)\Big \rangle dt+ \displaystyle \int \limits _0^1\Big \langle u_t(0,t),\;u^*(0,t)\Big \rangle dt. \end{aligned}$$
(31)

By substituting equality (31) in the inequality (30), we have

$$\begin{aligned}{} & {} J_*\Big ( u^*(x,t),\varphi ^*(x,t),\phi ^*(x,t)\Big )\le J[u(x,t)]+\displaystyle \int \limits _0^1\Big \langle u(x,1),\;u_{x}^*(x,1)\Big \rangle dx\\{} & {} \quad -\displaystyle \int \limits _0^1\Big \langle u(x,0),\;u_{x}^*(x,0)\Big \rangle dx\\{} & {} \quad -\displaystyle \int \limits _0^1\!\!\Big \langle u_t(1,t),\;u^*(1,t)\Big \rangle dt+ \displaystyle \int \limits _0^1\!\!\Big \langle u_t(0,t),\;u^*(0,t)\Big \rangle dt\\{} & {} \quad -\displaystyle \int \limits _0^1\!\!\Big \langle u_t^*(1,t),\;u(1,t)\Big \rangle dt-\displaystyle \int \limits _0^1\!\!\Big \langle u_x^*(x,1),\;u(x,1)\Big \rangle dx \end{aligned}$$

or in more convenient form

$$\begin{aligned}{} & {} J_*\Big ( u^*(x,t),\varphi ^*(x,t),\phi ^*(x,t)\Big )\le J[u(x,t)]-\displaystyle \int \limits _0^1\Big \langle u(x,0),\;u_{x}^*(x,0)\Big \rangle dx\nonumber \\{} & {} \quad -\displaystyle \int \limits _0^1\dfrac{d}{dt}\Big [\Big \langle u(1,t),\;u^*(1,t)\Big \rangle \Big ] dt+ \displaystyle \int \limits _0^1\Big \langle u_t(0,t),\;u^*(0,t)\Big \rangle dt . \end{aligned}$$
(32)

Then we rewrite the inequality (32) as follows

$$\begin{aligned}{} & {} J_*\Big ( u^*(x,t),\varphi ^*(x,t),\phi ^*(x,t)\Big )\le J[u(x,t)] \\{} & {} \quad -\displaystyle \int \limits _0^1\Big \langle u(x,0),\;u_{x}^*(x,0)\Big \rangle dx- \Big \langle u(1,1),\;u^*(1,1)\Big \rangle \\{} & {} \quad + \Big \langle u(1,0),\;u^*(1,0)\Big \rangle +\displaystyle \int \limits _0^1\dfrac{d}{dt}\Big [\Big \langle u(0,t),\;u^*(0,t)\Big \rangle \Big ] dt- \displaystyle \int \limits _0^1\Big \langle u(0,t),\;u^*_t(0,t)\Big \rangle dt. \end{aligned}$$

From the inequality above we get the following

$$\begin{aligned}{} & {} J_*\Big ( u^*(x,t),\varphi ^*(x,t),\phi ^*(x,t)\Big )\\{} & {} \quad \le J[u(x,t)]-\displaystyle \int \limits _0^1\Big \langle u(x,0),\;u_{x}^*(x,0)\Big \rangle dx\\{} & {} \quad - \displaystyle \int \limits _0^1\Big \langle u(0,t),\;u^*_t(0,t)\Big \rangle dt\\{} & {} \quad - \Big \langle u(1,1),\;u^*(1,1)\Big \rangle + \Big \langle u(1,0),\;u^*(1,0)\Big \rangle \\{} & {} \quad +\Big \langle u(0,1),\;u^*(0,1)\Big \rangle -\Big \langle u(0,0),\;u^*(0,0)\Big \rangle . \end{aligned}$$

Since the solutions u(x, t) and \(\{u^*(x,t),\varphi ^*(x,t),\phi ^*(x,t)\}\) are feasible solutions of the primal problem and dual problem, respectively, and \(u(x,0)=0\;,\;u(0,t)=0\), \(u^*(0,0)=u^*(1,1)=0\), we obtain that

$$\begin{aligned}{} & {} J_*\Big ( u^*(x,t),\varphi ^*(x,t),\phi ^*(x,t)\Big )\le J[u(x,t)]. \end{aligned}$$
(33)

This concludes the proof. \(\square\)

Theorem 4.2

Suppose that the function \(\widetilde{u}(x,t)\) satisfy the conditions \((a)-(c)\) of the Corollary 3.2, and \(\{u^*(x,t),\varphi ^*(x,t),\phi ^*(x,t)\}\) hold the relation \(W_{\Psi (x,t)}(\varphi ^*(x,t))= \Big \langle \widetilde{u}(x,t)\;,\;\varphi ^*(x,t)\Big \rangle\) and \(\phi ^*(x,t)\in \partial f(\widetilde{u}(x,t),x,t)\). Then \(\widetilde{u}(x,t)\) and \(\{u^*(x,t),\varphi ^*(x,t),\phi ^*(x,t)\}\) are optimal solutions of the primal and dual problems, respectively and their values are equal.

Proof

Suppose that \(\widetilde{u}(x,t)\) is a solution of the primal problem. Now let’s prove that \(\{u^*(x,t),\varphi ^*(x,t),\phi ^*(x,t)\}\) is an optimal solution. By definition of LAM, the condition (a) of the Corollary 3.2, we have

$$\begin{aligned} 0\le \Big \langle u_{xt}^*(x,t)-\varphi ^*(x,t)+\phi ^*(x,t),\; u-\widetilde{u}(x,t)\Big \rangle -\Big \langle u^*(x,t),\;v-\widetilde{u}_{xt}(x,t)\Big \rangle , \end{aligned}$$
(34)

\(v\in F(u)\) and \(W_{\Psi (x,t)}(\varphi ^*(x,t))= \Big \langle \widetilde{u}(x,t)\;,\;\varphi ^*(x,t)\Big \rangle\). This means that

$$\begin{aligned} \Big (u_{xt}^*(x,t)-\varphi ^*(x,t)+\phi ^*(x,t),\;-u^*(x,t)\Big )\in dom M_F \end{aligned}$$
(35)

where \(dom F=\{(u^*,-v^*):\;M_F(u^*,v^*)>-\infty \}\). Further, since \(\partial f(u,x,t)\subset dom f^*(\cdot ,x,t)\), it is clear that \(\phi ^*(x,t)\in dom f^*(\cdot ,x,t)\). Thus the triplet \(\{u^*(x,t),\varphi ^*(x,t),\phi ^*(x,t)\}\) is a feasible solution. It remains to show that it is optimal. On the other hand, we get

$$\begin{aligned}{} & {} M_F\Big (u_{xt}^*(x,t)-\varphi ^*(x,t)+\phi ^*(x,t),\;u^*(x,t)\Big )\nonumber \\{} & {} \quad =\Big \langle u_{xt}^*(x,t)-\varphi ^*(x,t)+\phi ^*(x,t),\;\widetilde{u}(x,t)\Big \rangle \nonumber \\{} & {} \quad -H_F\Big (\widetilde{u}(x,t),\;u^*(x,t)\Big ). \end{aligned}$$
(36)

By conditions (b) and (c) of the Corollary 3.2, we can write

$$\begin{aligned}{} & {} W_{\Psi (1,t)}(u_t^*(1,t))= \Big \langle u_t^*(1,t),\;\widetilde{u}(1,t)\Big \rangle \; t\in [0,1]\nonumber \\{} & {} W_{\Psi (x,1)}(u_x^*(x,1))= \Big \langle u_x^*(x,1),\;\widetilde{u}(x,1)\Big \rangle \; x\in [0,1]\nonumber \\{} & {} H_F\Big (\widetilde{u}(x,t),\;u^*(x,t)\Big )=\Big \langle \widetilde{u}_{xt}(x,t),\;u^*(x,t)\Big \rangle \end{aligned}$$
(37)

Since \(\phi ^*(x,t)\in \partial f(\widetilde{u}(x,t),x,t)\), we have

$$\begin{aligned} f^*(\phi ^*(x,t),x,t)= \Big \langle \phi ^*(x,t),\; \widetilde{u}(x,t)\Big \rangle -f(\widetilde{u}(x,t),x,t). \end{aligned}$$
(38)

Then in view of equations (36), (37) and (38), having instead of the inequalities in equations (24), (25) and (26), and following the proof of Theorem 4.1, we derive that

$$\begin{aligned} J_*\Big ( u^*(x,t),\varphi ^*(x,t),\phi ^*(x,t)\Big )=J[\widetilde{u}(x,t)] \end{aligned}$$
(39)

and so \(\{u^*(x,t),\varphi ^*(x,t),\phi ^*(x,t)\}\) is an optimal solution. The proof is complete. \(\square\)

Now, we illustrate the applicability of Theorems 4.1 and 4.2 for the problem (20). Based on duality theorems for the problem (13), we can obtain the following duality result.

Corollary 4.3

The dual problem to the continuous convex problem (20) is as follows:

$$\begin{aligned} \sup \limits _{\begin{array}{c} u^*, \phi ^* \\ u^*(0,0)=u^*(1,1) \end{array}} J_*\Big ( u^*(x,t),\phi ^*(x,t)\Big ) \end{aligned}$$
(40)

where

$$\begin{aligned}{} & {} J_*\Big ( u^*(x,t), \phi ^*(x,t)\Big )=\displaystyle \iint \limits _D\Big [M_F\Big (u_{xt}^*(x,t) +\phi ^*(x,t),\;u^*(x,t)\Big )-f^*(\phi ^*(x,t),x,t)\Big ]dxdt\nonumber \\{} & {} \quad -\displaystyle \int _0^1\Big \langle u^*_t(1,t),\;u(1,t)\Big \rangle dt -\displaystyle \int _0^1\Big \langle u^*_x(x,1),u(x,1)\Big \rangle dx. \end{aligned}$$
(41)

Corollary 4.4

Dual problem to the primal continuous problem (20) in polyhedral case is as follows:

$$\begin{aligned}{} & {} \sup _{\lambda (\cdot ,\cdot )}\Big \{-\displaystyle \iint \limits _D\Big \langle d,\lambda (x,t)\Big \rangle +f^*\Big ( Q^*\lambda _{xt}(x,t)-P^*\lambda (x,t),\;x,t\Big )dxdt\nonumber \\{} & {} \quad +\displaystyle \int _0^1\Big \langle Q^*\lambda _t(1,t),\;u(1,t)\Big \rangle dt+\displaystyle \int _0^1\Big \langle Q^*\lambda _x(x,1),u(x,1)\Big \rangle dx\Big \}. \end{aligned}$$

Proof

Indeed, it can be easily seen that, denoting \(w=(x,y)\in {\mathbb {R}}^{2n}\), \(w^*=(\xi ^*,-\eta ^*)\in {\mathbb {R}}^{2n}\) we have a linear programming problem

$$\begin{aligned} \inf \{\langle w,w^*\rangle :\; Cw\le d\} \end{aligned}$$
(42)

where \(C=[P:-Q]\) is \(m\times 2n\) block matrix. Then for a solution \(\widetilde{w}=(\widetilde{x},\widetilde{y})\) of the problem (42) there exists m-dimensional vector \(\lambda \ge 0\) such that \(w^*=-C^*\lambda\), \(\langle A\widetilde{x}-B\widetilde{y}-d,\lambda \rangle =0\). And vice versa, if these conditions are satisfied, then \(\widetilde{w}=(\widetilde{x},\widetilde{y})\) is a solution of the problem (42). Therefore \(w^*=-C^*\lambda\) implies that \(\xi ^*=-P^*\lambda\), \(\eta ^*=-Q^*\lambda\), \(\lambda \ge 0\). Thus we find that

$$\begin{aligned} M_F(\xi ^*,\eta ^*)= & {} \langle \widetilde{x},-P^*\lambda \rangle -\langle \widetilde{y},-Q^*\lambda \rangle =-\langle P\widetilde{x},\lambda \rangle +\langle Q\widetilde{y},\lambda \rangle \nonumber \\{} & {} \quad =-\langle P\widetilde{x}-Q\widetilde{y},\lambda \rangle =-\langle d,\lambda \rangle . \end{aligned}$$
(43)

Therefore by using the relation \(u_{xt}^*(x,t)+\phi ^*(x,t)=\xi ^*(x,t)=-P^*\lambda (x,t)\) and \(u^*(x,t)=\eta ^*(x,t)=-Q^*\lambda (x,t)\), we derive the following dual problem, to the primal continuous polyhedral problem (20).

$$\begin{aligned}{} & {} \sup _{\lambda (\cdot ,\cdot )}\Big \{-\displaystyle \iint \limits _D\Big \langle d,\lambda (x,t)\Big \rangle +f^*\Big ( Q^*\lambda _{xt}(x,t)-P^*\lambda (x,t),\;x,t\Big )dxdt\nonumber \\{} & {} \quad +\displaystyle \int _0^1\Big \langle Q^*\lambda _t(1,t),\;u(1,t)\Big \rangle dt+\displaystyle \int _0^1\Big \langle Q^*\lambda _x(x,1),u(x,1)\Big \rangle dx\Big \}. \end{aligned}$$
(44)

\(\square\)

Corollary 4.5

Suppose that \(\widetilde{u}(x,t)\) is an optimal solution of problem (20) with polyhedral differential inclusions. Then, in order for \(\lambda (x,t)\) to be an optimal solution to the dual problem (44), it is necessary and sufficient that the conditions of Corollary 3.4 are satisfied. Moreover, the optimal values in the primal problem (20) with polyhedral differential inclusions and the dual problem (44) are equal.

Example 4.6

Let us now construct the dual problem to the polyhedral differential problem in Example 3.5 and calculate its value. Taking into account the dual problem in the formula (44), we write the dual problem of the problem in Example 3.5 as follows:

$$\begin{aligned}{} & {} \sup _{\lambda (\cdot ,\cdot )}\Big \{-\displaystyle \iint \limits _D f^*\Big ( \lambda _{xt}(x,t)+\lambda (x,t),\;x,t\Big )dxdt\nonumber \\{} & {} \quad +\displaystyle \int _0^1\Big \langle \lambda _t(1,t),\;u(1,t)\Big \rangle dt+\displaystyle \int _0^1\Big \langle \lambda _x(x,1),u(x,1)\Big \rangle dx\Big \}. \end{aligned}$$
(45)

By definition of the conjugate function, we find that

$$\begin{aligned}{} & {} f^*(\beta ^*)=\sup \limits _{\beta } \Big \{\langle \beta ,\beta ^*\rangle -f(\beta )\Big \}=\sup \limits _{\beta } \Big \{ \beta \beta ^*-\beta -1\Big \}\nonumber \\{} & {} \quad =\sup \limits _{\beta } \Big \{ \beta (\beta ^*-1)-1\Big \}=\left\{ \begin{array}{cc} -1,&{} \beta ^*=1,\\ +\infty ,&{}\;\text{ otherwise. }\\ \end{array} \right. \end{aligned}$$
(46)

Therefore, we obtain that \(f^*\Big ( \lambda _{xt}(x,t)+\lambda (x,t)\;,\;x,t\Big )=f^*(1)=-1\). Moreover, since \(\widetilde{u}(x,t)=0\) in Example 3.5, we derive that dual problem as follows

$$\begin{aligned} \sup \Big \{-\displaystyle \iint \limits _D -1dxdt\Big \}=1. \end{aligned}$$

Consequently, we find the value of the dual problem in Example 3.5 is 1. Thus accordingly to Corollary 4.5, we have showed that under the conditions of Corollary 3.4, the optimal values of primal (20) and dual problem (45) coincide. We prove that if \(\eta =\iint \limits _{D}f(\widetilde{u}(x,t),x,t)dx dt=1\) and \(\eta ^*=1\) are the values of polyhedral differential problem in Example 3.5 and its dual problem, respectively, then \(\eta =\eta ^*=1\) for an optimal pair primal and dual problems.