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Frequency theorem and inertial manifolds for neutral delay equations

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Abstract

We study the infinite-horizon quadratic regulator problem for linear control systems in Hilbert spaces, where the cost functional is in some sense unbounded. Our motivation comes from delay equations with the feedback part containing discrete delays or, in other words, measurements given by \(\delta \)-functionals, which are unbounded in \(L_{2}\). Working in an abstract context in which such (and many others, including parabolic boundary control problems) equations can be treated, we obtain a version of the Frequency Theorem. It guarantees the existence of a unique optimal process and shows that the optimal cost is given by a quadratic Lyapunov-like functional. In our adjacent works it is shown that such functionals can be used to construct inertial manifolds and allow to treat and extend many works in the field in a unified manner. Here we concentrate on applications to delay equations and especially mention the works of R.A. Smith on developments of convergence theorems and the Poincaré-Bendixson theory; and also the works of Yu.A. Ryabov, R.D. Driver and C. Chicone on inertial manifolds for equations with small delays and their recent generalization for equations of neutral type given by S. Chen and J. Shen.

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Notes

  1. We use the convention that a sesquilinear form is linear in the first and conjugate-linear in the second argument.

  2. Note that in this case the solution \(v(\cdot )\) is classical and, in particular, we have \(v(\cdot ) \in C([0,T];\mathcal {D}(A))\) (see Theorem 6.5, Chapter I in [33]).

  3. A proper argument for this should use the (non-quadratic!) functionals

    figure a

    which can be also defined by continuity on \(\mathcal {Z}^{T}_{0}\).

  4. Note that Theorem 3 from [9] requires a clarification of its statement which is concerned with replacing formula (3.6) therein to an analog of (1.24) from the present work.

  5. Note that R.A. Smith used quadratic functionals and a “pocket” version of the Frequency Theorem in his works for ODEs [46], but he probably abandoned this approach for infinite-dimensional systems due to inability of obtaining optimal conditions for their existence [47]. Returning to this approach is our main motivation for this and adjacent works.

  6. Recall that the multiplication by a scalar in \(\mathbb {E}^{*}_{0}\) is given by the multiplication of functional’s values by scalar’s complex-conjugate.

  7. Here and below, the complexification of a real vector space \(\mathbb {E}_{0}\) is denoted by \(\mathbb {E}_{0}^\mathbb {C}\). For convenience, we use the same notations for the corresponding complexifcations of the operators.

  8. In fact, the Hilbert space structure is required only when we are studying the Fourier transform.

  9. Recall that \(\langle v, f \rangle := f(v)\) for any \(v \in \mathbb {E}\) and \(f \in \mathbb {E}^{*}\).

  10. This name comes from the Ancient Greek word \(\alpha \gamma \alpha \lambda \mu \alpha \) (agalma) that means an offering to a deity that, by its worth or artistic value, gives him/her special significance. So, “agalmanated” semantically can be understood as “glorified” or “adorned with glory”.

  11. In fact, the consideration of the operator S from (4.18) with \(\eta (\cdot ) = 0\) in \(L_{2}(-\tau ,T;\mathbb {R}^{n})\) and applying similar arguments to investigate its fixed point along with the fact that \(S^{2}\) is a uniform (in initial conditions) contraction show that A generates a \(C_{0}\)-semigroup.

  12. Here, for convenience, we put \((\mathcal {I}_{C}v)(\cdot ):= (\mathcal {I}_{C}\varPi _{2}v)(\cdot ) = (\mathcal {I}_{C}\phi )(\cdot )\) for \(v(\cdot )=(y(\cdot ),\phi (\cdot ))\). Moreover, as always, we omit emphasizing complexifications of the operators.

  13. That is \(\vartheta \) is the family of mappings \(\vartheta ^{t} :\mathbb {R} \rightarrow \mathbb {R}\) such that \(\vartheta ^{t}(s):=t+s\) for any \(t, s \in \mathbb {R}\)

  14. If there exists at least one bounded complete trajectory.

  15. For this it is required to satisfy (4.71) for two distinct values of \(\nu _{0}\) separating the same j roots. Clearly, if (4.71) is satisfied for some \(\nu _{0}\), it is also satisfied for all sufficiently close to \(\nu _{0}\) values due to the first resolvent identity combined with (4.31) and (4.38).

  16. Strictly speaking, the operator \(D_{0}\) also depends on \(\tau \) since it acts in the space \(C([-\tau ,0];\mathbb {R}^{n})\). In applications, we usually have \(M:=\sup _{\tau }\Vert D_{0}(\tau )\Vert < 1\). For example, \(D_{0}\phi = \alpha \phi (-\tau )\) with \(|\alpha | < 1\). This allows to vary \(\tau \rightarrow 0\) and \(\nu _{0} \rightarrow +\infty \) such that \(\tau \nu _{0} < -\ln M\).

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Acknowledgements

The author is grateful to the anonymous referee for careful reading and valuable suggestions, which particularly revealed technical flaws and led to significant improvements in the overall exposition.

Funding

The reported study was funded by the Russian Science Foundation (Project 22-11-00172).

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  1. Department of Applied Cybernetics, Faculty of Mathematics and Mechanics, St Petersburg University, 28 Universitetskiy prospekt, Peterhof, Russia, 198504

    Mikhail Anikushin

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Anikushin, M. Frequency theorem and inertial manifolds for neutral delay equations. J. Evol. Equ. 23, 66 (2023). https://doi.org/10.1007/s00028-023-00915-w

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