Earlier the author proposed a rather general form of describing controlled initial–boundary value problems (IBVPs) by means of Volterra functional equations (VFEs)
\[ z(t)=f\left(t,A[z](t),v(t)\right) ,\quad t\equiv \{t^{1},\ldots,t^{N}\} \in \Pi\subset{\mathbb R}^N ,\quad z\in L_p^m \equiv \left(L_p\left( \Pi \right)\right)^m, \]
where $ f (.,.,.): \Pi \times {\mathbb R}^l \times {\mathbb R}^s \rightarrow {\mathbb R}^m$, $ v(.) \in {\cal D} \subset L_k^s $ is a control function, and $ A:L_p^m\left( \Pi \right) \rightarrow L_q^l\left( \Pi \right) $ is a linear operator that is Volterra for some system $\mathbf{T}$ of subsets of $\Pi$ in the following sense: for any $H \in \mathbf{T}$, the restriction $\left. A \left[z \right] \right |_H $ does not depend on the values of $z|_{\Pi \backslash H}$, $p,q,k\in \left[ 1,+\infty \right]$ (this definition of a Volterra operator is a direct multidimensional generalization of the well-known Tikhonov definition of a functional Volterra type operator). Various IBVPs (for nonlinear hyperbolic and parabolic equations, integro-differential equations, equations with delay, etc.) are reduced by the method of inversion of the main part to such functional equations. The transition to an equivalent VFE-description of IBVPs is adequate to many problems of distributed optimization (obtaining conditions for maintaining the global solvability of equations under perturbed controls, substantiation of numerical methods of optimal control, derivation of necessary optimality conditions, study of singular controls for necessary optimality conditions, etc.). In particular, the author proposed (using such a description) a scheme for obtaining sufficient stability conditions (under perturbations of control) for the existence of global solutions for IBVPs. In the present paper, the effectiveness (for the theory of optimal control) of such a description of IBVPs is demonstrated by an example of a controlled semilinear parabolic equation. The problems of obtaining sufficient conditions for the preservation (under perturbation of control) of the global solvability of IBVPs and the derivation of necessary optimality conditions for singular in the sense of J.-L. Lions optimal control problems are considered. It is shown that some optimization problems that were classified as singular can in fact be classified as nonsingular, since the necessary optimality conditions for them may be derived by bringing the problems to the classical form and varying the controls.
Keywords: Volterra functional equations, controlled initial-boundary value problems, conditions for maintaining global solvability, singular optimality systems
Received June 15, 2022
Revised July 15, 2022
Accepted July 18, 2022
Funding Agency: This work was supported by the Russian Foundation for Basic Research (project no. 20-01-00199_a).
Vladimir Iosifovich Sumin, Dr. Phys.-Math. Sci., Prof., Nizhny Novgorod State University named after N.I. Lobachevsky, Nizhny Novgorod, 603950 Russia; Derzhavin Tambov State University, Tambov, 392000 Russia, e-mail: v_sumin@mail.ru
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Cite this article as: V.I. Sumin. Volterra functional equations in the theory of optimization of distributed systems. On the problem of singularity of controlled initial–boundary value problems. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2022, vol. 28, no. 3, pp. 188–201; Proceedings of the Steklov Institute of Mathematics (Suppl.), 2022, Vol. 319, Suppl. 1, pp. S257–S270.
