Approximation of optimal control for functional-differential equations by optimal control of equations without delay in infinite-dimensional spaces

O.V. Perehuda; O.A. Kapustian; V.V. Mogylova

doi:10.15421/242607

Approximation of optimal control for functional-differential equations by optimal control of equations without delay in infinite-dimensional spaces

O.V. Perehuda (Taras Shevchenko National University of Kyiv), https://orcid.org/0000-0002-7465-3173
O.A. Kapustian (Taras Shevchenko National University of Kyiv), https://orcid.org/0000-0002-2629-0750
V.V. Mogylova (National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute"), https://orcid.org/0000-0003-3757-4561

Abstract

The problem of optimal control for a functional-differential equation of parabolic type in a Banach space is studied. An approximation procedure is proposed, which consists of replacing the original equation by the system of parabolic equations without delay. It is proved that the optimal control for such a system weakly converges to the optimal control of the original problem.

Keywords

$$$C_0$$$-semigroup; functional-differential equation; delay; weak convergence; mild solution; optimal control; approximation system

MSC 2020

34K05; 34K35; 49K21

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References

Bayractar E., Keller C. "Part-dependent Hamilton-Jacobi equations in infinite dimensionals", J. Funct. Analys., 2018; 275(8): pp. 2096-2161. doi:10.1016/j.jfa.2018.07.010

Casas E., Wachsmuth D. "A note on existence of solution to control problems of semilinear partial differential equations", SIAM J. Control Optim., 2023; 61(3): pp. 1095-1112. doi:10.1137/22M1486418

Casas E., Tröltzch F. "Sparce optimal control for a semilinear heat with mixed control-state constraints-regularity of Lagrange multipliers", ESAIM Control Optim. Calc. Var., 2021; 27. doi:10.1051/cocv/2020084

Cherevko I., Piddubna L. "Approximations of differential difference equations and calculation of nonasymptotic roots of quasipolynomial", Revue d'Analise Numèrique et de Thèorie de L'approximation, 1999; 28(1): pp. 15-21.

Lark J., Misiats O., Stanzhytskiy O., Mogylova V. "Asymptotic behavior of stochastic functional differential evolution equation", El. J. Diff. Equat., 2023; 2023(35): pp. 1-21. doi:10.58997/ejde.2023.35

Cosso A., Federico S., Gozzi F., Rosestolato M., Touzi N. "Part-dependent equations and viscosity solutions in infinite dimension", Ann. Probab., 2018; 46(1): pp. 126-174. doi:10.1214/17-AOP1183

Ekren I., Touzi N., Zhang J. "Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part 1", Ann. Probab., 2016; 44(2): pp. 1212-1253. doi:10.1214/15-AOP1004

Evans L.C. Partial Differential Equations, Part of: Graduate Studies in Mathematics, Vol.19, Amer. Math. Soc., 2010.

Henry D. Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics. Springer-Verlag, 1981.

Kolmanovskii V., Shaikhet L. Control of systems with after-effect, Amer. Math. Soc., 1992.

Latysh A., Kichmarenko O. "Optimal control of functional-differential equations of parabolic type in Banach spaces", J. Math. Sci., 2024; 278: pp. 1034-1054. doi:10.1007/s10958-024-06983-8

Li X., Yong J. Optimal control theory for infinite dimensional systems, Birkhäuser, 2012.

Matvii O.V., Cherevko I.M. "On approximation of systems with delay and their stability", Nonlin. Oscill., 2004; 7(2): pp. 207-215. doi:10.1007/s11072-005-0006-x

Neitsel I., Vexler V. "A priori error estimates for space-time finite element discretization of semilinear parabolic optimal control problems", Numer. Math., 2012; 120: pp. 345-386. doi:10.1007/s00211-011-0414-y

Perehuda O.V., Stanzhytskiy O.M., Klimchuk T.V., Ospanov M.N. "Optimal Control for Functional-Differential Equation of Neutral Type in Banach Spaces", J. Optim., Diff. Equat. Applic., 2025; 33(1): pp. 182-207. doi:10.15421/142509

Da Prato G., Zabczyk J. Stochastic Equations in Infinite Dimensions, Cambridge Univ. Press, 1992.

Stanzhytskyi A., Misiats O., Stanzhytskyi O. "Invariant measure for neutral stochastic functional differential equations with non-Lipschitz coefficients", Evolut. Equat. Control Theory, 2022; 11(5): pp. 1929-1953. doi:10.3934/eect.2022005

Yosida K. Functional Analysis, Springer, 1995. doi:10.1007/978-3-642-61859-8