Abstract:
The problem is considered of minimizing an integral functional with integrand that is not convex in the control, on solutions of a control system described by a first-order non-linear evolution equation with mixed non-convex constraints on the control. A relaxation problem is treated along with the original problem. Under appropriate assumptions it is proved that the relaxation problem has an optimal solution and that for each optimal solution there is a minimizing sequence for the original problem that converges to the optimal solution. Moreover, in the appropriate topologies the convergence is uniform simultaneously for the trajectory, the control, and the functional. The converse also holds. An example of a non-linear parabolic control system is treated in detail.
Citation:
A. A. Tolstonogov, “Relaxation in non-convex optimal control problems described by first-order evolution equations”, Sb. Math., 190:11 (1999), 1689–1714
\Bibitem{Tol99}
\by A.~A.~Tolstonogov
\paper Relaxation in non-convex optimal control problems described by first-order evolution equations
\jour Sb. Math.
\yr 1999
\vol 190
\issue 11
\pages 1689--1714
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Linking options:
https://www.mathnet.ru/eng/sm441
https://doi.org/10.1070/sm1999v190n11ABEH000441
https://www.mathnet.ru/eng/sm/v190/i11/p135
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Caijing Jiang, Fengzhen Long, “Existence and convergence for multiple feedback control problems governed by evolution equations”, Applicable Analysis, 2024, 1
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Charles Castaing, Christiane Godet-Thobie, Soumia Saïdi, Manuel D. P. Monteiro Marques, “Various Perturbations of Time Dependent Maximal Monotone/Accretive Operators in Evolution Inclusions with Applications”, Appl Math Optim, 87:2 (2023)
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Jiang Y.-r. Huang N.-j. Zhang Q.-f. Shang Ch.-ch., “Relaxation in Nonconvex Optimal Control Problems Governed By Evolution Inclusions With the Difference of Two Clarke'S Subdifferentials”, Int. J. Control, 94:2 (2021), 534–547
Li J., Bin M., “Control Systems Described By a Class of Fractional Semilinear Evolution Hemivariational Inequalities and Their Relaxation Property”, Optimization, 2021
Bin M., Deng H., Li Yu., Zhao J., “Properties of the Set of Admissible “State Control” Pair For a Class of Fractional Semilinear Evolution Control Systems”, Fract. Calc. Appl. Anal., 24:4 (2021), 1275–1298
A. A. Tolstonogov, “Bogolyubov's theorem for a controlled system related to a variational inequality”, Izv. Math., 84:6 (2020), 1192–1223
Papageorgiou N.S., Radulescu V.D., Repovs D.D., “Relaxation Methods For Optimal Control Problems”, Bull. Math. Sci., 10:1 (2020), UNSP 2050004
Bin M., Liu Zh., “Relaxation in Nonconvex Optimal Control For Nonlinear Evolution Hemivariational Inequalities”, Nonlinear Anal.-Real World Appl., 50 (2019), 613–632
Bin M., Liu Zh., “On the “Bang-Bang” Principle For Nonlinear Evolution Hemivariational Inequalities Control Systems”, J. Math. Anal. Appl., 480:1 (2019), UNSP 123364
Krejci P., Timoshin S.A., Tolstonogov A.A., “Relaxation and Optimisation of a Phase-Field Control System With Hysteresis”, Int. J. Control, 91:1 (2018), 85–100
Bayraktar E., Keller Ch., “Path-Dependent Hamilton–Jacobi Equations in Infinite Dimensions”, J. Funct. Anal., 275:8 (2018), 2096–2161
Tolstonogov A.A., “Existence and relaxation of solutions for a subdifferential inclusion with unbounded perturbation”, J. Math. Anal. Appl., 447:1 (2017), 269–288
Aiki T., Timoshin S.A., “Relaxation For a Control Problem in Concrete Carbonation Modeling”, SIAM J. Control Optim., 55:6 (2017), 3489–3502
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