Abstract
This paper is devoted to the study of a Mayer-type optimal control problem for semilinear unbounded evolution inclusions in reflexive and separable Banach spaces subject to endpoint constraints described by finitely many Lipschitzian equalities and inequalities. First we construct a sequence of discrete approximations to the optimal control problem for evolution inclusions and prove that optimal solutions to discrete approximation problems uniformly converge to a given optimal solution for the original continuous-time problem. Then, based on advanced tools of variational analysis and generalized differentiation, we derive necessary optimality conditions for discrete-time problems under fairly general assumptions. Combining these results with recent achievements of variational analysis in infinite-dimensional spaces, we establish new necessary optimality conditions for continuous-time evolution inclusions by passing to the limit from discrete approximations.
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References
Vinter, R.B.: Optimal Control. Birkhäuser, Boston (2000)
Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, I: Basic Theory. Springer, Berlin (2006)
Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, II: Applications. Springer, Berlin (2006)
Clarke, F.H.: Necessary conditions in dynamic optimization. Mem. Amer. Math. Soc. 173, 816 (2005)
Ahmed, N.U.: Semigroup Theory with Applications to Systems and Control. Longman, Harlow (1991)
Fattorrini, H.O.: Infinite-Dimensional Optimization and Control Theory. Cambridge University Press, Cambridge (1999)
Lasiecka, I., Triggiani, R.: Control Theory for Partial Differential Equations. Cambridge University Press, Cambridge (2000). Published in two volumes
Li, X.J., Yong, J.: Optimal Control Theory for Infinite-Dimensional Systems. Birkhäuser, Boston (1995)
Smirnov, G.V.: Introduction to the Theory of Differential Inclusions. American Mathematical Society, Providence (2002)
Donchev, T., Farkhi, E.M., Mordukhovich, B.S.: Discrete approximations, relaxation, and optimization of one-sided Lipschitzian differential inclusions in Hilbert spaces. J. Diff. Eqs. 243, 301–328 (2007)
Mordukhovich, B.S.: Optimal control of evolution inclusions. Nonlinear Anal. 63, 775–784 (2004)
Mordukhovich, B.S.: Variational analysis of evolution inclusions. SIAM J. Optim. 18, 752–777 (2007)
Tolstonogov, A.A.: Differential Inclusions in a Banach Spaces. Kluwer, Dordrecht (2000)
Mordukhovich, B.S., Wang, D.: Optimal control of semilinear unbounded differential inclusions. Nonlinear Anal. 63, 847–853 (2005)
Mordukhovich, B.S., Wang, D.: Optimal control of semilinear evolution inclusions via discrete approximations. Control and Cybernet. 34, 849–870 (2005)
Colombo, G., Henrion, R., Hoang, D.N., Mordukhovich, B.S.: Optimal control of the sweeping process. Dynam. Contin. Discrete Impuls. Syst., Ser. B 19, 117–159 (2012)
Mordukhovich, B.S.: Discrete approximations and refined Euler–Lagrange conditions for nonconvex differential inclusions. SIAM J. Control Optim. 33, 882–915 (1995)
Din, K., Donchev, T.: Discrete approximations and optimization of evolution inclusions. Set-Valued Var. Anal. 20, 15–30 (2012)
Dontchev, A.L., Farkhi, E.M.: Error estimates for discretized differential inclusions. Computing 41, 349–358 (1989)
Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)
Bounkhel, M., Thibault, L.: Further characterizations of regular sets in Hilbert spaces and their applications to nonconvex sweeping processes. J. Nonlinear Convex Anal. 6, 359–374 (2005)
Diestel, J., Uhl, J.J.: Vector Measures. American Mathematical Society, Providence (1977)
Acknowledgements
The authors are thankful to anonymous referees for their suggestions and remarks that allowed us to improve the original presentation. Research of B.S. Mordukhovich was partly supported by the National Science Foundation under grant DMS-1007132, by the Australian Research Council under grant DP-12092508, and by the Portuguese Foundation of Science and Technologies under grant MAT/11109. Research of D. Wang was supported by the Fayetteville State University Office of Academic Affairs.
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Mordukhovich, B.S., Wang, D. Optimal Control of Semilinear Unbounded Evolution Inclusions with Functional Constraints. J Optim Theory Appl 167, 821–841 (2015). https://doi.org/10.1007/s10957-013-0301-0
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DOI: https://doi.org/10.1007/s10957-013-0301-0