Abstract
The paper is devoted to a generalization of a necessary optimality condition in the form of the Feedback Minimum Principle for a nonconvex discrete-time free-endpoint control problem. The approach is based on an exact formula for the increment of the cost functional. This formula is completely defined through a solution of the adjoint system corresponding to a reference process. By minimizing that increment in control variable for a fixed adjoint state, we define a multivalued map, whose selections are feedback controls with the property of potential “improvement” of the reference process. As a result, we derive a necessary optimality condition (optimal process does not admit feedback controls of a “potential descent” in the cost functional). In the case when the well-known Discrete Maximum Principle holds, our condition can be further strengthened. Note that obtained optimality condition is quite constructive and may lead to an iterative algorithm for discrete-time optimal control problems. Finally, we present sufficient optimality conditions for problems, where Discrete Maximum Principle does not make sense.
Partially supported by the Russian Foundation for Basic Research, projects nos 17-01-00733, 18-31-20030, 18-31-00425.
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References
Sorokin, S.P.: Necessary feedback optimality conditions and nonstandard duality in problems of discrete system optimization. Autom. Remote Control 75(9), 1556–1564 (2014)
Dykhta, V.A.: Variational necessary optimality conditions with feedback descent controls for optimal control problems. Doklady Math. 91(3), 394–396 (2015)
Dykhta, V.A.: Positional strengthenings of the maximum principle and sufficient optimality conditions. Proc. Steklov Inst. Math. 293(1), S43–S57 (2016)
Gabasov, R., Kirillova, F.M.: Qualitative Theory of Optimal Processes. Nauka, Moscow (1971). [in Russian]
Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation I-II. Fundamental Principles of Mathematical Sciences, vol. 330-331. Springer, Heidelberg (2006)
Propoi, A.I.: Elements of the theory of optimal discrete processes. Nauka, Moscow (1973). [in Russian]
Mordukhovich, B.S.: Approximation Methods in Optimization and Control Problems. Nauka, Moscow (1988). [in Russian]
Boltyanskiy, V.G.: Optimal Control of Discrete Systems. Nauka, Moscow (1973). [in Russian]
Sorokin, S.P., Staritsyn, M.V.: Numeric algorithm for optimal impulsive control based on feedback maximum principle. Optim. Lett. (2018). https://doi.org/10.1007/s11590-018-1344-9
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Dykhta, V., Sorokin, S. (2019). Feedback Minimum Principle for Optimal Control Problems in Discrete-Time Systems and Its Applications. In: Khachay, M., Kochetov, Y., Pardalos, P. (eds) Mathematical Optimization Theory and Operations Research. MOTOR 2019. Lecture Notes in Computer Science(), vol 11548. Springer, Cham. https://doi.org/10.1007/978-3-030-22629-9_31
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DOI: https://doi.org/10.1007/978-3-030-22629-9_31
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