# American Institute of Mathematical Sciences

September  2020, 16(5): 2503-2520. doi: 10.3934/jimo.2019066

## Optimal control of Sturm-Liouville type evolution differential inclusions with endpoint constraints

 1 Department of Mathematics, Istanbul Technical University, 34469, Maslak, Istanbul, Turkey 2 Azerbaijan National Academy of Sciences Institute of Control Systems, Baku, Azerbaijan

* Corresponding author: Elimhan N. Mahmudov

Received  November 2018 Revised  February 2019 Published  May 2019

The present paper studies a new class of problems of optimal control theory with linear second order self-adjoint Sturm-Liouville type differential operators and with functional and non-functional endpoint constraints. Sufficient conditions of optimality, containing both the second order Euler-Lagrange and Hamiltonian type inclusions are derived. The presence of functional constraints generates a special second order transversality inclusions and complementary slackness conditions peculiar to inequality constraints; this approach and results make a bridge between optimal control problem with Sturm-Liouville type differential differential inclusions and constrained mathematical programming problems in finite-dimensional spaces.The idea for obtaining optimality conditions is based on applying locally-adjoint mappings to Sturm-Liouville type set-valued mappings. The result generalizes to the problem with a second order non-self-adjoint differential operator. Furthermore, practical applications of these results are demonstrated by optimization of some semilinear optimal control problems for which the Pontryagin maximum condition is obtained. A numerical example is given to illustrate the feasibility and effectiveness of the theoretic results obtained.

Citation: Elimhan N. Mahmudov. Optimal control of Sturm-Liouville type evolution differential inclusions with endpoint constraints. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2503-2520. doi: 10.3934/jimo.2019066
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