# American Institute of Mathematical Sciences

December  2017, 7(4): 507-535. doi: 10.3934/mcrf.2017019

## Necessary conditions for a weak minimum in a general optimal control problem with integral equations on a variable time interval

 1 Russian Academy of Sciences, Central Economics and Mathematics Institute, Lomonosov Moscow State University, Moscow, Russia 2 Moscow Institute of Physics and Technology, Dolgoprudny, Russia, Russia 117418, Moscow, Nakhimovskii prospekt, 47, Russia 3 Systems Research Institute, Polish Academy of Sciences, Warszawa, Moscow State University of Civil Engineering, Russia 4 University of Technology and Humanities in Radom, Poland, 26-600 Radom, ul. Malczewskiego 20A, Poland

Received  December 2016 Revised  June 2017 Published  September 2017

We study an optimal control problem with a nonlinear Volterra-type integral equation considered on a nonfixed time interval, subject to endpoint constraints of equality and inequality type, mixed state-control constraints of inequality and equality type, and pure state constraints of inequality type. The main assumption is the linear–positive independence of the gradients of active mixed constraints with respect to the control. We obtain first-order necessary optimality conditions for an extended weak minimum, the notion of which is a natural generalization of the notion of weak minimum with account of variations of the time. The conditions obtained generalize the corresponding ones for problems with ordinary differential equations.

Citation: Andrei V. Dmitruk, Nikolai P. Osmolovski. Necessary conditions for a weak minimum in a general optimal control problem with integral equations on a variable time interval. Mathematical Control & Related Fields, 2017, 7 (4) : 507-535. doi: 10.3934/mcrf.2017019
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##### References:
 [1] Andrei V. Dmitruk, Nikolai P. Osmolovskii. Proof of the maximum principle for a problem with state constraints by the v-change of time variable. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2189-2204. doi: 10.3934/dcdsb.2019090 [2] H. O. Fattorini. The maximum principle for linear infinite dimensional control systems with state constraints. Discrete & Continuous Dynamical Systems, 1995, 1 (1) : 77-101. doi: 10.3934/dcds.1995.1.77 [3] Andrei V. Dmitruk, Alexander M. Kaganovich. Quadratic order conditions for an extended weak minimum in optimal control problems with intermediate and mixed constraints. Discrete & Continuous Dynamical Systems, 2011, 29 (2) : 523-545. doi: 10.3934/dcds.2011.29.523 [4] Md. Haider Ali Biswas, Maria do Rosário de Pinho. A nonsmooth maximum principle for optimal control problems with state and mixed constraints - convex case. Conference Publications, 2011, 2011 (Special) : 174-183. doi: 10.3934/proc.2011.2011.174 [5] Eduardo Casas, Fredi Tröltzsch. Sparse optimal control for the heat equation with mixed control-state constraints. Mathematical Control & Related Fields, 2020, 10 (3) : 471-491. doi: 10.3934/mcrf.2020007 [6] Huaiqiang Yu, Bin Liu. Pontryagin's principle for local solutions of optimal control governed by the 2D Navier-Stokes equations with mixed control-state constraints. Mathematical Control & Related Fields, 2012, 2 (1) : 61-80. doi: 10.3934/mcrf.2012.2.61 [7] Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020110 [8] Vincenzo Basco, Piermarco Cannarsa, Hélène Frankowska. Necessary conditions for infinite horizon optimal control problems with state constraints. Mathematical Control & Related Fields, 2018, 8 (3&4) : 535-555. doi: 10.3934/mcrf.2018022 [9] Luís Tiago Paiva, Fernando A. C. C. Fontes. Adaptive time--mesh refinement in optimal control problems with state constraints. Discrete & Continuous Dynamical Systems, 2015, 35 (9) : 4553-4572. doi: 10.3934/dcds.2015.35.4553 [10] Thomas Lorenz. Partial differential inclusions of transport type with state constraints. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1309-1340. doi: 10.3934/dcdsb.2019018 [11] M. Arisawa, P.-L. Lions. Continuity of admissible trajectories for state constraints control problems. Discrete & Continuous Dynamical Systems, 1996, 2 (3) : 297-305. doi: 10.3934/dcds.1996.2.297 [12] Mikhail Gusev. On reachability analysis for nonlinear control systems with state constraints. Conference Publications, 2015, 2015 (special) : 579-587. doi: 10.3934/proc.2015.0579 [13] Maria do Rosário de Pinho, Ilya Shvartsman. Lipschitz continuity of optimal control and Lagrange multipliers in a problem with mixed and pure state constraints. Discrete & Continuous Dynamical Systems, 2011, 29 (2) : 505-522. doi: 10.3934/dcds.2011.29.505 [14] Gui-Dong Li, Chun-Lei Tang. Existence of positive ground state solutions for Choquard equation with variable exponent growth. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2035-2050. doi: 10.3934/dcdss.2019131 [15] Andrei V. Dmitruk, Nikolai P. Osmolovskii. Necessary conditions for a weak minimum in optimal control problems with integral equations on a variable time interval. Discrete & Continuous Dynamical Systems, 2015, 35 (9) : 4323-4343. doi: 10.3934/dcds.2015.35.4323 [16] J.-P. Raymond, F. Tröltzsch. Second order sufficient optimality conditions for nonlinear parabolic control problems with state constraints. Discrete & Continuous Dynamical Systems, 2000, 6 (2) : 431-450. doi: 10.3934/dcds.2000.6.431 [17] Nguyen Thi Hoai. Asymptotic approximation to a solution of a singularly perturbed linear-quadratic optimal control problem with second-order linear ordinary differential equation of state variable. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020040 [18] Heinz Schättler, Urszula Ledzewicz, Helmut Maurer. Sufficient conditions for strong local optimality in optimal control problems with $L_{2}$-type objectives and control constraints. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2657-2679. doi: 10.3934/dcdsb.2014.19.2657 [19] Elena K. Kostousova. On polyhedral control synthesis for dynamical discrete-time systems under uncertainties and state constraints. Discrete & Continuous Dynamical Systems, 2018, 38 (12) : 6149-6162. doi: 10.3934/dcds.2018153 [20] Yuefen Chen, Yuanguo Zhu. Indefinite LQ optimal control with process state inequality constraints for discrete-time uncertain systems. Journal of Industrial & Management Optimization, 2018, 14 (3) : 913-930. doi: 10.3934/jimo.2017082

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