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
References:
 [1] V. L. Bakke, A maximum principle for an optimal control problem with integral constraints, J. Optim. Theory Appl., 13 (1974), 32-55. doi: 10.1007/BF00935608. [2] J. F. Bonnans and C. De La Vega, Optimal control of state constrained integral equations, Set-Valued Var. Anal., 18 (2010), 307-326. doi: 10.1007/s11228-010-0154-8. [3] J. F. Bonnans, X. Dupuis and C. De La Vega, First and second order optimality conditions for optimal control problems of state constrained integral equations, J. Optim. Theory Applic., 159 (2013), 1-40. doi: 10.1007/s10957-013-0299-3. [4] D. A. Carlson, An elementary proof of the maximum principle for optimal control problems governed by a Volterra integral equation, J. Optim. Theory Appl., 54 (1987), 43-61. doi: 10.1007/BF00940404. [5] C. De la Vega, Necessary conditions for optimal terminal time control problems governed by a Volterra integral equation, J. Optim. Theory Appl., 130 (2006), 79-93. doi: 10.1007/s10957-006-9087-7. [6] A. Dmitruk, A. A. Milyutin and N. P. Osmolovskii, Lyusternik's theorem and the theory of extrema, Russian Math. Surveys, 35 (1980), 11-46,215. [7] A. Dmitruk, Maximum principle for a general optimal control problem with state and regular mixed constraints, Computation Math. and Modeling, 4 (1993), 364-377. doi: 10.1007/BF01128760. [8] A. Dmitruk and N. P. Osmolovskii, Necessary conditions for a weak minimum in optimal control problems with integral equations subject to state and mixed constraints, SIAM J. on Control and Optimization, 52 (2014), 3437-3462. doi: 10.1137/130921465. [9] A. Dmitruk and N. P. Osmolovskii, Necessary conditions for a weak minimum in optimal control problems with integral equations on a variable time interval, Discrete and Continuous Dynamical Systems, Ser. A, 35 (2015), 4323-4343. doi: 10.3934/dcds.2015.35.4323. [10] D. Filatova, M. Grzywaczewski and N. Osmolovskii, Optimal control problems with integral equation as control object, Nonlinear Analysis, 72 (2010), 1235-1246. doi: 10.1016/j.na.2009.08.008. [11] R. F. Hartl and S. P. Sethi, Optimal control on a class of systems with continuous lags: Dynamic programming approach and economic interpretations, J. Optim. Theory Appl., 43 (1984), 73-88. doi: 10.1007/BF00934747. [12] M. I. Kamien and E. Muller, Optimal control with integral state equations, The Review of Economic Studies, 43 (1976), 469-473. doi: 10.2307/2297225. [13] L. Kantorovich and G. P. Akilov, Functional Analysis Pergamon Press, 1982. [14] A. A. Milyutin, A. Dmitruk and N. P. Osmolovskii, Maximum Principle in Optimal Control Moscow State University, Faculty of Mechanics and Mathematics, Moscow, 2004 (in Russian). [15] L. W. Neustadt, Optimization: A Theory of Necessary Conditions Princeton University Press, Princeton, New Jersey, 1976. [16] L. S. Pontryagin, G. Boltyanskii, R. Gamkrelidze and E. F. Mischenko, The Mathematical Theory of Optimal Processes Moscow, Nauka, 1961 (in Russian), English translation: John Wiley & Sons, New York/London, 1962. [17] R. Vinokurov, Optimal control of processes described by integral equations, SIAM J. on Control, 7 (1969), 324-355.

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References:
 [1] V. L. Bakke, A maximum principle for an optimal control problem with integral constraints, J. Optim. Theory Appl., 13 (1974), 32-55. doi: 10.1007/BF00935608. [2] J. F. Bonnans and C. De La Vega, Optimal control of state constrained integral equations, Set-Valued Var. Anal., 18 (2010), 307-326. doi: 10.1007/s11228-010-0154-8. [3] J. F. Bonnans, X. Dupuis and C. De La Vega, First and second order optimality conditions for optimal control problems of state constrained integral equations, J. Optim. Theory Applic., 159 (2013), 1-40. doi: 10.1007/s10957-013-0299-3. [4] D. A. Carlson, An elementary proof of the maximum principle for optimal control problems governed by a Volterra integral equation, J. Optim. Theory Appl., 54 (1987), 43-61. doi: 10.1007/BF00940404. [5] C. De la Vega, Necessary conditions for optimal terminal time control problems governed by a Volterra integral equation, J. Optim. Theory Appl., 130 (2006), 79-93. doi: 10.1007/s10957-006-9087-7. [6] A. Dmitruk, A. A. Milyutin and N. P. Osmolovskii, Lyusternik's theorem and the theory of extrema, Russian Math. Surveys, 35 (1980), 11-46,215. [7] A. Dmitruk, Maximum principle for a general optimal control problem with state and regular mixed constraints, Computation Math. and Modeling, 4 (1993), 364-377. doi: 10.1007/BF01128760. [8] A. Dmitruk and N. P. Osmolovskii, Necessary conditions for a weak minimum in optimal control problems with integral equations subject to state and mixed constraints, SIAM J. on Control and Optimization, 52 (2014), 3437-3462. doi: 10.1137/130921465. [9] A. Dmitruk and N. P. Osmolovskii, Necessary conditions for a weak minimum in optimal control problems with integral equations on a variable time interval, Discrete and Continuous Dynamical Systems, Ser. A, 35 (2015), 4323-4343. doi: 10.3934/dcds.2015.35.4323. [10] D. Filatova, M. Grzywaczewski and N. Osmolovskii, Optimal control problems with integral equation as control object, Nonlinear Analysis, 72 (2010), 1235-1246. doi: 10.1016/j.na.2009.08.008. [11] R. F. Hartl and S. P. Sethi, Optimal control on a class of systems with continuous lags: Dynamic programming approach and economic interpretations, J. Optim. Theory Appl., 43 (1984), 73-88. doi: 10.1007/BF00934747. [12] M. I. Kamien and E. Muller, Optimal control with integral state equations, The Review of Economic Studies, 43 (1976), 469-473. doi: 10.2307/2297225. [13] L. Kantorovich and G. P. Akilov, Functional Analysis Pergamon Press, 1982. [14] A. A. Milyutin, A. Dmitruk and N. P. Osmolovskii, Maximum Principle in Optimal Control Moscow State University, Faculty of Mechanics and Mathematics, Moscow, 2004 (in Russian). [15] L. W. Neustadt, Optimization: A Theory of Necessary Conditions Princeton University Press, Princeton, New Jersey, 1976. [16] L. S. Pontryagin, G. Boltyanskii, R. Gamkrelidze and E. F. Mischenko, The Mathematical Theory of Optimal Processes Moscow, Nauka, 1961 (in Russian), English translation: John Wiley & Sons, New York/London, 1962. [17] R. Vinokurov, Optimal control of processes described by integral equations, SIAM J. on Control, 7 (1969), 324-355.
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