September  2015, 35(9): 4323-4343. doi: 10.3934/dcds.2015.35.4323

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

1. 

Russian Academy of Sciences, Central Economics and Mathematics Institute and Lomonosov Moscow State University, Russia 117418, Moscow, Nakhimovskii prospekt, 47, Russian Federation

2. 

Systems Research Institute, Polish Academy of Sciences, Warszawa, Moscow State University of Civil Engineering, Russian Federation

Received  April 2014 Revised  October 2014 Published  April 2015

We study an optimal control problem with Volterra-type integral equation, considered on a nonfixed time interval, subject to endpoint constraints of equality and inequality type. 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 Euler--Lagrange equation and transversality conditions for the Lagrange problem in the classical calculus of variations with ordinary differential equations.
Citation: 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 and Continuous Dynamical Systems, 2015, 35 (9) : 4323-4343. doi: 10.3934/dcds.2015.35.4323
References:
[1]

V. M. Alekseev, V. M. Tikhomirov and S. V. Fomin, Optimal Control, (Translated from Russian). Consultants Bureau. New York etc., 1987. doi: 10.1007/978-1-4615-7551-1.

[2]

V. L. Bakke, A maximum principle for an optimal control problem with integral constraints, JOTA, 13 (1974), 32-55. doi: 10.1007/BF00935608.

[3]

J. F. Bonnans and C. De La Vega, Optimal control of state constrained integral equations, Set-Valued Var. Analysis, 18 (2010), 307-326. doi: 10.1007/s11228-010-0154-8.

[4]

J. F. Bonnans, C. De La Vega and X. Dupuis, First and second order optimality conditions for optimal control problems of state constrained integral equations, JOTA, 159 (2013), 1-40. doi: 10.1007/s10957-013-0299-3.

[5]

D. A. Carlson, An elementary proof of the maximum principle for optimal control problems governed by a Volterra integral equation, JOTA, 54 (1987), 43-61. doi: 10.1007/BF00940404.

[6]

A. V. Dmitruk and A. M. Kaganovich, Quadratic order conditions for a weak minimum in optimal control problems with intermediate and mixed constraints, Discrete and Continuous Dynamical Systems, Ser. A, 29 (2011), 523-545. doi: 10.3934/dcds.2011.29.523.

[7]

A. V. Dmitruk, A. A. Milyutin and N. P. Osmolovskii, Lyusternik's theorem and the theory of extrema, Russian Math. Surveys, 35 (1980), 11-51.

[8]

A. V. 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]

I. V. Girsanov, Lectures on Mathematical Theory of Extremum Problems, (Translated from Russian), Lecture Notes in Econ. and Math. Systems, v. 67, Berlin-Heidelberg-New York: Springer-Verlag, 1972.

[10]

R. F. Hartl and S. P. Sethi, Optimal control on a class of systems with continuous lags: Dynamic programming approach and economic interpretations, JOTA, 43 (1984), 73-88. doi: 10.1007/BF00934747.

[11]

A. D. Ioffe and V. M. Tikhomirov, Theory of Extremal Problems, (Translated from Russian), North-Holland, New York, 1979.

[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]

C. De la Vega, Necessary conditions for optimal terminal time control problems governed by a Volterra integral equation, JOTA, 130 (2006), 79-93. doi: 10.1007/s10957-006-9087-7.

[14]

V. R. Vinokurov, Optimal control of processes described by integral equations, SIAM J. on Control, 7 (1969), 324-355.

show all references

References:
[1]

V. M. Alekseev, V. M. Tikhomirov and S. V. Fomin, Optimal Control, (Translated from Russian). Consultants Bureau. New York etc., 1987. doi: 10.1007/978-1-4615-7551-1.

[2]

V. L. Bakke, A maximum principle for an optimal control problem with integral constraints, JOTA, 13 (1974), 32-55. doi: 10.1007/BF00935608.

[3]

J. F. Bonnans and C. De La Vega, Optimal control of state constrained integral equations, Set-Valued Var. Analysis, 18 (2010), 307-326. doi: 10.1007/s11228-010-0154-8.

[4]

J. F. Bonnans, C. De La Vega and X. Dupuis, First and second order optimality conditions for optimal control problems of state constrained integral equations, JOTA, 159 (2013), 1-40. doi: 10.1007/s10957-013-0299-3.

[5]

D. A. Carlson, An elementary proof of the maximum principle for optimal control problems governed by a Volterra integral equation, JOTA, 54 (1987), 43-61. doi: 10.1007/BF00940404.

[6]

A. V. Dmitruk and A. M. Kaganovich, Quadratic order conditions for a weak minimum in optimal control problems with intermediate and mixed constraints, Discrete and Continuous Dynamical Systems, Ser. A, 29 (2011), 523-545. doi: 10.3934/dcds.2011.29.523.

[7]

A. V. Dmitruk, A. A. Milyutin and N. P. Osmolovskii, Lyusternik's theorem and the theory of extrema, Russian Math. Surveys, 35 (1980), 11-51.

[8]

A. V. 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]

I. V. Girsanov, Lectures on Mathematical Theory of Extremum Problems, (Translated from Russian), Lecture Notes in Econ. and Math. Systems, v. 67, Berlin-Heidelberg-New York: Springer-Verlag, 1972.

[10]

R. F. Hartl and S. P. Sethi, Optimal control on a class of systems with continuous lags: Dynamic programming approach and economic interpretations, JOTA, 43 (1984), 73-88. doi: 10.1007/BF00934747.

[11]

A. D. Ioffe and V. M. Tikhomirov, Theory of Extremal Problems, (Translated from Russian), North-Holland, New York, 1979.

[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]

C. De la Vega, Necessary conditions for optimal terminal time control problems governed by a Volterra integral equation, JOTA, 130 (2006), 79-93. doi: 10.1007/s10957-006-9087-7.

[14]

V. R. Vinokurov, Optimal control of processes described by integral equations, SIAM J. on Control, 7 (1969), 324-355.

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