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    Lipschitz continuity of optimal control and Lagrange multipliers in a problem with mixed and pure state constraints
April  2011, 29(2): 523-545. doi: 10.3934/dcds.2011.29.523

Quadratic order conditions for an extended weak minimum in optimal control problems with intermediate and mixed constraints

Andrei V. Dmitruk 1, and  Alexander M. Kaganovich 2,

1. 

Central Economics and Mathematics Institute of the Russian Academy of Sciences, Nakhimovskii prospekt, 47, Moscow 117418, Russian Federation
2. 

Moscow State University, Faculty of Computational Mathematics and Cybernetics, GSP-2, Leninskie Gory, VMK MGU, Moscow 119992, Russian Federation

Received  August 2009 Revised  April 2010 Published  October 2010

We consider a general optimal control problem with intermediate and mixed constraints. Using a natural transformation (replication of the state and control variables), this problem is reduced to a standard optimal control problem with mixed constraints, which makes it possible to obtain quadratic order conditions for an "extended" weak minimum. The conditions obtained are applied to the problem of light refraction.
Keywords: Intermediate constraints, weak minimum, mixed constraints, replication of state and control variables., Euler--Lagrange equation, second variation.
Mathematics Subject Classification: Primary: 49K15; Secondary: 93B1.
Citation: Andrei V. Dmitruk, Alexander M. Kaganovich. Quadratic order conditions for an extended weak minimum in optimal control problems with intermediate and mixed constraints. Discrete and Continuous Dynamical Systems, 2011, 29 (2) : 523-545. doi: 10.3934/dcds.2011.29.523
References:
[1]

A. V. Arutyunov and A. I. Okoulevitch, Necessary optimality conditions for optimal control problems with intermediate constraints, J. of Dynamical and Control Systems, 4 (1998), 49-58. doi: oi:10.1023/A:1022820900022.

[2]

D. Augustin and H. Maurer, Second order sufficient conditions and sensitivity analysis for optimal multiprocess control problems, Control and Cybernetics, 29 (2000), 11-31.

[3]

G. A. Bliss, "Lectures on the Calculus of Variations," The Univ. of Chicago Press, 1946.

[4]

F. H. Clarke and R. B. Vinter, Optimal multiprocesses, SIAM J. on Control and Optimization, 27 (1989), 1072-1091. doi: doi:10.1137/0327057.

[5]

C. H. Denbow, "A Generalized Form of the Problem of Bolza," Dissertation, The University of Chicago Press, 1937.

[6]

A. V. Dmitruk, Jacobi type conditions for singular extremals, Control and Cybernetics, 37 (2008), 285-306.

[7]

A. V. Dmitruk and A. M. Kaganovich, Maximum principle for optimal control problems with intermediate constraints, in "Nonlinear Dynamics and Control" (eds. S. V. Emeljanov and S. K. Korovin), Moscow, Fizmatlit, 2008, 101-136, (in Russian).

[8]

A. V. Dmitruk and A. M. Kaganovich, The hybrid maximum principle is a consequence of Pontryagin maximum principle, Systems and Control Letters, 57 (2008), 964-970. doi: doi:10.1016/j.sysconle.2008.05.006.

[9]

A. Ja. Dubovitskii and A. A. Milyutin, Extremal problems in the presence of constraints, USSR Comput. Math. and Math. Physics, 5 (1965), 395-453.

[10]

A. D. Ioffe and V. M. Tikhomirov, "Theory of Extremal Problems," Nauka, Moscow, 1974.

[11]

A. M. Kaganovich, Quadratic weak minimum conditions for optimal control problems with intermediate constraints, J. of Math. Sciences, 165 (2010), 710-731. doi: doi:10.1007/s10958-010-9836-x.

[12]

C. Y. Kaya and J. L. Noakes, Computational method for time-optimal switching control, JOTA, 117 (2003), 69-92. doi: doi:10.1023/A:1023600422807.

[13]

A. A. Milyutin, A. V. Dmitruk and N. P. Osmolovskii, "Maximum Principle in Optimal Control," Mech-Math. Faculty of MSU, 2004, (in Russian), available at www.milyutin.ru.

[14]

A. A. Milyutin and N. P. Osmolovskii, "Calculus of Variations and Optimal Control," American Math. Society, Providence, 1998.

[15]

N. P. Osmolovskii, Second-order conditions for broken extremal, in "Calculus of Variations and Optimal Control" (ed. A. Ioffe, et al.), Chapman Hall/CRC Res. Notes in Math., 411 (2000), 198-216.

[16]

Yu. M. Volin and G. M. Ostrovskii, Maximum principle for discontinuous systems and its application to problems with state constraints, Izvestia Vuzov. Radiofizika, 12 (1969), 1609-1621, (in Russian).

[17]

X. Xu and P. J. Antsaklis, Optimal control of switched systems based on parametrization of the swithing instants, IEEE Trans. Automatic Control, 49 (2004), 2-16. doi: doi:10.1109/TAC.2003.821417.

show all references

References:
[1]

A. V. Arutyunov and A. I. Okoulevitch, Necessary optimality conditions for optimal control problems with intermediate constraints, J. of Dynamical and Control Systems, 4 (1998), 49-58. doi: oi:10.1023/A:1022820900022.

[2]

D. Augustin and H. Maurer, Second order sufficient conditions and sensitivity analysis for optimal multiprocess control problems, Control and Cybernetics, 29 (2000), 11-31.

[3]

G. A. Bliss, "Lectures on the Calculus of Variations," The Univ. of Chicago Press, 1946.

[4]

F. H. Clarke and R. B. Vinter, Optimal multiprocesses, SIAM J. on Control and Optimization, 27 (1989), 1072-1091. doi: doi:10.1137/0327057.

[5]

C. H. Denbow, "A Generalized Form of the Problem of Bolza," Dissertation, The University of Chicago Press, 1937.

[6]

A. V. Dmitruk, Jacobi type conditions for singular extremals, Control and Cybernetics, 37 (2008), 285-306.

[7]

A. V. Dmitruk and A. M. Kaganovich, Maximum principle for optimal control problems with intermediate constraints, in "Nonlinear Dynamics and Control" (eds. S. V. Emeljanov and S. K. Korovin), Moscow, Fizmatlit, 2008, 101-136, (in Russian).

[8]

A. V. Dmitruk and A. M. Kaganovich, The hybrid maximum principle is a consequence of Pontryagin maximum principle, Systems and Control Letters, 57 (2008), 964-970. doi: doi:10.1016/j.sysconle.2008.05.006.

[9]

A. Ja. Dubovitskii and A. A. Milyutin, Extremal problems in the presence of constraints, USSR Comput. Math. and Math. Physics, 5 (1965), 395-453.

[10]

A. D. Ioffe and V. M. Tikhomirov, "Theory of Extremal Problems," Nauka, Moscow, 1974.

[11]

A. M. Kaganovich, Quadratic weak minimum conditions for optimal control problems with intermediate constraints, J. of Math. Sciences, 165 (2010), 710-731. doi: doi:10.1007/s10958-010-9836-x.

[12]

C. Y. Kaya and J. L. Noakes, Computational method for time-optimal switching control, JOTA, 117 (2003), 69-92. doi: doi:10.1023/A:1023600422807.

[13]

A. A. Milyutin, A. V. Dmitruk and N. P. Osmolovskii, "Maximum Principle in Optimal Control," Mech-Math. Faculty of MSU, 2004, (in Russian), available at www.milyutin.ru.

[14]

A. A. Milyutin and N. P. Osmolovskii, "Calculus of Variations and Optimal Control," American Math. Society, Providence, 1998.

[15]

N. P. Osmolovskii, Second-order conditions for broken extremal, in "Calculus of Variations and Optimal Control" (ed. A. Ioffe, et al.), Chapman Hall/CRC Res. Notes in Math., 411 (2000), 198-216.

[16]

Yu. M. Volin and G. M. Ostrovskii, Maximum principle for discontinuous systems and its application to problems with state constraints, Izvestia Vuzov. Radiofizika, 12 (1969), 1609-1621, (in Russian).

[17]

X. Xu and P. J. Antsaklis, Optimal control of switched systems based on parametrization of the swithing instants, IEEE Trans. Automatic Control, 49 (2004), 2-16. doi: doi:10.1109/TAC.2003.821417.

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