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

Previous Article
Regularity of minimizers for second order variational problems in one independent variable
DCDS Home
This Issue
Next Article
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

Abstract
Related Papers

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.

[1]	Eduardo Casas, Fredi Tröltzsch. Sparse optimal control for the heat equation with mixed control-state constraints. Mathematical Control and Related Fields, 2020, 10 (3) : 471-491. doi: 10.3934/mcrf.2020007
[2]	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 and Continuous Dynamical Systems, 2011, 29 (2) : 505-522. doi: 10.3934/dcds.2011.29.505
[3]	Mourad Azi, Mohand Ouamer Bibi. Optimal control of a dynamical system with intermediate phase constraints and applications in cash management. Numerical Algebra, Control and Optimization, 2022, 12 (2) : 279-291. doi: 10.3934/naco.2021005
[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]	Elimhan N. Mahmudov. Optimal control of second order delay-discrete and delay-differential inclusions with state constraints. Evolution Equations and Control Theory, 2018, 7 (3) : 501-529. doi: 10.3934/eect.2018024
[6]	J.-P. Raymond, F. Tröltzsch. Second order sufficient optimality conditions for nonlinear parabolic control problems with state constraints. Discrete and Continuous Dynamical Systems, 2000, 6 (2) : 431-450. doi: 10.3934/dcds.2000.6.431
[7]	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
[8]	M. Arisawa, P.-L. Lions. Continuity of admissible trajectories for state constraints control problems. Discrete and Continuous Dynamical Systems, 1996, 2 (3) : 297-305. doi: 10.3934/dcds.1996.2.297
[9]	Nidhal Gammoudi, Hasnaa Zidani. A differential game control problem with state constraints. Mathematical Control and Related Fields, 2022 doi: 10.3934/mcrf.2022008
[10]	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 and Related Fields, 2012, 2 (1) : 61-80. doi: 10.3934/mcrf.2012.2.61
[11]	Piernicola Bettiol, Hélène Frankowska. Lipschitz regularity of solution map of control systems with multiple state constraints. Discrete and Continuous Dynamical Systems, 2012, 32 (1) : 1-26. doi: 10.3934/dcds.2012.32.1
[12]	Vincenzo Basco, Piermarco Cannarsa, Hélène Frankowska. Necessary conditions for infinite horizon optimal control problems with state constraints. Mathematical Control and Related Fields, 2018, 8 (3&4) : 535-555. doi: 10.3934/mcrf.2018022
[13]	H. O. Fattorini. The maximum principle for linear infinite dimensional control systems with state constraints. Discrete and Continuous Dynamical Systems, 1995, 1 (1) : 77-101. doi: 10.3934/dcds.1995.1.77
[14]	J.-P. Raymond. Nonlinear boundary control of semilinear parabolic problems with pointwise state constraints. Discrete and Continuous Dynamical Systems, 1997, 3 (3) : 341-370. doi: 10.3934/dcds.1997.3.341
[15]	Luís Tiago Paiva, Fernando A. C. C. Fontes. Adaptive time--mesh refinement in optimal control problems with state constraints. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 4553-4572. doi: 10.3934/dcds.2015.35.4553
[16]	Theodore Tachim-Medjo. Optimal control of a two-phase flow model with state constraints. Mathematical Control and Related Fields, 2016, 6 (2) : 335-362. doi: 10.3934/mcrf.2016006
[17]	Dmitry V. Zenkov, Anthony M. Bloch. Dynamics of generalized Euler tops with constraints. Conference Publications, 2001, 2001 (Special) : 398-405. doi: 10.3934/proc.2001.2001.398
[18]	Giovanni Bonfanti, Arrigo Cellina. The validity of the Euler-Lagrange equation. Discrete and Continuous Dynamical Systems, 2010, 28 (2) : 511-517. doi: 10.3934/dcds.2010.28.511
[19]	Matthias Gerdts, Martin Kunkel. Convergence analysis of Euler discretization of control-state constrained optimal control problems with controls of bounded variation. Journal of Industrial and Management Optimization, 2014, 10 (1) : 311-336. doi: 10.3934/jimo.2014.10.311
[20]	Matthias Gerdts, Martin Kunkel. A nonsmooth Newton's method for discretized optimal control problems with state and control constraints. Journal of Industrial and Management Optimization, 2008, 4 (2) : 247-270. doi: 10.3934/jimo.2008.4.247

2020 Impact Factor: 1.392

Tools

Metrics

Other articles
by authors

on AIMS
Andrei V. Dmitruk Alexander M. Kaganovich
on Google Scholar
Andrei V. Dmitruk Alexander M. Kaganovich

[Back to Top]