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April  2011, 29(2): 505-522. doi: 10.3934/dcds.2011.29.505

Lipschitz continuity of optimal control and Lagrange multipliers in a problem with mixed and pure state constraints

Maria do Rosário de Pinho 1, and  Ilya Shvartsman 2,

1. 

Faculdade de Engenharia da Universidade do Porto, Department of Electrical Engineering and Computers, Porto, Portugal
2. 

Department of Mathematics and Computer Science, Penn State Harrisburg, Middletown, PA 17057, United States

Received  September 2009 Revised  April 2010 Published  October 2010

In this paper we report conditions ensuring Lipschitz continuity of optimal control and Lagrange multipliers for a dynamic optimization problem with inequality pure state and mixed state-control constraints.
Keywords: Lipschitz control, necessary optimality conditions, Optimal control, regularity..
Mathematics Subject Classification: Primary: 49K15, 49K4.
Citation: 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
References:
[1]

J. F. Bonnans and A. Hermant, Second-order analysis for optimal control problems with pure state constraints and mixed control-state constraints, Ann. I. H. Poincaré - Analyse Non Linéaire, 26 (2009), 561-598.  Google Scholar

[2]

J. F. Bonnans and A. Shapiro, "Perturbation Analysis of Optimization Problems," Springer-Verlag, New York, 2000. Google Scholar

[3]

A. V. Dmitruk, Maximum principle for a general optimal control problem with state and regular mixed constraints, Comp. Math. and Modeling, 4 (1993), 364-377. doi: doi:10.1007/BF01128760.  Google Scholar

[4]

G. N. Galbraith and R. B. Vinter, Lipschitz Continuiuty of Optimal Controls for State Constrained Problems, SIAM J. Control and Optimization, 42 (2003), 1727-1744. doi: doi:10.1137/S0363012902404711.  Google Scholar

[5]

W. W. Hager, Lipschitz continuity for constrained processes, SIAM J. Control and Optimization, 17 (1979), 321-338. doi: doi:10.1137/0317026.  Google Scholar

[6]

K. Malanowski, On the regularity of solutions to optimal control problems for systems linear with respect to control variable, Arch. Auto. i Telemech., 23 (1978), 227-241.  Google Scholar

[7]

K. Malanowski, On normality of Lagrange multipliers for state constrained optimal control problems, Optimization, 52 (2003), 75-91. doi: doi:10.1080/0233193021000058940.  Google Scholar

[8]

A. A. Milyutin, A. V. Dmitruk and N. P. Osmolovsky, "Maximum Principle in Optimal Control" (Russian), Moscow State University Press, 2004. Available online at http://www.milyutin.ru/book.html. Google Scholar

[9]

S. M. Robinson and R. H. Day, A sufficient condition for continuity of optimal sets in mathematical programming, J. Math. Anal. Appl., 45 (1974), 506-511. doi: doi:10.1016/0022-247X(74)90089-4.  Google Scholar

[10]

S. M. Robinson, Strongly regular generalized equations, Mathematics of Operations Research, 5 (1980), 43-62. doi: doi:10.1287/moor.5.1.43.  Google Scholar

[11]

I. Shvartsman and R. B. Vinter, Regularity properties of optimal controls for state constrained problems with time-varying control constraints, Nonlinear Analysis: Theory, Methods and Applications, 65 (2006), 448-474.  Google Scholar

[12]

R. B. Vinter, "Optimal Control," Birkhäuser, Boston, 2000. Google Scholar

show all references

References:
[1]

J. F. Bonnans and A. Hermant, Second-order analysis for optimal control problems with pure state constraints and mixed control-state constraints, Ann. I. H. Poincaré - Analyse Non Linéaire, 26 (2009), 561-598.  Google Scholar

[2]

J. F. Bonnans and A. Shapiro, "Perturbation Analysis of Optimization Problems," Springer-Verlag, New York, 2000. Google Scholar

[3]

A. V. Dmitruk, Maximum principle for a general optimal control problem with state and regular mixed constraints, Comp. Math. and Modeling, 4 (1993), 364-377. doi: doi:10.1007/BF01128760.  Google Scholar

[4]

G. N. Galbraith and R. B. Vinter, Lipschitz Continuiuty of Optimal Controls for State Constrained Problems, SIAM J. Control and Optimization, 42 (2003), 1727-1744. doi: doi:10.1137/S0363012902404711.  Google Scholar

[5]

W. W. Hager, Lipschitz continuity for constrained processes, SIAM J. Control and Optimization, 17 (1979), 321-338. doi: doi:10.1137/0317026.  Google Scholar

[6]

K. Malanowski, On the regularity of solutions to optimal control problems for systems linear with respect to control variable, Arch. Auto. i Telemech., 23 (1978), 227-241.  Google Scholar

[7]

K. Malanowski, On normality of Lagrange multipliers for state constrained optimal control problems, Optimization, 52 (2003), 75-91. doi: doi:10.1080/0233193021000058940.  Google Scholar

[8]

A. A. Milyutin, A. V. Dmitruk and N. P. Osmolovsky, "Maximum Principle in Optimal Control" (Russian), Moscow State University Press, 2004. Available online at http://www.milyutin.ru/book.html. Google Scholar

[9]

S. M. Robinson and R. H. Day, A sufficient condition for continuity of optimal sets in mathematical programming, J. Math. Anal. Appl., 45 (1974), 506-511. doi: doi:10.1016/0022-247X(74)90089-4.  Google Scholar

[10]

S. M. Robinson, Strongly regular generalized equations, Mathematics of Operations Research, 5 (1980), 43-62. doi: doi:10.1287/moor.5.1.43.  Google Scholar

[11]

I. Shvartsman and R. B. Vinter, Regularity properties of optimal controls for state constrained problems with time-varying control constraints, Nonlinear Analysis: Theory, Methods and Applications, 65 (2006), 448-474.  Google Scholar

[12]

R. B. Vinter, "Optimal Control," Birkhäuser, Boston, 2000. Google Scholar

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