# American Institute of Mathematical Sciences

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

 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.
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 and 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. [2] J. F. Bonnans and A. Shapiro, "Perturbation Analysis of Optimization Problems," Springer-Verlag, New York, 2000. [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. [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. [5] W. W. Hager, Lipschitz continuity for constrained processes, SIAM J. Control and Optimization, 17 (1979), 321-338. doi: doi:10.1137/0317026. [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. [7] K. Malanowski, On normality of Lagrange multipliers for state constrained optimal control problems, Optimization, 52 (2003), 75-91. doi: doi:10.1080/0233193021000058940. [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. [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. [10] S. M. Robinson, Strongly regular generalized equations, Mathematics of Operations Research, 5 (1980), 43-62. doi: doi:10.1287/moor.5.1.43. [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. [12] R. B. Vinter, "Optimal Control," Birkhäuser, Boston, 2000.

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. [2] J. F. Bonnans and A. Shapiro, "Perturbation Analysis of Optimization Problems," Springer-Verlag, New York, 2000. [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. [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. [5] W. W. Hager, Lipschitz continuity for constrained processes, SIAM J. Control and Optimization, 17 (1979), 321-338. doi: doi:10.1137/0317026. [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. [7] K. Malanowski, On normality of Lagrange multipliers for state constrained optimal control problems, Optimization, 52 (2003), 75-91. doi: doi:10.1080/0233193021000058940. [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. [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. [10] S. M. Robinson, Strongly regular generalized equations, Mathematics of Operations Research, 5 (1980), 43-62. doi: doi:10.1287/moor.5.1.43. [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. [12] R. B. Vinter, "Optimal Control," Birkhäuser, Boston, 2000.
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