# 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 & Continuous Dynamical Systems - A, 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.   Google Scholar [2] J. F. Bonnans and A. Shapiro, "Perturbation Analysis of Optimization Problems,", Springer-Verlag, (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.  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.  doi: doi:10.1137/S0363012902404711.  Google Scholar [5] W. W. Hager, Lipschitz continuity for constrained processes,, SIAM J. Control and Optimization, 17 (1979), 321.  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.   Google Scholar [7] K. Malanowski, On normality of Lagrange multipliers for state constrained optimal control problems,, Optimization, 52 (2003), 75.  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).   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.  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.  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, 65 (2006), 448.   Google Scholar [12] R. B. Vinter, "Optimal Control,", Birkhäuser, (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.   Google Scholar [2] J. F. Bonnans and A. Shapiro, "Perturbation Analysis of Optimization Problems,", Springer-Verlag, (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.  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.  doi: doi:10.1137/S0363012902404711.  Google Scholar [5] W. W. Hager, Lipschitz continuity for constrained processes,, SIAM J. Control and Optimization, 17 (1979), 321.  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.   Google Scholar [7] K. Malanowski, On normality of Lagrange multipliers for state constrained optimal control problems,, Optimization, 52 (2003), 75.  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).   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.  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.  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, 65 (2006), 448.   Google Scholar [12] R. B. Vinter, "Optimal Control,", Birkhäuser, (2000).   Google Scholar
 [1] Sofia O. Lopes, Fernando A. C. C. Fontes, Maria do Rosário de Pinho. On constraint qualifications for nondegenerate necessary conditions of optimality applied to optimal control problems. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 559-575. doi: 10.3934/dcds.2011.29.559 [2] Jianxiong Ye, An Li. Necessary optimality conditions for nonautonomous optimal control problems and its applications to bilevel optimal control. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1399-1419. doi: 10.3934/jimo.2018101 [3] Shahlar F. Maharramov. Necessary optimality conditions for switching control problems. Journal of Industrial & Management Optimization, 2010, 6 (1) : 47-55. doi: 10.3934/jimo.2010.6.47 [4] Francis Clarke. A general theorem on necessary conditions in optimal control. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 485-503. doi: 10.3934/dcds.2011.29.485 [5] Vincenzo Basco, Piermarco Cannarsa, Hélène Frankowska. Necessary conditions for infinite horizon optimal control problems with state constraints. Mathematical Control & Related Fields, 2018, 8 (3&4) : 535-555. doi: 10.3934/mcrf.2018022 [6] Stepan Sorokin, Maxim Staritsyn. Feedback necessary optimality conditions for a class of terminally constrained state-linear variational problems inspired by impulsive control. Numerical Algebra, Control & Optimization, 2017, 7 (2) : 201-210. doi: 10.3934/naco.2017014 [7] M. Soledad Aronna. Second order necessary and sufficient optimality conditions for singular solutions of partially-affine control problems. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1233-1258. doi: 10.3934/dcdss.2018070 [8] Lucas Bonifacius, Ira Neitzel. Second order optimality conditions for optimal control of quasilinear parabolic equations. Mathematical Control & Related Fields, 2018, 8 (1) : 1-34. doi: 10.3934/mcrf.2018001 [9] Heinz Schättler, Urszula Ledzewicz, Helmut Maurer. Sufficient conditions for strong local optimality in optimal control problems with $L_{2}$-type objectives and control constraints. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2657-2679. doi: 10.3934/dcdsb.2014.19.2657 [10] 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 & Continuous Dynamical Systems - A, 2015, 35 (9) : 4323-4343. doi: 10.3934/dcds.2015.35.4323 [11] Hongwei Lou, Jiongmin Yong. Second-order necessary conditions for optimal control of semilinear elliptic equations with leading term containing controls. Mathematical Control & Related Fields, 2018, 8 (1) : 57-88. doi: 10.3934/mcrf.2018003 [12] Hongwei Lou. Second-order necessary/sufficient conditions for optimal control problems in the absence of linear structure. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1445-1464. doi: 10.3934/dcdsb.2010.14.1445 [13] Andrei V. Dmitruk, Nikolai P. Osmolovski. Necessary conditions for a weak minimum in a general optimal control problem with integral equations on a variable time interval. Mathematical Control & Related Fields, 2017, 7 (4) : 507-535. doi: 10.3934/mcrf.2017019 [14] Liangquan Zhang, Qing Zhou, Juan Yang. Necessary condition for optimal control of doubly stochastic systems. Mathematical Control & Related Fields, 2020, 10 (2) : 379-403. doi: 10.3934/mcrf.2020002 [15] Thierry Horsin, Peter I. Kogut, Olivier Wilk. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. II. Approximation of solutions and optimality conditions. Mathematical Control & Related Fields, 2016, 6 (4) : 595-628. doi: 10.3934/mcrf.2016017 [16] Ciro D'Apice, Olha P. Kupenko, Rosanna Manzo. On boundary optimal control problem for an arterial system: First-order optimality conditions. Networks & Heterogeneous Media, 2018, 13 (4) : 585-607. doi: 10.3934/nhm.2018027 [17] Vladimir Gaitsgory, Alex Parkinson, Ilya Shvartsman. Linear programming based optimality conditions and approximate solution of a deterministic infinite horizon discounted optimal control problem in discrete time. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1743-1767. doi: 10.3934/dcdsb.2018235 [18] Piernicola Bettiol, Hélène Frankowska. Lipschitz regularity of solution map of control systems with multiple state constraints. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 1-26. doi: 10.3934/dcds.2012.32.1 [19] Piernicola Bettiol, Nathalie Khalil. Necessary optimality conditions for average cost minimization problems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2093-2124. doi: 10.3934/dcdsb.2019086 [20] Omid S. Fard, Javad Soolaki, Delfim F. M. Torres. A necessary condition of Pontryagin type for fuzzy fractional optimal control problems. Discrete & Continuous Dynamical Systems - S, 2018, 11 (1) : 59-76. doi: 10.3934/dcdss.2018004

2019 Impact Factor: 1.338