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July  2019, 15(3): 1399-1419. doi: 10.3934/jimo.2018101

Necessary optimality conditions for nonautonomous optimal control problems and its applications to bilevel optimal control

1. 

College of Mathematics and Informatics & FJKLMAA, Fujian Normal University, Fuzhou 350108, Fujian, China

2. 

School of Mathematical Sciences, Xiamen University, Xiamen 361005, Fujian, China

* Corresponding author. E-mail address: anlee@xmu.edu.cn

Received  October 2017 Revised  February 2018 Published  July 2018

Fund Project: The research was partially supported by Natural Science Foundation of China (Grant Nos. 11301081, 11671335), Natural Science Foundation of Fujian Province, China (Grant Nos. 2018J01657, 2016J01013) and Fundamental Research Funds for the Central Universities (Grant No. 20720160036).

This paper focuses on the development of necessary optimality conditions for nonautonomous optimal control problems with nonsmooth mixed state and control constraints. In most of the existing results, the necessary optimality conditions for nonautonomous optimal control problems with mixed state and control constraints are derived under the Mangasarian-Fromovitz condition or even stronger. In this paper we derive the necessary optimality conditions for nonautonomous optimal control problems under constraint qualifications which are weaker than Mangasarian-Fromovitz condition. Moreover necessary optimality conditions with an Euler inclusion taking a bounded explicit multiplier form are derived for certain cases. Specifying these results to bilevel optimal control problems with finite-dimensional lower level we obtain necessary optimality conditions under weaker qualification conditions.

Citation: 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
References:
[1]

J. P. Aubin, Lipschitz behavior of solutions to convex minimization problems, Math. Oper. Res., 9 (1994), 87-111.  doi: 10.1287/moor.9.1.87.  Google Scholar

[2]

F. BenitaS. Dempe and P. Mehlitz, Bilevel optimal control problems with pure state constraints and finite-dimensional lower level, SIAM J. Optim., 26 (2016), 564-588.  doi: 10.1137/141000889.  Google Scholar

[3]

F. Benita and P. Mehlitz, Bilevel optimal control with final-state-dependent finite-dimensional lower level, SIAM J. Optim., 26 (2016), 718-752.  doi: 10.1137/15M1015984.  Google Scholar

[4]

P. BettiolA. Boccia and R. B. Vinter, Stratified necessary conditions for differential inclusions with state constraints in the presence of a penalty function, SIAM J. Control Optim., 51 (2013), 3903-3917.  doi: 10.1137/120880914.  Google Scholar

[5]

M. H. A. Biswas and M. R. de Pinho, A nonsmooth maximum principle for optimal control problems with state and mixed constraints-convex case, Discrete Cont. Dyn. Syst. Supplement, (2011), 174-183.   Google Scholar

[6]

F. H. Clarke, Necessary conditions in dynamic optimization, Mem. Amer. Math. Soc., 173 (2005), ⅹ+113 pp. doi: 10.1090/memo/0816.  Google Scholar

[7]

F. H. Clarke, A general theorem on necessary conditions in optimal control, Discrete Cont. Dyn., 29 (2011), 485-503.  doi: 10.3934/dcds.2011.29.485.  Google Scholar

[8]

F. H. Clarke and M. R. de Pinho, Optimal control problems with mixed constraints, SIAM J. Control Optim., 48 (2010), 4500-4524.  doi: 10.1137/090757642.  Google Scholar

[9]

F. H. Clarke, Y. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Springer, New York, 1998.  Google Scholar

[10]

M. R. de Pinho, Mixed constrained control problems, J. Math. Anal. Appl., 278 (2003), 293-307.  doi: 10.1016/S0022-247X(02)00351-7.  Google Scholar

[11]

M. R. de PinhoM. M. A. Ferreira and F. A. C. C. Fontes, Unmaximized inclusion necessary conditions for nonconvex constrained optimal control problems, Optim. Calc. Var., 11 (2005), 614-632.  doi: 10.1051/cocv:2005020.  Google Scholar

[12]

M. R. de Pinho and A. Ilchmann, Weak maximum principle for optimal control problems with mixed constraints, Nonlinear Anal., 48 (2002), 1179-1196.  doi: 10.1016/S0362-546X(01)00094-3.  Google Scholar

[13]

M. R. de Pinho and J. F. Rosenblueth, Necessary conditions for constrained problems under Mangasarian-Fromowitz conditions, SIAM J. Control Optim., 47 (2008), 535-552.  doi: 10.1137/060663623.  Google Scholar

[14]

M. R. de Pinho and R. B. Vinter, An Euler-Lagrange inclusion for optimal control problems, IEEE Trans. Automat. Control, 40 (1995), 1191-1198.  doi: 10.1109/9.400492.  Google Scholar

[15]

M. R. de PinhoR. B. Vinter and H. Zheng, A maximum principle for optimal control problems with mixed constraints, IMA J. Math. Control Inform., 18 (2001), 189-205.  doi: 10.1093/imamci/18.2.189.  Google Scholar

[16]

S. Dempe and J. Dutta, Is bilevel programming a special case of a mathematical program with complementarity constraints, Math. Program., 131 (2012), 37-48.  doi: 10.1007/s10107-010-0342-1.  Google Scholar

[17]

H. Gfrerer and J. V. Outrata, On Lipschitzian properties of implicit multifunctions, SIAM J. Optim., 26 (2016), 2160-2189.  doi: 10.1137/15M1052299.  Google Scholar

[18]

L. GuoJ. J. Ye and J. Zhang, Mathematical programs with geometric constraints in Banach spaces: enhanced optimality, exact penalty, and sensitivity, SIAM J. Optim., 23 (2013), 2295-2319.  doi: 10.1137/130910956.  Google Scholar

[19]

L. GuoG. H. LinJ. J. Ye and J. Zhang, Sensitivity Analysis of the value function for parametric mathematical programs with equilibrium constraints, SIAM J. Optim., 24 (2014), 1206-1237.  doi: 10.1137/130929783.  Google Scholar

[20]

S. C. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis, Vol. I: Theory, Springer, 1997. doi: 10.1007/978-1-4615-6359-4.  Google Scholar

[21]

A. D. Ioffe and J. V. Outrata, On metric and calmness qualification in subdifferential calculus, Set-Valued Anal., 16 (2008), 199-227.  doi: 10.1007/s11228-008-0076-x.  Google Scholar

[22]

A. Li and J. J. Ye, Necessary optimality conditions for optimal control problems with nonsmooth mixed state and control constraints, Set-Valued Var. Anal., 24 (2016), 449-470.  doi: 10.1007/s11228-015-0358-z.  Google Scholar

[23]

A. Li and J. J. Ye, Necessary optimality conditions for implicit control systems with applications to control of differential algebraic equations, Set-Valued Var. Anal., 26 (2018), 179-203.  doi: 10.1007/s11228-017-0444-5.  Google Scholar

[24]

B. S. Mordukhovich, Generalized differential calculus for nonsmooth and set-valued mappings, J. Math. Anal. Appl., 183 (1994), 250-288.  doi: 10.1006/jmaa.1994.1144.  Google Scholar

[25]

B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, Vol. I: Basic Theory, Springer-Verlag, Berlin, 2006.  Google Scholar

[26]

R. T. Rockafellar and R. J. -B. Wets, Variational Analysis, Springer, Berlin, 1998. doi: 10.1007/978-3-642-02431-3.  Google Scholar

[27]

J. J. Ye, Necessary and sufficient optimality conditions for mathematical programs with equilibrium constraints, J. Math. Anal. Appl., 307 (2005), 350-369.  doi: 10.1016/j.jmaa.2004.10.032.  Google Scholar

[28]

J. J. Ye and D. L. Zhu, New necessary optimality conditions for bilevel programs by combining MPEC and the value function approaches, SIAM J. Optim., 20 (2010), 1885-1905.  doi: 10.1137/080725088.  Google Scholar

show all references

References:
[1]

J. P. Aubin, Lipschitz behavior of solutions to convex minimization problems, Math. Oper. Res., 9 (1994), 87-111.  doi: 10.1287/moor.9.1.87.  Google Scholar

[2]

F. BenitaS. Dempe and P. Mehlitz, Bilevel optimal control problems with pure state constraints and finite-dimensional lower level, SIAM J. Optim., 26 (2016), 564-588.  doi: 10.1137/141000889.  Google Scholar

[3]

F. Benita and P. Mehlitz, Bilevel optimal control with final-state-dependent finite-dimensional lower level, SIAM J. Optim., 26 (2016), 718-752.  doi: 10.1137/15M1015984.  Google Scholar

[4]

P. BettiolA. Boccia and R. B. Vinter, Stratified necessary conditions for differential inclusions with state constraints in the presence of a penalty function, SIAM J. Control Optim., 51 (2013), 3903-3917.  doi: 10.1137/120880914.  Google Scholar

[5]

M. H. A. Biswas and M. R. de Pinho, A nonsmooth maximum principle for optimal control problems with state and mixed constraints-convex case, Discrete Cont. Dyn. Syst. Supplement, (2011), 174-183.   Google Scholar

[6]

F. H. Clarke, Necessary conditions in dynamic optimization, Mem. Amer. Math. Soc., 173 (2005), ⅹ+113 pp. doi: 10.1090/memo/0816.  Google Scholar

[7]

F. H. Clarke, A general theorem on necessary conditions in optimal control, Discrete Cont. Dyn., 29 (2011), 485-503.  doi: 10.3934/dcds.2011.29.485.  Google Scholar

[8]

F. H. Clarke and M. R. de Pinho, Optimal control problems with mixed constraints, SIAM J. Control Optim., 48 (2010), 4500-4524.  doi: 10.1137/090757642.  Google Scholar

[9]

F. H. Clarke, Y. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Springer, New York, 1998.  Google Scholar

[10]

M. R. de Pinho, Mixed constrained control problems, J. Math. Anal. Appl., 278 (2003), 293-307.  doi: 10.1016/S0022-247X(02)00351-7.  Google Scholar

[11]

M. R. de PinhoM. M. A. Ferreira and F. A. C. C. Fontes, Unmaximized inclusion necessary conditions for nonconvex constrained optimal control problems, Optim. Calc. Var., 11 (2005), 614-632.  doi: 10.1051/cocv:2005020.  Google Scholar

[12]

M. R. de Pinho and A. Ilchmann, Weak maximum principle for optimal control problems with mixed constraints, Nonlinear Anal., 48 (2002), 1179-1196.  doi: 10.1016/S0362-546X(01)00094-3.  Google Scholar

[13]

M. R. de Pinho and J. F. Rosenblueth, Necessary conditions for constrained problems under Mangasarian-Fromowitz conditions, SIAM J. Control Optim., 47 (2008), 535-552.  doi: 10.1137/060663623.  Google Scholar

[14]

M. R. de Pinho and R. B. Vinter, An Euler-Lagrange inclusion for optimal control problems, IEEE Trans. Automat. Control, 40 (1995), 1191-1198.  doi: 10.1109/9.400492.  Google Scholar

[15]

M. R. de PinhoR. B. Vinter and H. Zheng, A maximum principle for optimal control problems with mixed constraints, IMA J. Math. Control Inform., 18 (2001), 189-205.  doi: 10.1093/imamci/18.2.189.  Google Scholar

[16]

S. Dempe and J. Dutta, Is bilevel programming a special case of a mathematical program with complementarity constraints, Math. Program., 131 (2012), 37-48.  doi: 10.1007/s10107-010-0342-1.  Google Scholar

[17]

H. Gfrerer and J. V. Outrata, On Lipschitzian properties of implicit multifunctions, SIAM J. Optim., 26 (2016), 2160-2189.  doi: 10.1137/15M1052299.  Google Scholar

[18]

L. GuoJ. J. Ye and J. Zhang, Mathematical programs with geometric constraints in Banach spaces: enhanced optimality, exact penalty, and sensitivity, SIAM J. Optim., 23 (2013), 2295-2319.  doi: 10.1137/130910956.  Google Scholar

[19]

L. GuoG. H. LinJ. J. Ye and J. Zhang, Sensitivity Analysis of the value function for parametric mathematical programs with equilibrium constraints, SIAM J. Optim., 24 (2014), 1206-1237.  doi: 10.1137/130929783.  Google Scholar

[20]

S. C. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis, Vol. I: Theory, Springer, 1997. doi: 10.1007/978-1-4615-6359-4.  Google Scholar

[21]

A. D. Ioffe and J. V. Outrata, On metric and calmness qualification in subdifferential calculus, Set-Valued Anal., 16 (2008), 199-227.  doi: 10.1007/s11228-008-0076-x.  Google Scholar

[22]

A. Li and J. J. Ye, Necessary optimality conditions for optimal control problems with nonsmooth mixed state and control constraints, Set-Valued Var. Anal., 24 (2016), 449-470.  doi: 10.1007/s11228-015-0358-z.  Google Scholar

[23]

A. Li and J. J. Ye, Necessary optimality conditions for implicit control systems with applications to control of differential algebraic equations, Set-Valued Var. Anal., 26 (2018), 179-203.  doi: 10.1007/s11228-017-0444-5.  Google Scholar

[24]

B. S. Mordukhovich, Generalized differential calculus for nonsmooth and set-valued mappings, J. Math. Anal. Appl., 183 (1994), 250-288.  doi: 10.1006/jmaa.1994.1144.  Google Scholar

[25]

B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, Vol. I: Basic Theory, Springer-Verlag, Berlin, 2006.  Google Scholar

[26]

R. T. Rockafellar and R. J. -B. Wets, Variational Analysis, Springer, Berlin, 1998. doi: 10.1007/978-3-642-02431-3.  Google Scholar

[27]

J. J. Ye, Necessary and sufficient optimality conditions for mathematical programs with equilibrium constraints, J. Math. Anal. Appl., 307 (2005), 350-369.  doi: 10.1016/j.jmaa.2004.10.032.  Google Scholar

[28]

J. J. Ye and D. L. Zhu, New necessary optimality conditions for bilevel programs by combining MPEC and the value function approaches, SIAM J. Optim., 20 (2010), 1885-1905.  doi: 10.1137/080725088.  Google Scholar

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