Here we derive a variant of the nonsmooth maximum principle for optimal control problems with both pure state and mixed state and control constraints. Our necessary conditions include a Weierstrass condition together with an Euler adjoint inclusion involving the joint subdifferentials with respect to both state and control, generalizing previous results in [M.d.R. de Pinho, M.M.A. Ferreira, F.A.C.C. Fontes, Unmaximized inclusion necessary conditions for nonconvex constrained optimal control problems. ESAIM: COCV 11 (2005) 614–632]. A notable feature is that our main results are derived combining old techniques with recent results. We use a well known penalization technique for state constrained problem together with an appeal to a recent nonsmooth maximum principle for problems with mixed constraints.

DOI: 10.1051/cocv/2014047

Keywords: Optimal control, state and mixed constraints, maximum principle

^{1}; do Rosario de Pinho, Maria

^{2}

@article{COCV_2015__21_4_939_0, author = {Haider Ali Biswas, Md. and do Rosario de Pinho, Maria}, title = {A maximum principle for optimal control problems with state and mixed constraints}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {939--957}, publisher = {EDP-Sciences}, volume = {21}, number = {4}, year = {2015}, doi = {10.1051/cocv/2014047}, mrnumber = {3395750}, zbl = {1330.49018}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2014047/} }

TY - JOUR AU - Haider Ali Biswas, Md. AU - do Rosario de Pinho, Maria TI - A maximum principle for optimal control problems with state and mixed constraints JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2015 SP - 939 EP - 957 VL - 21 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2014047/ DO - 10.1051/cocv/2014047 LA - en ID - COCV_2015__21_4_939_0 ER -

%0 Journal Article %A Haider Ali Biswas, Md. %A do Rosario de Pinho, Maria %T A maximum principle for optimal control problems with state and mixed constraints %J ESAIM: Control, Optimisation and Calculus of Variations %D 2015 %P 939-957 %V 21 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2014047/ %R 10.1051/cocv/2014047 %G en %F COCV_2015__21_4_939_0

Haider Ali Biswas, Md.; do Rosario de Pinho, Maria. A maximum principle for optimal control problems with state and mixed constraints. ESAIM: Control, Optimisation and Calculus of Variations, Volume 21 (2015) no. 4, pp. 939-957. doi : 10.1051/cocv/2014047. http://archive.numdam.org/articles/10.1051/cocv/2014047/

Pontryagin Maximum Principle revisited with feedbacks. Eur. J. Control 17 (2011) 46–54. | DOI | MR | Zbl

,A.V. Arutyunov, Optimality Conditions. Abnormal and Degenerate Problems, 1st edition. Kluwer Academic Publishers, Dordrecht (2000). | MR | Zbl

Maximum principle in problems with mixed constraints under weak assumptions of regularity. J. Optim. 59 (2010) 1067–1083. | DOI | MR | Zbl

, and ,Lipschitz regularity of solution map of control systems with multiple state constraints. Discrete Contin. Dyn. Syst. A 32 (2012) 1–26. | DOI | MR | Zbl

and ,Stratified necessary conditions for differential inclusions with state constraints. SIAM J. Control Optim. 51 (2013) 3903–3917. | DOI | MR | Zbl

, and ,M.H.A. Biswas and M.d.R. de Pinho, A Variant of Nonsmooth Maximum Principle for State Constrained Problems. IEEE Proc. of 51th CDC (CDC12) (2012) 7685–7690.

M.H.A. Biswas, Necessary Conditions for Optimal Control Problems with State Constraints: Theory and Applications. Ph.D. thesis, University of Porto, Faculty of Engineering, DEEC, PDEEC (2013).

F. Clarke, Optimization and Nonsmooth Analysis. John Wiley, New York (1993). | MR | Zbl

F. Clarke, Y. Ledyaev, R.J. Stern and P.R. Wolenski, Nonsmooth Analysis and Control Theory. Springer-Verlag, New York (1998). | MR | Zbl

F. Clarke, Necessary conditions in dynamic optimization. Mem. Amer. Math. Soc. (2005). | MR | Zbl

The Nonsmooth Maximum Principle. Control Cybern. 38 (2009) 1151–1168. | MR | Zbl

and ,Optimal control problems with mixed constraints. SIAM J. Control Optim. 48 (2010) 4500–4524. | DOI | MR | Zbl

and ,An extension of the Schwarzkopf multiplier rule in optimal control. SIAM J. Control Optim. 49 (2011) 599–610. | DOI | MR | Zbl

, and ,An Euler-Lagrange inclusion for optimal control problems. IEEE Trans. Automat. Control 40 (1995) 1191–1198. | DOI | MR | Zbl

and ,An Euler-Lagrange inclusion for optimal control problems with state constraints. Dyn. Control Syst. 8 (2002) 23–45. | DOI | MR | Zbl

, and ,Unmaximized inclusion necessary conditions for nonconvex constrained optimal control problems. ESAIM: COCV 11 (2005) 614–632. | Numdam | MR | Zbl

, and ,A weak maximum principle for optimal control problems with nonsmooth mixed constraints. Set-Valued Var. Anal. 17 (2009) 203–221. | DOI | MR | Zbl

, and ,Maximum principle for implicit control systems. Appl. Math. Optim. 40 (1999) 79–103. | DOI | MR | Zbl

and ,Maximum principle for the general optimal control problem with phase and regular mixed constraints. Comput. Math. Model. 4 (1993) 364–377. | DOI | MR | Zbl

,M.R. Hestenes, Calculus of Variations and Optimal Control Theory. John Wiley, New York (1966). | MR | Zbl

Nondegenerate necessary conditions for nonconvex optimal control problems with state constraints. J. Math. Anal. Appl. 233 (1999) 116–129. | DOI | MR | Zbl

, and ,Normal forms of necessary conditions for dynamic optimization problems with pathwise inequality constraints. J. Math. Anal. Appl. 399 (2013) 27–37. | DOI | MR | Zbl

and ,Regularity of minimizers and of adjoint states for optimal control problems under state constraints. J. Convex Anal. 13 (2006) 299–328. | MR | Zbl

,I. Kornienko and M.d.R. de Pinho, Differential inclusion approach for mixed constrained problems revisited. Prepublished in: Set Valued Var. Anal. (2014) DOI:. | DOI | MR

I. Kornienko and M.d.R. de Pinho, Properties of some control systems with mixed constraints in the form of inequalities. Report, ISR, DEEC, FEUP (2013). Available at http://paginas.fe.up.pt/˜mrpinho/

B. Mordukhovich, Variational analysis and generalized differentiation. Basic Theory. Fundamental Principles of Mathematical Sciences 330. Springer-Verlag, Berlin (2006). | MR | Zbl

The Maximum Principle of optimal control: A history of ingenious ideas and missed opportunities. Control Cybern. 38 (2009) 973–995. | MR | Zbl

and ,L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze and E.F. Mischenko, The Mathematical Theory of Optimal Processes. John Wiley, New York (1962). | MR

R.T. Rockafellar and B. Wets, Variational Analysis. Vol. 317 of Grundlehren Math. Wiss. Springer-Verlag, Berlin (1998). | MR | Zbl

A maximum principle for nonsmooth optimal-control problems with state constraints. J. Math. Anal. Appl. 89 (1982) 212–232. | DOI | MR | Zbl

and ,R.B. Vinter, Optimal Control. Birkhäuser, Boston (2000). | MR | Zbl

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