Unmaximized inclusion necessary conditions for nonconvex constrained optimal control problems
ESAIM: Control, Optimisation and Calculus of Variations, Tome 11 (2005) no. 4, pp. 614-632.

Necessary conditions of optimality in the form of Unmaximized Inclusions (UI) are derived for optimal control problems with state constraints. The conditions presented here generalize earlier optimality conditions to problems that may be nonconvex. The derivation of UI-type conditions in the absence of the convexity assumption is of particular importance when deriving necessary conditions for constrained problems. We illustrate this feature by establishing, as an application, optimality conditions for problems that in addition to state constraints incorporate mixed state-control constraints.

DOI : https://doi.org/10.1051/cocv:2005020
Classification : 49K15
Mots clés : optimal control, state constraints, nonsmooth analysis, Euler-Lagrange inclusion
@article{COCV_2005__11_4_614_0,
     author = {de Pinho, Maria Do Ros\'ario and Ferreira, Maria Margarida and Fontes, Fernando},
     title = {Unmaximized inclusion necessary conditions for nonconvex constrained optimal control problems},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {614--632},
     publisher = {EDP-Sciences},
     volume = {11},
     number = {4},
     year = {2005},
     doi = {10.1051/cocv:2005020},
     zbl = {1081.49016},
     mrnumber = {2167877},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv:2005020/}
}
de Pinho, Maria Do Rosário; Ferreira, Maria Margarida; Fontes, Fernando. Unmaximized inclusion necessary conditions for nonconvex constrained optimal control problems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 11 (2005) no. 4, pp. 614-632. doi : 10.1051/cocv:2005020. http://www.numdam.org/articles/10.1051/cocv:2005020/

[1] K.E. Brenen, S.L. Campbell and L.R. Petzold, Numerical Solution of Initial-Value Problems in Differential Algebraic Equations. Classics Appl. Math. SIAM, Philadelphia (1996). | MR 1363258 | Zbl 0844.65058

[2] F.H. Clarke, Optimization and Nonsmooth Analysis. Wiley, New York (1983). Reprinted as Vol. 5 of Classics Appl. Math. SIAM, Philadelphia (1990). | MR 709590 | Zbl 0696.49002

[3] M.D.R. De Pinho, M.M.A. Ferreira and F.A.C.C. Fontes, An Euler-Lagrange inclusion for optimal control problems with state constraints. J. Dynam. Control Syst. 8 (2002) 23-45. | Zbl 1027.49019

[4] M.D.R. De Pinho, M.M.A. Ferreira and F.A.C.C. Fontes, Necessary conditions in Euler-Lagrange inclusion form for constrained nonconvex optimal control problems, in Proc. of the 10th Mediterranean Conference on Control and Automation. Lisbon, Portugal (2002). | Zbl 1027.49019

[5] M.D.R. De Pinho and A. Ilchmann, Weak maximum principle for optimal control problems with mixed constraints. Nonlinear Anal. Theory Appl. 48 (2002) 1179-1196. | Zbl 1019.49024

[6] M.D.R. De Pinho and R.B. Vinter, An Euler-Lagrange inclusion for optimal control problems. IEEE Trans. Aut. Control 40 (1995) 1191-1198. | Zbl 0827.49014

[7] M.D.R. De Pinho and R.B. Vinter, Necessary conditions for optimal control problems involving nonlinear differential algebraic equations. J. Math. Anal. Appl. 212 (1997) 493-516. | Zbl 0891.49013

[8] M.D.R. De Pinho, R.B. Vinter and H. Zheng, A maximum principle for optimal control problems with mixed constraints. IMA J. Math. Control Inform. 18 (2001) 189-205. | Zbl 1103.49307

[9] B.S. Mordukhovich, Maximum principle in problems of time optimal control with nonsmooth constraints. J. Appl. Math. Mech. 40 (1976) 960-969. | Zbl 0362.49017

[10] B.S. Mordukhovich, Approximation Methods in Problems of Optimization and Control. Nakua, Moscow; the 2nd edition to appear in Wiley-Interscience (1988). | MR 945143 | Zbl 0643.49001

[11] R.T. Rockafellar and B. Wets, Variational Analysis. Springer, Berlin (1998). | MR 1491362 | Zbl 0888.49001

[12] R.B. Vinter, Optimal Control. Birkhauser, Boston (2000). | MR 1756410 | Zbl 0952.49001