# Unmaximized inclusion necessary conditions for nonconvex constrained optimal control problems

Maria Do Rosário de Pinho; Maria Margarida Ferreira; Fernando Fontes^{[1]}

- [1] Officina Mathematica, Universidade do Minho, 4800-058 Guimarães, Portugal

ESAIM: Control, Optimisation and Calculus of Variations (2005)

- Volume: 11, Issue: 4, page 614-632
- ISSN: 1292-8119

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topde Pinho, Maria Do Rosário, Ferreira, Maria Margarida, and Fontes, Fernando. "Unmaximized inclusion necessary conditions for nonconvex constrained optimal control problems." ESAIM: Control, Optimisation and Calculus of Variations 11.4 (2005): 614-632. <http://eudml.org/doc/245247>.

@article{dePinho2005,

abstract = {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.},

affiliation = {Officina Mathematica, Universidade do Minho, 4800-058 Guimarães, Portugal},

author = {de Pinho, Maria Do Rosário, Ferreira, Maria Margarida, Fontes, Fernando},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {optimal control; state constraints; nonsmooth analysis; Euler-Lagrange inclusion},

language = {eng},

number = {4},

pages = {614-632},

publisher = {EDP-Sciences},

title = {Unmaximized inclusion necessary conditions for nonconvex constrained optimal control problems},

url = {http://eudml.org/doc/245247},

volume = {11},

year = {2005},

}

TY - JOUR

AU - de Pinho, Maria Do Rosário

AU - Ferreira, Maria Margarida

AU - Fontes, Fernando

TI - Unmaximized inclusion necessary conditions for nonconvex constrained optimal control problems

JO - ESAIM: Control, Optimisation and Calculus of Variations

PY - 2005

PB - EDP-Sciences

VL - 11

IS - 4

SP - 614

EP - 632

AB - 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.

LA - eng

KW - optimal control; state constraints; nonsmooth analysis; Euler-Lagrange inclusion

UR - http://eudml.org/doc/245247

ER -

## References

top- [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). Zbl0844.65058MR1363258
- [2] F.H. Clarke, Optimization and Nonsmooth Analysis. Wiley, New York (1983). Reprinted as Vol. 5 of Classics Appl. Math. SIAM, Philadelphia (1990). Zbl0696.49002MR709590
- [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. Zbl1027.49019
- [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).
- [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. Zbl1019.49024
- [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. Zbl0827.49014
- [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. Zbl0891.49013
- [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. Zbl1103.49307
- [9] B.S. Mordukhovich, Maximum principle in problems of time optimal control with nonsmooth constraints. J. Appl. Math. Mech. 40 (1976) 960–969. Zbl0362.49017
- [10] B.S. Mordukhovich, Approximation Methods in Problems of Optimization and Control. Nakua, Moscow; the 2nd edition to appear in Wiley-Interscience (1988). Zbl0643.49001MR945143
- [11] R.T. Rockafellar and B. Wets, Variational Analysis. Springer, Berlin (1998). Zbl0888.49001MR1491362
- [12] R.B. Vinter, Optimal Control. Birkhauser, Boston (2000). Zbl0952.49001MR1756410

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