The paper deals with an optimal control problem with a scalar first-order state constraint and a scalar control. In presence of (nonessential) touch points, the arc structure of the trajectory is not stable. Under some reasonable assumptions, we show that boundary arcs are structurally stable, and that touch point can either remain so, vanish or be transformed into a single boundary arc. Assuming a weak second-order optimality condition (equivalent to uniform quadratic growth), stability and sensitivity results are given. The main tools are the study of a quadratic tangent problem and the notion of strong regularity. Those results enable us to design a new continuation algorithm, presented at the end of the paper, that handles automatically changes in the structure of the trajectory.
Mots-clés : optimal control, first-order state constraint, strong regularity, sensitivity analysis, touch point, homotopy method
@article{COCV_2008__14_4_825_0,
author = {Bonnans, Joseph Fr\'ed\'eric and Hermant, Audrey},
title = {Stability and sensitivity analysis for optimal control problems with a first-order state constraint and application to continuation methods},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {825--863},
publisher = {EDP-Sciences},
volume = {14},
number = {4},
year = {2008},
doi = {10.1051/cocv:2008016},
mrnumber = {2451799},
zbl = {1148.49026},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/cocv:2008016/}
}
TY - JOUR AU - Bonnans, Joseph Frédéric AU - Hermant, Audrey TI - Stability and sensitivity analysis for optimal control problems with a first-order state constraint and application to continuation methods JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2008 SP - 825 EP - 863 VL - 14 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv:2008016/ DO - 10.1051/cocv:2008016 LA - en ID - COCV_2008__14_4_825_0 ER -
%0 Journal Article %A Bonnans, Joseph Frédéric %A Hermant, Audrey %T Stability and sensitivity analysis for optimal control problems with a first-order state constraint and application to continuation methods %J ESAIM: Control, Optimisation and Calculus of Variations %D 2008 %P 825-863 %V 14 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv:2008016/ %R 10.1051/cocv:2008016 %G en %F COCV_2008__14_4_825_0
Bonnans, Joseph Frédéric; Hermant, Audrey. Stability and sensitivity analysis for optimal control problems with a first-order state constraint and application to continuation methods. ESAIM: Control, Optimisation and Calculus of Variations, Tome 14 (2008) no. 4, pp. 825-863. doi : 10.1051/cocv:2008016. http://archive.numdam.org/articles/10.1051/cocv:2008016/
[1] and , Numerical continuation methods, Springer Series in Computational Mathematics 13. Springer-Verlag, Berlin (1990). | MR | Zbl
[2] , and , Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York (2000). | MR | Zbl
[3] and , Abort landing in windshear: optimal control problem with third-order state constraint and varied switching structure. J. Optim. Theory Appl. 85 (1995) 21-57. | MR | Zbl
[4] and , Conditions d'optimalité du second ordre nécessaires ou suffisantes pour les problèmes de commande optimale avec une contrainte sur l'état et une commande scalaires. C. R. Math. Acad. Sci. Paris 343 (2006) 473-478. | MR | Zbl
[5] and , Second-order analysis for optimal control problems with pure state constraints and mixed control-state constraints. Ann. Inst. H. Poincaré Anal. Non Linéaire (to appear). | EuDML | Numdam | MR | Zbl
[6] and , Well-posedness of the shooting algorithm for state constrained optimal control problems with a single constraint and control. SIAM J. Control Optim. 46 (2007) 1398-1430. | MR | Zbl
[7] and , No gap second order optimality conditions for optimal control problems with a single state constraint and control. Math. Programming, Ser. B (2007) DOI: 10.1007/s10107-007-0167-8. | MR | Zbl
[8] and , Perturbation analysis of optimization problems. Springer-Verlag, New York (2000). | MR | Zbl
[9] , and , Optimal programming problems with inequality constraints I: Necessary conditions for extremal solutions. AIAA Journal 1 (1963) 2544-2550. | MR | Zbl
[10] , and , Abort landing in the presence of windshear as a minimax optimal control problem. II. Multiple shooting and homotopy. J. Optim. Theory Appl. 70 (1991) 223-254. | MR | Zbl
[11] , Newton methods for nonlinear problems, Affine invariance and adaptive algorithms, Springer Series in Computational Mathematics 35. Springer-Verlag, Berlin (2004). | MR | Zbl
[12] and , Lipschitzian stability for state constrained nonlinear optimal control. SIAM J. Control Optim. 36 (1998) 698-718 (electronic). | MR | Zbl
[13] and , Linear operators, Vols. I and II. Interscience, New York (1958), (1963). | MR | Zbl
[14] and , Homotopy method for minimum consumption orbit transfer problem. ESAIM: COCV 12 (2006) 294-310 (electronic). | EuDML | Numdam | MR | Zbl
[15] , Lipschitz continuity for constrained processes. SIAM J. Control Optim. 17 (1979) 321-338. | MR | Zbl
[16] , How to differentiate the projection on a convex set in Hilbert space. Some applications to variational inequalities. J. Math. Soc. Japan 29 (1977) 615-631. | MR | Zbl
[17] , and , A survey of the maximum principles for optimal control problems with state constraints. SIAM Review 37 (1995) 181-218. | MR | Zbl
[18] and , Theory of Extremal Problems. North-Holland Publishing Company, Amsterdam (1979). Russian Edition: Nauka, Moscow (1974). | MR | Zbl
[19] , and , New necessary conditions of optimality for control problems with state-variable inequality contraints. J. Math. Anal. Appl. 35 (1971) 255-284. | MR | Zbl
[20] , Two-norm approach in stability and sensitivity analysis of optimization and optimal control problems. Adv. Math. Sci. Appl. 2 (1993) 397-443. | MR | Zbl
[21] , Stability and sensitivity of solutions to nonlinear optimal control problems. J. Appl. Math. Optim. 32 (1995) 111-141. | MR | Zbl
[22] , Sufficient optimality conditions for optimal control subject to state constraints. SIAM J. Control Optim. 35 (1997) 205-227. | MR | Zbl
[23] and , Sensitivity analysis for state constrained optimal control problems. Discrete Contin. Dynam. Systems 4 (1998) 241-272. | MR | Zbl
[24] and , An application of PL continuation methods to singular arcs problems, in Recent advances in optimization, Lect. Notes Econom. Math. Systems 563, Springer, Berlin (2006) 163-186. | MR | Zbl
[25] , On the minimum principle for optimal control problems with state constraints. Schriftenreihe des Rechenzentrum 41, Universität Münster (1979).
[26] and , Solution differentiability for nonlinear parametric control problems. SIAM J. Control Optim. 32 (1994) 1542-1554. | MR | Zbl
[27] , Contrôle dans les inéquations variationnelles elliptiques. J. Funct. Anal. 22 (1976) 130-185. | MR | Zbl
[28] , , and , The mathematical theory of optimal processes. Translated from the Russian by K.N. Trirogoff; L.W. Neustadt Ed., Interscience Publishers John Wiley & Sons, Inc. New York-London (1962). | MR | Zbl
[29] , First order conditions for general nonlinear optimization. SIAM J. Appl. Math. 30 (1976) 597-607. | MR | Zbl
[30] , Stability theorems for systems of inequalities, part II: Differentiable nonlinear systems. SIAM J. Numer. Anal. 13 (1976) 497-513. | MR | Zbl
[31] , Strongly regular generalized equations. Math. Oper. Res. 5 (1980) 43-62. | MR | Zbl
[32] , Sensitivity analysis of control constrained optimal control problems for distributed parameter systems. SIAM J. Control Optim. 25 (1987) 1542-1556. | MR | Zbl
[33] and , Introduction to Numerical Analysis. Springer-Verlag, New York (1993). | MR | Zbl
Cité par Sources :
