Pontryagin Maximum Principle for coupled slow and fast systems

• Autorzy: Artstein Z.
• Języki publikacji: EN

Abstrakty

EN

When slow and fast controlled dynamics are coupled, the variational limit, as the ratio of time scales grows, is best depicted as a trajectory in a probability measures space. The effective control is then an invariant measure on the fast state-control space. The paper presents the form of the Pontryagin Maximum Principle for this variational limit and examines its relation to the Maximum Principle of the perturbed system.

Słowa kluczowe

EN

optimal control; singular perturbations; maximum principle

• Wydawca: Systems Research Institute, Polish Academy of Sciences
• Czasopismo: Control and Cybernetics
• Rocznik: 2009
• Tom: Vol. 38, no 4A
• Strony: 1003--1019
• Opis fizyczny: Bibliogr. 31 poz.

Twórcy

autor

Artstein Z.

• Department of Mathematics, The Weizmann Institute of Science, Rehovot 76100, Israel,

Bibliografia

• ALVAREZ, O. and BARDI, M. (2009) Ergodicity, stabilization and singular perturbations for Bellman-Isaacs equations. Memoirs of the Amer. Math. Soc., to appear.
• ARTSTEIN, Z. (1993) Chattering variational limits of control systems. Forum Mathematicum 5, 369-403.
• ARTSTEIN, Z. (2000) The chattering limit of singularly perturbed optimal control problems. Proceedings of the 39th IEEE Conference on Decision and Control, Sydney, Australia, December 2000, 564-569.
• ARTSTEIN, Z. (2002a) On singularly perturbed ordinary differential equations with measure-valued limits. Mathematica Bohemica 127, 139-152.
• ARTSTEIN, Z. (2002b) An occupational measure solution to a singularly perturbed optimal control problem. Control and Cybernetics 31, 623-642.
• ARTSTEIN, Z. (2004a) On the value function of singularly perturbed optimal control systems. Proceedings of the 43rd IEEE Conference on Decision and Control, Paradise Island, Bahamas, 432-437.
• ARTSTEIN, Z. (2004b) Invariant measures and their projections in nonautonomous dynamical systems. Stochastics and Dynamics 4, 439-459.
• ARTSTEIN, Z. and GAITSGORY, V. (1997) Tracking fast trajectories along a slow dynamics: A singular perturbations approach. SIAM J. Control and Optimization 35, 1487-1507.
• ARTSTEIN, Z. and GAITSGORY, V. (2000) The value function of singularly perturbed control systems. Applied Mathematics and Optimization 41, 425-445.
• ARTSTEIN, Z. and LEIZAROWITZ, A. (2002) Singularly perturbed control systems with one-dimensional fast dynamics. SIAM J. Control and Optimization 41, 641-658.
• ARTSTEIN, Z. and VIGODNER, A. (1996) Singularly perturbed ordinary differential equations with dynamic limits. Proceedings of the Royal Society of Edinburgh 126A, 541-569.
• CLARKE, F. (1983) Optimization and Nonsmooth Analysis. Wiley-Interscience Publication, New York. Reprinted as Classics in Applied Mathematics 5, SIAM Publications, Philadelphia, 1990.
• CLARKE, F. (2005a) Necessary conditions in dynamic optimization. Memoirs Amer. Math. Soc. 173, 816.
• CLARKE, F. (2005b) The maximum principle in optimal control, then and now. Control and Cybernetics 34, 3, 709-722.
• DONTCHEV, A.L. (1983) Perturbations, Approximations and Sensitivity Analysis of Optimal Control Systems. Lecture Notes in Control and Information Sciences 52, Springer-Verlag, Berlin.
• DONTCHEV, A.L. and VELIOV, V.M. (1983) Singular perturbation in Mayer’s problem for linear systems. SIAM J. Control Optimization 21, 566-581.
• DONTCHEV, A.L. and VELIOV, V.M. (1985) Singular perturbations in linear control systems with weakly coupled stable and unstable fast subsystems. J. Math. Anal. Appl. 110, 1-30.
• GAITSGORY, V. (1986) On the use of the averaging method in control problems. Differential Equations 22, 1290-1299.
• GAITSGORY, V. (1991) Control of systems with fast and slow motions (in Russian). Nauka, Moscow.
• GAITSGORY, V. (1992) Suboptimization of singularly perturbed control systems. SIAM J. Control Optimization 30, 1228-1249.
• GAITSGORY, V. (2004) On a representation of the limit occupational measures set of a control system with applications to singularly perturbed control systems. SIAM J. Control Optim. 43, 1, 325-340.
• GAITSGORY, V. and GRAMMEL, G. (1997) On the construction of asymptotically optimal controls for singularly perturbed systems. Systems Control Letters 30, 139-147.
• GAITSGORY, V. and LEIZAROWITZ, A. (1999) Limit occupational measures set for a control system and averaging of singularly perturbed control systems. J. Math. Anal. Appl. 233, 461-475.
• KOKOTOVIC, P.V., KHALIL, H.K. and O’REILLY, J. (1999) (1986) Singular Perturbation Methods in Control: Analysis and Design. Academic Press, London. Reprinted as Classics in Applied Mathematics 25, SIAM Publications, Philadelphia,
• LEE, E.B. and MARKUS, L. (1967) Foundation of Optimal Control Theory. Wiley, New York.
• LEIZAROWITZ, A. (2002a) Order reduction is invalid for singularly perturbed control problems with vector fast variable. Math. Control Signals and Systems 15, 101-119.
• LEIZAROWITZ, A. (2002b) On the order reduction approach for singularly perturbed optimal control systems. Set-valued Analysis 10, 185-207.
• LOEWEN, P.D. and VINTER, R.B. (1987) Pontryagin-type necessary conditions for differential inclusion problems. Systems Control Lett. 9, 263-265.
• VELIOV, V. (1996) On the model simplification of control/uncertain systems with multiple time scales. IIASA Working Paper, WP-96-82, Laxenburg.
• VIGODNER, A. (1997) Limits of singularly perturbed control problems with statistical dynamics of fast motions. SIAM J. Control Optimization 35, 1-28.
• VINTER, R. (2000) Optimal Control. Birkhäuser, Boston.