The maximum principle in optimal control, then and now

• Autorzy: Clarke F.
• Języki publikacji: EN

Abstrakty

EN

We discuss the evolution of the Pontryagin maximum principle, focusing primarily on the hypotheses required for its validity. We proceed to describe briefly a unifying result giving rise to both classical and new versions, a recent theorem of the author giving necessary conditions for optimal control problems formulated in terms of differential inclusions. We conclude with a new application of this result for the case in which mixed constraints on the state and control are imposed in terms of equalities, inequalities, and unilateral set constraints. In order to lighten the exposition, the discussion is limited to differentiable data, thereby avoiding mention of generalized gradients or normal cones, except, in the technical section on differential inclusions.

Słowa kluczowe

EN

optimal control; necessary conditions; maximum principle; nonsmooth analysis

PL

sterowanie optymalne; warunek konieczny; zasada maksimum; analiza niegładka

• Wydawca: Systems Research Institute, Polish Academy of Sciences
• Czasopismo: Control and Cybernetics
• Rocznik: 2005
• Tom: Vol. 34, no 3
• Strony: 709--722
• Opis fizyczny: Bibliogr. 14 poz.

Twórcy

autor

Clarke F.

• Institut Camille Jordan UMR CNRS 5208, Batiment Jean Braconnier Universite Lyon I 43, Boulevard du 11 novembre 1918, 69622 Villeurbanne, France

Bibliografia

• Clarke. F. (2005) Necessary conditions in dynamic optimization. Memoirs of the Amer. Math. Soc. 173 (816).
• Clarke, F.H. (1976) The generalized problem of Bolza. SIAM J. Control Optim. 14, 682–699.
• Clarke, F.H. (1976) The maximum principle under minimal hypotheses. SIAM J. Control Optim. 14, 1078–1091.
• Clarke, F.H. (1983) Optimization and Nonsmooth Analysis. Wiley-Interscience, New York. Republished as vol. 5 of Classics in Applied Mathematics, SIAM, 1990.
• Clarke, F.H., Ledyaev, Yu.S., Stern, R.J. and Wolenski, P.R. (1998) Nonsmooth Analysis and Control Theory. Graduate Texts in Mathematics, vol. 178. Springer-Verlag, New York.
• Clarke, F.H. and Vinter, R.B. (1984) On the conditions under which the Euler equation or the maximum principle hold. Applied Math. and Optimization 12, 73–79.
• Dmitruk, A.V. (1993) Maximum principle for a general optimal control problem with state and regular mixed constraints. Comp. Math. and Modeling 4, 364–377.
• Dubovitskiı, A.Ya. and Milyutin, A.A. (1981) Theory of the principle of the maximum (in Russian). In: Methods of the Theory of Extremal Problems in Economics. Nauka, Moscow.
• Hestenes, M.R.(1966) Calculus of Variations and Optimal Control Theory. Wiley, New York.
• Ioffe, A.D. and Tikhomirov, V. (1974) Theory of Extremal Problems. Nauka, Moscow. English translation, North-Holland, 1979.
• Milyutin, A.A. and Osmolovskii, N.P. (1998) Calculus of Variations and Optimal Control. American Math. Soc., Providence.
• Pontryagin, L.S., Boltyanskii, R.V., Gamkrelidze, R.V. and Mischenko, E.F. (1962) The Mathematical Theory of Optimal Processes. Wiley-Interscience, New York.
• Vinter, R.B. (2000) Optimal Control. Birkhauser, Boston.
• Warga, J. (1972) Optimal Control of Differential and Functional Equations. Academic Press, New York.