On a nonlocal metric regularity of nonlinear operators

• Autorzy: Dmitruk A. V.
• Języki publikacji: EN

Abstrakty

EN

We consider some versions and generalizations of the classical Lyusternik theorem on the covering property (metric regularity) of nonlinear mappings, study some related properties, and propose nonlocal theorems of the given type, which then are used in the proof of a relaxation theorem for a nonlinear control system with sliding modes and terminal equality constraints.

Słowa kluczowe

EN

non-linear mapping; covering and metric regularity; Lyusternik iteration process; uniform covering; combined operator; sliding mode controls; weak-* convergence

PL

odwzorowanie nieliniowe; pokrycie jednostajne; sterowanie ślizgowe; słaba zbieżność

• Wydawca: Systems Research Institute, Polish Academy of Sciences
• Czasopismo: Control and Cybernetics
• Rocznik: 2005
• Tom: Vol. 34, no 3
• Strony: 723--746
• Opis fizyczny: Bibliogr. 24 poz.

Twórcy

autor

Dmitruk A. V.

• Central Economics & Mathematics Institute, Russian Academy of Sciences Nakhimovskii prospekt 47, Moscow 117418, Russian Federation

Bibliografia

• Artstein, Z. (1989) Rapid oscillations, chattering systems, and relaxed controls. SIAM J. on Control and Opt. 27 (5), 940–948.
• Balder, E.J. (1984) A general denseness result for relaxed control theory. Bull. Aust. Math. Soc. 30, 463–475.
• Borwein, J.M. and Zhuang, D.M. (1988) Verifiable necessary and sufficient conditions for openness and regularity of set-valued and single-valued maps. J. Math. Anal. Appl. 134 (2), 441–459.
• Bulgakov, A.I. and Vasilyev, V.V. (2002) On the theory of functionaldifferential inclusions of neutral type. Georgian Math. J. 9 (1), 33–52.
• Chukanov, S.V. (1990) Maximum principle for optimal control problems with integral equations. In: A.A. Milyutin, ed., Necessary Condition in Optimal Control. Nauka, Ch. 6.
• Dmitruk, A.V. (1976) The justification of the sliding mode method to optimal control problems with mixed constraints. Functional Analysis and its Appl. 10, 197–201.
• Dmitruk, A.V. (1993) Maximum principle for a general optimal control problem with state and regular mixed constraints. Computational Mathematics and Modeling 4 (4), 364–377.
• Dmitruk, A.V. (2002) A nonlocal Lyusternik estimate and its application to control systems with sliding modes. In: A.B. Kurzhanski and A.L. Fradkov, eds., Nonlinear Control Systems 2001, 2, 1061–1064, Elsevier.
• Dmitruk, A.V., Milyutin, A.A. and Osmolovskii, N.P. (1980) Luysternik’s theorem and the theory of extrema. Russian Math. Surveys 35 (6), 11–51.
• Dontchev, A.L. and Rockafellar, R.T. (2004) Regularity and conditioning of solution mappings in variational analysis. Set-Valued Analysis 12 (1), 79–109.
• Gamkrelidze, R.V. (1962) Optimal sliding states. Soviet Math. Dokl. 3, 559–562.
• Ioffe, A.D. (2000) Metric regularity and subdiffertential calculus. Russian Math. Surveys 55 (3), 501–558.
• Ioffe, A.D. (2001) On perturbation stability of metric regularity. Set-Valued Analysis 9 (1-2), 101–109.
• Ioffe, A.D. and Tikhomirov, V.M. (1974) Theory of Extremal Problems. M., Nauka, 1974; English translation: Amsterdam, North-Holland, 1979.
• Lyusternik, L.A. (1934) On the conditional extrema of functionals. Mat. Sbornik 41, 390–401 (in Russian).
• McShane, E.J. (1967) Relaxed controls and variational problems. SIAM J. on Control 5, 438–485.
• Milyutin, A.A., Dmitruk, A.V. and Osmolovskii, N.P. (2004) The Maximum Principle in Optimal Control. Mech.-Math. Faculty of Moscow State University, (in Russian).
• Olech, C. (1976) Existence theory in optimal control. In: Control theory and topics in functional analysis, I, Vienna, 291–328.
• Penot, J.-P. (1989) Metric regularity, openness and Lipschitzian behavior of multifunctions. Nonlinear Analysis, TMA 13 (6), 629–643.
• Rosenblueth, J.F. and Vinter, R.B. (1991) Relaxation procedures for time delay systems. J. Math. Anal. Appl. 162 (2), 542–563.
• Roubicek, T. (1997) Relaxation in Optimization Theory and Variational Calculus. de Gruyter, Berlin.
• Tolstonogov, A.A. (2000) Differential Inclusions in a Banach Space. Dordrecht, Kluwer Academic Publishers.
• Warga, J. (1972) Optimal Control of Differential and Functional Equations. New York, Academic Press.
• Young, L.C. (1969) Lectures on the Calculus of Variations and Optimal Control Theory. Saunders, Philadelphia.