On robustness of the regularity property of maps

• Autorzy: Ioffe A. D.
• Języki publikacji: EN

Abstrakty

EN

The problem considered in the paper can be described as follows. We are given a continuous mapping from one metric space into another which is regular (in the sense of metric regularity or, equivalently, controllability at a linear rate) near a certain point. How small may be an additive perturbation of the mapping which destroys regularity? The paper contains a new proof of a recent theorem of Dontchev-Lewis-Rockafellar for linear perturbations of maps between finite-dimensional Banach spaces and an exact estimate for Lipschitz perturbations of maps between complete metric spaces.

Słowa kluczowe

EN

metric regularity; openness at a linear rate; pseudo-Lipschitz (Aubin) property; regularity criterion

PL

prawidłowość metryczna; nachylenie; kryterium prawidłowości

• Wydawca: Systems Research Institute, Polish Academy of Sciences
• Czasopismo: Control and Cybernetics
• Rocznik: 2003
• Tom: Vol. 32, no 3
• Strony: 543--554
• Opis fizyczny: Bibliogr. 11 poz.

Twórcy

autor

Ioffe A. D.

• Department of Mathematics, Technion, Haifa 32000, Israel

Bibliografia

• Aze D., Corvellec J-N. ´ and Lucchetti, R. (2002) Variational pairs and application to stability in nonsmooth analysis, Nonlinear Analysis TMA, 49, 643-670.
• Giorgi, De.E., Marino A. and Tosques, M. (1980) Problemi di evoluzione in spazi metrici e curve di massima pendenza, Atti Acad. Nat. Lincei, Rend. Cl. Sci. Fiz. Mat. Natur., 68, 180-187.
• Dmitruk, A.V., Miljutin, A.A. and Osmolovskii, N.P. (1980) The theorem of Ljusternik and theory of extremum, Uspehi Matem. Nauk, 35:6, 11-46 (in Russian); English translation: Russian Math. Surveys, 35:6, 11-51.
• Dontchev, A.L., Lewis, A.S. and Rockafellar, R.T. (2002) The radius of metric regularity, Trans. Amer. Math. Soc., 355, 493-517.
• Ioffe, A.D. (1987) On the local surjection property, Nonlinear Anal. TMA, 11, 565-592.
• Ioffe, A.D. (2000) Metric regularity and subdifferential calculus, Uspehi Matem. Nauk, 55:3, 103-162 (in Russian); English translation: Russian Math. Surveys, 55:3.
• Ioffe, A.D. (2001) Towards metric theory of metric regularity, in Approximation, Optimization and mathematical Economics, M. Lassond, ed. Physica Verlag , 165-177.
• Ioffe, A.D. On stability estimates for the regularity property of maps. Preprint, 2002.
• Kruger, A. (1988) A covering theorem for multifunctions, Optimization, 19, 763-780.
• Mordukhovich, B.S. (1993) Complete characterization of openness, metric regularity and Lipschitz properties of multifunctions, Trans. Amer. Math. Soc., 340, 1-36.
• Mordukhovich, B.S. (2003) Coderivative analysis of variational systems, J. Global Optimization, to appear