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On the stability of sets defined by a finite number of equalities and inequalities. (English) Zbl 0794.93096
Summary: Let a set be defined by a finite number of equalities and inequalities. For smooth data, the condition of Mangasarian and Fromovitz is known to be equivalent to the local stability – in a strong sense – of the set. We study here weaker forms of stability. Namely, we state a condition generalizing the one of Mangasarian and Fromovitz that, for some weak form of stability, is necessary. If the gradients of the equality constraints are linearly independent or if there is no equality constraint, this condition is also sufficient.

MSC:
93D99 Stability of control systems
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[1] Daniel, J. W.,On Perturbations in Systems of Linear Inequalities, SIAM Journal on Numerical Analysis, Vol. 10, pp. 299-307, 1973. · Zbl 0268.90039 · doi:10.1137/0710029
[2] Daniel, J. W.,Remarks on Perturbations in Linear Systems, SIAM Journal on Numerical Analysis, Vol. 12, pp. 770-772, 1975. · Zbl 0317.90036 · doi:10.1137/0712057
[3] Robinson, S. M.,Stability Theory for Systems of Inequalities, Part 1: Linear Systems, SIAM Journal on Numerical Analysis, Vol. 12, pp. 754-769, 1975. · Zbl 0317.90035 · doi:10.1137/0712056
[4] Mangasarian, O. L., andShiau, T. H.,Lipschitz Continuity of Solutions of Linear Equalities, SIAM Journal on Control and Optimization, Vol. 25, pp. 583-595, 1987. · Zbl 0613.90066 · doi:10.1137/0325033
[5] Robinson, S. M.,Stability Theory for Systems of Inequalities, Part 2: Differentiable Nonlinear Systems, SIAM Journal on Numerical Analysis, Vol. 13, pp. 497-513, 1976. · Zbl 0347.90050 · doi:10.1137/0713043
[6] Mangasarian, O. L., andFromovitz, S.,The Fritz John Necessary Optimality Conditions in the Presence of Equality and Inequality Constraints, Journal of Mathematical Analysis and Applications, Vol. 17, pp. 37-47, 1967. · Zbl 0149.16701 · doi:10.1016/0022-247X(67)90163-1
[7] Frankowska, H.,Some Inverse Mappings Theorems, Annales de l’Institut Henri Poincar?, Vol. 7, pp. 183-234, 1990. · Zbl 0727.26014
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