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Journal of Convex Analysis, Vol. 4, No. 2, pp. 189-219 (1997)
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Complements, Approximations, Smoothings and Invariance Properties

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F. H. Clarke and Yu. S. Ledyaev and R. J. Stern

Institut Desargues, Universite de Lyon I, 69622 Villeurbanne, France, and Steklov Institute of Mathematics, Moscow 117966, Russia, and Department of Mathematics and Statistics, Concordia University, Montreal, Quebec H4B 1R6, Canada, stern@vax2.concordia.ca

**Abstract:** Proximal methods are used to determine the relationship between normal cones to a closed set in $\Re^n$ and those to the closure of its complement. The geometry of outer and inner set approximations is explored as well. In the context of differential inclusions, we study the extent to which approximations and associated smoothings inherit invariance and tangentiality properties posited for the original set. This analysis leads to the construction of a Lipschitz feedback law which achieves penetration of the interior of compact sets satisfying hypotheses which include a strict tangentiality condition. Analytic versions of some of the geometric results are provided, when the set of interest is the epigraph of a continuous function. In that setting, invariance properties correspond to types of monotonicity along trajectories, and we obtain smoothing results which are couched in terms of convolution operations and Hamilton-Jacobi inequalities, as well as a result on the existence of a universal Lipschitz feedback law which achieves monotonicty along trajectories in an approximate sense.

**Keywords:** Nonsmooth analysis, epi-Lipschitz, complement, approximations, invariance, differential inclusion, smoothing, proximal smoothness, tangentiality, feedback, convolution, Hamiltonian-Jacobi inequalities

**Classification (MSC2000):** 26B05

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