Approximate weak invariance and relaxation for fully nonlinear differential inclusions
The first part of the paper is devoted to establishing necessary and sufficient conditions for the approximate weak invariance of a set with respect to fully nonlinear differential inclusions. The criterion for weak invariance is given in terms of new tangency condition involving the concept of \(A\)-quasi tangency, introduced in [\textit{O. Cârja} et al., Set-Valued Anal. 16, No. 5--6, 701--731 (2008; Zbl 1179.34068)]. In the second part, the author proves a relaxation type theorem, without assuming that the multifunction is Lipschitz. To be more precise, it is shown that the set of approximate solutions to the considered differential inclusion is dense in the set of solutions to the relaxed one. The proof is in the spirit of \textit{F. H. Clarke} et al. [J. Dyn. Control Syst. 3, No. 4, 493--518 (1997; Zbl 0951.49007)]. The problem is treated as a viability problem for a suitably constructed set. The main tools are viability results adapted from [O. Cârja et al., loc. cit.], and an interesting-by-itself result, which states that a subset of a separable Banach space with uniformly convex dual is \(A\)-quasi tangent if and only if its convex hull is \(A\)-quasi tangent.
relaxed nonlinear differential inclusion
scientific article; zbMATH DE number 6147182