Near viability for fully nonlinear differential inclusions
The authors consider the differential inclusion \[ x'(t)\in Ax(t)+F(x(t))\tag{1} \] in a separable Banach space \(X\) where \(A\) and \(F\) are multivalued. \ The goal of this article is to prove a near viability result with respect to \(C^{0}\)-solutions of (1). \ By ``near viability'', the authors mean the existence of solutions which stay arbitrarily near a given set \(K\subseteq \overline{D(A)}\) beginning from initial states in \(K\). The authors point out that ``near viability'' is not the same as ``approximate viability'' or ``approximate weak invariance'' which has been used in other articles. Along the way, the authors prove several results for the problem \[ x'(t)\in Ax(t)+F(t,x(t))\tag{2} \] {\parindent=8mm\begin{itemize}\item[(i)] Theorem 3.1, similar to a result from [\textit{A. F. Filippov}, SIAM J. Control 5, 609--621 (1967); translation from Vestnik Moskov. Univ., Ser. I 22, No. 3, 16--26 (1967; Zbl 0238.34010)] in which the distance from a given function \(y\) to the set of solutions from (2) is estimated, \item[(ii)] Corollary 3.2, on Lipschitz dependence of solutions of (2) on the initial state, \item[(iii)] Theorem 3.3, on compactness of the solution set for (2) and \item[(iv)] Theorem 3.5, a ``tracking'' result for (2). \end{itemize}} These results extend ones for semilinear differential inclusions. \ The near viability result for (1) assumes \(X\) is separable, \(K\) is locally closed, \(A\) is an m-dissipative operator, \(F\) is Lipschitz on bounded sets, closed- and bounded-valued and satisfies the tangency condition from [\textit{O. Cârjă} et al., Set-Valued Anal. 16, No. 5--6, 701--731 (2008; Zbl 1179.34068)]. \ No compactness assumptions are made, and \(F\) is not required to be convex-valued. The proof makes use of Theorem 3.1. \ Finally the authors apply this near viability result to prove an approximate null controllability result for a certain control system, meaning the existence of a solution which starts near the origin that gets arbitrarily close to the origin in finite time.
scientific article; zbMATH DE number 6335853