Controllability of differential inclusions
An autonomous differential inclusion on a Rimannian compact manifold with a convex-valued Lipschitz-continuous right-hand side is studied. A control set is defined as a maximal viable (in some sources also called weakly invariant) subset of the state space. A control set \(C\) is called locally attractive if attainable sets emanating from a small neighborhood of \(C\) asymptotically converges to \(C\) in the Hausdorff metric. The objective of the study is the behavior of control sets under perturbations of the attainable set semiflow. A differential inclusion on a manifold is characterized by a set field, i.e. at each point \(x\) there is specified a subset \(F(x)\) of the tangential space. However, a small perturbation of a set field is defined in terms of the respective perturbation of the attainable set semiflow. Such a trick allows the author to avoid a requirement of uniform Lipschitz-continuity of the perturbed set fields. Two main results are obtained. The first theorem asserts that control sets with nonempty interiors are lower semi-continuous with respect to small perturbations. The second theorem claims that the lower continuity can be upgraded to continuity in the Hausdorff metric under the additional assumption that the control set is locally attractive. The continuity is understood in the following weak sense: given a locally attractive control set \(C\) and an arbitrary positive \(\varepsilon\) there exists a positive \(\delta_0\) such that for any perturbation parameter \(\delta\), \(0<\delta<\delta_0\) there exists a control set \(C_\delta\) of the perturbed system, for which \(d_H(C_\delta,C)\leq\varepsilon\). As an illustration, the continuity result is applied to a singularly perturbed nonlinear control system. A sequence of invariant control sets of the perturbed systems which approximate the entire state space as the parameter of perturbation tends to zero is constructed.
scientific article; zbMATH DE number 1431802
scientific article