Stability notions for a class of nonlinear systems with measure controls

Abstract

We consider the problem of stability in a class of differential equations which are driven by a differential measure associated with inputs of locally bounded variation. After discussing some existing notions of solution for such systems, we derive Lyapunov-based conditions on the system's vector fields for asymptotic stability under a specific class of inputs. These conditions are based on the stability margin of the Lebesgue-integrable and the measure-driven components of the system. For more general inputs which do not necessarily lead to asymptotic stability, we then derive conditions such that the maximum norm of the resulting trajectory is bounded by some function of the total variation of the input, which generalizes the notion of integral input-to-state stability in measure-driven systems.

Well-Posedness and Output Regulation for Implicit Time-Varying Evolution Variational Inequalities

“…To prove this result as an application of Theorem 1, we basically show that the assumptions (A3), (A4) and (A5) (resp. (A5-BV)) hold for system (33). It is first seen that the condition (39) implies (A4), because Λ x (t) = −(∂σ S(t) + J) −1 (Hx).…”

“…We can write e + − e − = G η, so that e − P e − = e + P e + − 2e + P G η + η G P G η ≥ e + P e + where the middle term is nonnegative due to the same arguments as used in the proof of Theorem 2. The formal arguments to deduce asymptotic stability for systems with BV solutions can be found in [32,33,34].…”

“…and reformulate the dynamics of (33) in the form of (5). Towards this end, we recall the following fundamental relation from convex analysis [17, Proposition 1.1.3]:…”

“…If h is locally absolutely continuous (respectively, right-continuous BV), then there exists a unique weak solution to (33) which is continuous (resp. right-continuous BV).…”