Relaxed Controls and the Dynamics of Control Systems

Abstract

The relationship between relaxed controls and the family of processes or flows generated by ordinary controls is studied. We find that the flows generated by the relaxed controls form a completion of the space of flows generated by ordinary controls. With the aid of this completion we study the asymptotic and limiting behavior of the dynamics of the control system. Invariance properties of the to-limiting sets of admissible solutions are established. Stability, eventual stability and finite time stability properties of solutions with respect to ordinary and relaxed controls are investigated. Invariance properties of to-limit sets for autonomous and nonautonomous systems are treated in 4 and 5 respectively. The invariance in the autonomous case *

Input-Output-to-State Stability

“…Condition (8) is a dissipation inequality in the sense of [52]. Remark 2.3 A smooth function V : X → R ≥0 , satisfying (7) on X with some α 1 , α 2 of class K ∞ , is an UIOSS-lyapunov function for a system (5) if and only if there exist functions α 3 of class K ∞ , and γ, and χ 1 of class K such that…”

“…A few particular cases of the UIOSS property have been studied in the literature. If the system (5) in consideration has no outputs and no disturbances, UIOSS reduces to the wellknown ISS property, whose Lyapunov characterization was obtained in [42]. In case (5) is autonomous, UIOSS becomes OSS.…”

“…This is proven in section 4. The construction of a UIOSS-Lyapunov function for an original system (5) is reduced, via a small gain argument, to the construction of a UOSS-Lyapunov function for a special system (17) related to the original system (5). This reduction is done in section 3.1, and section 3.2 completes the construction of UIOSS-Lyapunov functions.…”

“…Define ϕ(r) to be a locally Lipschitz K ∞ -function, which minorizes γ 1 −1 ( 1 4 α −1 (r)) and can be extended as a Lipschitz function to a neighborhood of [0, ∞). To prove the lemma we will show that ϕ is a stability margin for (5).…”

Uniform controllability via the limiting systems

“…The proof that Conti [6] [10]. Limiting control systems were used in Artstein [3]. The constructions in these papers address, primarily, nonlinear equations.…”