Constructions of Strict Lyapunov Functions

Abstract

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Uniform global position feedback tracking control of mechanical systems without friction

“…Then, a direct computation shows that [q,q, q c ] = [0, 0, 0] is a unique equilibrium of the closedloop equations (13a), (14). The stability proof is divided in three main steps which establish the three properties listed in Definition 4.…”

“…Proposition 1 The closed-loop trajectories of the system (1), (13) Proof. We analyze the solutions to (14), (15) …”

“…In a general nonlinear context, the state of the art in constructing Lyapunov functions for nonlinear time-varying systems relies on Lyapunov functions that have negative semi-definite derivatives -see [14]. That is, in the present context, the methods in the latter reference require thaṫ V 1 ≤ 0 as opposed to (17).…”

“…Consider a function V : R ≥0 × B R → R defined as in (27). Under Assumption 1 its total time-derivative along the trajectories of (14), (15) satisfies, for all (t, x) ∈ R ≥0 × B R ,…”

Stability analysis of PE systems via Steklov's averaging technique

“…The theory of converse Lyapunov functions guarantees existence of a Lyapunov function in many situations where stability does hold; however, this theory often offers only an implicit, hardly tractable, and abstract function of such a kind. Meanwhile by itself, a closed form of the strict Lyapunov functional brings substantial benefits in engineering applications [5]; for example, with such a functional at hand, many robustness and stabilization issues can be explicitly treated via standard feedback designs or robustness arguments. To the best knowledge of the authors, no explicit form of the Lyapunov functional that proves stability of the aforementioned PE systems is currently available.…”