Remarks on the state convergence of nonlinear systems given any Lp input

Abstract

Abstract-In this paper, we study the state convergence of L pstable systems and input-to-state stable (ISS) systems given any L p input signal where p ∈ [1, ∞). Under some conditions on the system equations, it is shown that if the system is dissipative with a proper storage function H and supply rate s(y, u) = u p − k y p , k > 0 and it is zero-state detectable then the state x of the system converges to zero for any L p input. This implies also that the system is also L p -stable.Using the same technique, if the system is ISS, under some conditions on the system equations and on the smooth ISS Lyapunov function, then the state x of the system converges to zero for any L p input.

iISS gain of dissipative systems

“…Using the fact that γ = σ , we can recover the state convergence results of [4] via the iISS approach of the present paper.…”

“…In view of recent results on L p -input state-convergence, we briefly discuss the various assumptions on f used in [1], [4], [7] and compare these with assumption (A) of the present paper. We focus on the case wherein σ (s) = s p for all s ≥ 0 for p ∈ [1, ∞).…”

“…Motivated by the state convergence result from [4], we show that (2) holds with γ = σ under a mild assumption on f which is weaker than those posited in [4], [7] and is also weaker than the condition that, for some c…”

“…Under certain conditions on f , it is shown in Jayawardhana [4] that, if the equilibrium x = 0 of P (with zero input u = 0) is GAS and there exists a continuously differentiable function H such that (3) holds with σ (s) = s p for all s ≥ 0, then the state x(t) converges to zero as t → ∞ for every initial state x 0 and for every L p input u. Note that the argument in [4] utilizes infinite-dimensional systems theory.…”

State Convergence of Dissipative Nonlinear Systems Given Bounded-Energy Input Signal

“…An earlier version of this paper is presented in [4], where the condition on f is stronger. In this paper, we extend the result in [4] by assuming a weaker condition on f , allowing a different type of supply rate and allowing nonproper storage functions.…”