On Bounded Strictly Positive Operators of Closed Range and Some Applications to Asymptotic Hyperstability of Dynamic Systems

M De La Sen

doi:10.1155/2013/639576

Abstract

The problem discussed is the stability of two input-output feedforward and feedback relations, under an integral-type constraint defining an admissible class of feedback controllers. Sufficiency-type conditions are given for the positive, bounded and of closed range feed-forward operator to be strictly positive and then boundedly invertible, with its existing inverse being also a strictly positive operator. The general formalism is first established and the linked to properties of some typical contractive and pseudocontractive mappings while some real-world applications and links of the above formalism to asymptotic hyperstability of dynamic systems are discussed later on.

Highlights

The properties of absolute stability and hyperstability and asymptotic hyperstability of dynamic systems are very important tools in dynamic systems since they are associated with the positivity and boundedness of the energy for all feedback controllers within a wide class characterized by a Popovtype integral inequality, implying global Lyapunov’s stability [1–8]
The extended Popov-type control inequality of the controller ⟨Pty, Pt(φΓ(y))⟩ ≥ −γ > −∞ and G ≻ 0 implies that 0 ≤ E(t) ≤ γ < ∞, (0 < E ≤ γ < ∞ for any nonzero control input with compact support); ∀t ∈ Γ and all φΓ(y) satisfying the assumption 6 of Theorem 5; that is the input-output energy is nonnegative and bounded; ∀t ∈ Γ
The use of such a constraint allows the simultaneous investigation of the maintenance of the positivity and stability properties of (1) under a class of nonlinear time-varying controllers rather than for a particular controller device belonging to such a class

Summary

Introduction

The properties of absolute stability and hyperstability and asymptotic hyperstability of dynamic systems are very important tools in dynamic systems since they are associated with the positivity and boundedness of the energy for all feedback controllers within a wide class characterized by a Popovtype integral inequality, implying global Lyapunov’s stability [1–8]. Further links with technical results and some real-world examples are established through the paper related to the relevant problems of absolute stability and asymptotic hyperstability of continuous-time and discrete-time dynamic systems [1–8] Such dynamic systems possess the significant physical property that their associate input-output energy is non-negative and finite for all time. On the other hand, Abstract and Applied Analysis that the crucial property for the boundedness and stability of the operator restricted to the Hilbert space of interest is that it will be stable on its whole definition domain

Problem Statement and Main Results

Application Examples