Time-Delayed Feedback Control of a Hydraulic Model Governed by a Diffusive Wave System

Boumediène Chentouf,Nejib Smaoui

doi:10.1155/2020/4986026

Abstract

This paper is concerned with the feedback flow control of an open-channel hydraulic system modeled by a diffusive wave equation with delay. Firstly, we put forward a feedback flow control subject to the action of a constant time delay. Thereafter, we invoke semigroup theory to substantiate that the closed-loop system has a unique solution in an energy space. Subsequently, we deal with the eigenvalue problem of the system. More importantly, exponential decay of solutions of the closed-loop system is derived provided that the feedback gain of the control is bounded. Finally, the theoretical findings are validated via a set of numerical results.

Highlights

Introduction e dynamics of irrigation canals are often described by nonlinear complex partial differential equations derived from the conservation of mass and momentum
Among several models available in the literature, the de Saint-Venant system [1, 2] has been standing at the forefront of the modeling and analysis of irrigation canals for decades. e system consists of two nonlinear coupled hyperbolic partial differential equations: zZ zq zt + zx Ld, (1)
An irrigation system has been considered when a constant time delay occurs in the flow control

Summary

Problem Set-Up and Preliminaries

The closed-loop system is formulated as a differential equation in an appropriate functional space. Given a linear operator P on a Hilbert space Y with D(P) Y, if P is dissipative and there is a scalar λ0 such that λ0I − P is onto Y, P generates a C0 semigroup of contractions on Y. Assume that P be a closed densely defined operator on a Hilbert space Y satisfying the following property: there exists η ∈ R such that. A direct consequence of eorem 4 is that the C0-semigroup S(t) of the closed-loop system is not necessarily uniformly bounded for all values of the feedback gain κ > 0 One can explain this property by recalling that the generator A of S(t) is not dissipative for all values of the feedback gain κ > 0, unlike the uncontrolled system whose operator A0 is dissipative and its semigroup is contractive (see Proposition 1)

Eigenvalue Problem

Conclusion