Approximate controllability conditions for some linear 1D parabolic systems with space-dependent coefficients

Franck Boyer,Guillaume Olive

doi:10.3934/mcrf.2014.4.263

Abstract

In this article we are interested in the controllability with one single control force of parabolic systems with space-dependent zero-order coupling terms. We particularly want to emphasize that, surprisingly enough for parabolic problems, the geometry of the control domain can have an important influence on the controllability properties of the system, depending on the structure of the coupling terms. &nbsp Our analysis is mainly based on a criterion given by Fattorini in [12] (and systematically used in [22] for instance), that reduces the problem to the study of a unique continuation property for elliptic systems. We provide several detailed examples of controllable and non-controllable systems. This work gives theoretical justifications of some numerical observations described in [9].

Highlights

This paper deals with the controllability properties at time T > 0 of the following class of 1D linear parabolic systems
The domain is Ω = (0, 1), y ∈ C0([0, T ], L2(Ω)n) is the state, y0 ∈ L2(Ω)n is the initial data, A(x) is a n × n real matrix with entries in L∞(Ω), B is a constant vector in Rn and v ∈ L2((0, T ) × Ω) is the control which is acting only on the control domain ω, a non-empty open subset of Ω
We have given some checkable necessary and sufficient conditions for the approximate controllability of some 1D coupled parabolic systems with space-dependent coefficients. These conditions have been illustrated on many simple examples to show that the controllability issue for those systems can be an intricate problem depending on the geometry of the control domain and of the characteristics of the coupling terms in the system

Summary

Introduction

This paper deals with the controllability properties at time T > 0 of the following class of 1D linear parabolic systems. The domain is Ω = (0, 1), y ∈ C0([0, T ], L2(Ω)n) is the state, y0 ∈ L2(Ω)n is the initial data, A(x) is a n × n real matrix with entries in L∞(Ω), B is a constant vector in Rn and v ∈ L2((0, T ) × Ω) is the (scalar-valued) control which is acting only on the control domain ω, a non-empty open subset of Ω. L = L Id operates on vector-valued functions component-wise through the scalar elliptic operator L defined by. The coefficients of L are supposed to satisfy the standard uniform ellipticity assumptions γ, γ0 ∈ L∞(Ω), with infΩ γ > 0

Objectives

Conclusion