Remark on the global null controllability for a viscous Burgers-particle system with particle supported control

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Abstract

This paper is devoted to study the controllability of a one-dimensional fluid-particle interaction model where the fluid follows the viscous Burgers equation and the point mass obeys Newton’s second law. We prove the null controllability for the velocity of the fluid and the particle and an approximate controllability for the position of the particle with a control variable acting only on the particle. One of the novelties of our work is the fact that we achieve this controllability result in a uniform time for all initial data and without any smallness assumptions on the initial data.

Section snippets

Introduction and main result

In this work, we analyze the global null controllability of a simplified one-dimensional model of fluid-particle interaction. Here the fluid is governed by the viscous Burgers equation and the particle follows the Newton law. More precisely, we consider the following control problem: ∂tu(t,x)−∂xxu(t,x)+u(t,x)∂xu(t,x)=0,t∈(0,T),x∈(0,1)∖{h(t)},u(t,0)=0=u(t,1),t∈(0,T),u(t,h(t))=h′(t),t∈(0,T),mh′′(t)=〚∂xu〛(t,h(t))+g(t),t∈(0,T),u(0,x)=u0(x),x∈(0,1),h(0)=h0,h′(0)=ℓ0.In the above equation, u(t,x)

Preliminaries

In Theorem 1.2 or in Theorem 1.1, we have used the notion of weak solutions to (1.1). We give here the precise definition of such solutions:

Definition 2.1

Given T>0, u0∈L2(0,1), h0∈(0,1), ℓ0∈R and g∈L2(0,T), we say that (h,u) is a weak solution of (1.1) if h∈H1(0,T),u∈C0([0,T];L2(0,1))∩L2(0,T;H01(0,1))if h(0)=h0,h′(t)=ℓ(t)=u(t,h(t)),h(t)∈(0,1)for almost everyt∈[0,T]and if ∫01u(t,x)ψ(t,x)dx−∫01u0(x)ψ(0,x)dx−∫0tℓ(s)ξ′(s)ds+ℓ(t)ξ(t)−ℓ0ξ(0)−∫0t∫01u(s,x)ψ′(s,x)dxds+∫0t∫01∂xu(s,x)∂xψ(s,x)dxds−12∫0t∫01u2(s,x)∂xψ(s,x)

Proof of Theorem 1.2

As explained in the previous section, in order to apply Theorem 2.3, we first consider h1∈S such that (2.1) holds and we are going to show that there exists a time T>0 such that for any h0∈(0,1), ℓ0∈R and u0∈L2(0,1), there exists a control g∈L2(0,T) such that the solution of the system (1.1) satisfies h(T)=h1,h′(T)=0,u(T,⋅)=0.

We are now in a position to prove Theorem 1.2:

Proof of Theorem 1.2

The proof is divided into several steps:

Step 1: Parabolic smoothing of (1.1) with g=0. Using Proposition 2.2, for g=0, there

Burgers equation in a time varying domain

We recall in this section a standard result on the viscous Burgers equation in a moving domain since we use it in the proof of Theorem 1.2. In this section, we thus consider a given h∈H2(0,T;(0,1)) and we consider the following Burgers system: ∂tu(t,x)−∂xxu(t,x)+u(t,x)∂xu(t,x)=0t∈(0,T),x∈(0,1)∖{h(t)},u(t,0)=u(t,1)=0,t∈(0,T),u(t,h(t))=h′(t),t∈(0,T),u(0,x)=u0(x),x∈(0,1).

Theorem 4.1

Let h∈H2(0,T;(0,1)) and u0∈H1(0,1) with u0(h(0))=h′(0).Then, for any T>0, the problem (4.1) admits a unique solution u∈C0([0,T];H

Acknowledgments

AR and TT were partially supported by the ANR research project IFSMACS (ANR-15-CE40-0010). The three authors were partially supported by the IFCAM project “Analysis, Control and Homogenization of Complex Systems”.

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