Abstract
In this article we study a controllability problem for a parabolic and a hyperbolic partial differential equations in which the control is the shape of the domain where the equation holds. The quantity to be controlled is the trace of the solution into an open subdomain and at a given time, when the right hand side source term is known. The mapping that associates this trace to the shape of the domain is nonlinear. We show (i) an approximate controllability property for the linearized parabolic problem and (ii) an exact local controllability property for the linearized and the nonlinear equations in the hyperbolic case. We then address the same questions in the context of a finite difference spatial semi-discretization in both the parabolic and hyperbolic problems. In this discretized case again we prove a local controllability result for the parabolic problem, and an exact controllability for the hyperbolic case, applying a local surjectivity theorem together with a unique continuation property of the underlying adjoint discrete system.
Highlights
The problem of characterizing the shape of a domain where a certain dynamical phenomenon is partially observable is a model for a wide class of applications
We only address the case of the heat and wave equations as paradigmatic examples of, respectively, parabolic and hyperbolic equations and will restrict the study to the case of Dirichlet boundary conditions, the problem might arise for other differential operators and different boundary conditions and the methods used here may apply in most regular cases
We show that dΛ(0) is surjective on H01(ω) × L2(ω), ensuring an exact controllability property of the wave equation using a local surjectivity theorem
Summary
The problem of characterizing the shape of a domain where a certain dynamical phenomenon is partially observable is a model for a wide class of applications. Aim at proving existence and uniqueness of a bounded, possibly time-varying, open set Ω(t) ⊂ Ên for t ∈ [0, T ] with ω ⋐ Ω(t) for all t and such that the solution yΩ(·) of the equation: LyΩ = f on Ω(t) yΩ(t, x) = 0 x ∈ ∂Ω(t) satisfies yΩ(T, ·)|ω = yd It is a shape identification problem, which can be seen as a controllability problem in the sense that the domain Ω has to be determined so that yΩ(T, ·)|ω = yd holds. The question we ask is the following: given yd a function close from y0(T )|ω in a suitable topology, is there an open set Ω∗(t) close from Ω (in a sense that will be defined in the sequel) such that yΩ∗ (T )|ω = yd This problem is referred to as the local controllability problem. | |det(id + ∇φ)| − 1| ≤ 1/2, and |det(id + ∇φ)| ≥ 1/2 B(φ) ≤ 3/2
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