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Geometric control in the presence of a black box. (English) Zbl 1050.35058
The purpose of this paper is to show how ideas coming from scattering theory (resolvent estimates) lead to results in control theory and to some closely related eigenfunction estimates. The point of view is the following: a black box in a confined system is replaced by a scattering problem. That permits having isolated dynamical phenomena (such as only one closed orbit) impossible in confined systems. It also permits using some finer results of scattering theory directly. We present two typical applications.
1. In geometric control theory for the Schrödinger equation we are concerned with the following mixed problem: $(i\partial_t+ \Delta)u= 0\quad \text{in }\Omega,\qquad u|_{[0,T]\times\Omega}= g^1_{[0,T]\times\Gamma},\;u|_{t=0}= u_0,\tag{1}$ where $$\Omega$$ is an open set of $$\mathbb{R}^d$$, $$\partial\Omega$$ is its boundary and $$\Gamma$$ is an open subset of $$\partial\Omega$$. The question is to determine a (large) class of functions $$u_0$$ for which there exists a control $$g$$ such that $$u|_{t=T}= 0$$. The following result was established by N. Burg [Mémoire de la S.M.F., 55 (1993):
Theorem 1. Consider $$\Theta= \bigcup_{1\leq j\leq N}\Theta_j\subset\mathbb{R}^d$$, a union of mutually disjoint closed sets with strictly convex smooth boundaries and satisfying some geometric assumptions. Let $$\widetilde\Omega$$ be a bounded domain with a smooth boundary and containing $$\text{convhull}(\Theta)$$. Denote $$\Omega= \widetilde\Omega\setminus\Theta$$ and $$\Gamma= \partial\widetilde\Omega$$. Then for any $$T$$, $$\varepsilon> 0$$ and any $$u_0\in H^{1+\varepsilon}_0(\Omega)$$ there exists $$g\in L^2([0, T]\times \Gamma)$$ such that in (1) we have $$u|_{t> T}= 0$$.
The authors show how Theorem 1 can be obtained directly from estimates on the resolvent of the Laplace operator, which in turn can be deduced from semiclassical microlocal analysis or from known results in scattering theory.
2. The second application generalizes a result of Y. Colin de Verdière and B. Parisse [Commun. Partial Differ. Equations 19, 9–10, 1535–1563 (1994; Zbl 0819.35116)], who considered a special case of an isolated trajectory lying on a segment of a constant negative curvature cylinder in dimension two:
Theorem 2: Suppose that $$(X,g)$$ is a compact Riemannian manifold with a (possibly empty) boundary and $$\gamma\subset X$$ is a closed real hyperbolic geodesic not intersecting the boundary. If $$\chi\in{\mathcal C}^\infty(X,[0,1])$$ is supported in a sufficiently small neighbourhood of $$\gamma$$, then there exists a constant $$C= C(\gamma)$$ such that for any eigenfunction $$u$$ of the Laplacian $$\Delta_g$$ with Dirichlet or Neumann boundary conditions, we have $C\int_X| u(x)|^2(1- \chi)(x)\,d\,\text{vol}_g\geq {1\over\log\lambda} \int_X| u(x)|^2\,d\,\text{vol}_g,\;-\Delta_g u=\lambda u.\tag{2}$ The proof is based on putting the closed hyperbolic orbit into a microlocal black box, where that orbit becomes the only trapped orbit in a scattering problem. One can than use scattering estimates to obtain (2).

##### MSC:
 35P25 Scattering theory for PDEs 35B37 PDE in connection with control problems (MSC2000) 35Q40 PDEs in connection with quantum mechanics 35P20 Asymptotic distributions of eigenvalues in context of PDEs 81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
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