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Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time. (English) Zbl 1053.93010

It is considered a control problem for a parabolic Dirichlet-Cauchy problem for a parabolic system \[ \partial_tu- \Delta u = {\mathbf 1}_{(0,T)\times\Omega}\;g\;\text{ on }(0,T)\times M,\quad u=0\;\text{ on }(0,T)\times \partial M \] with Cauchy data \(u=u_0\) at \(t=0\), \(M\) being a smooth connected compact \(n\)-dimensional Riemannian manifold. The null controllability problem is concerned with finding the control \(g\in L^2({\mathbb R}\times M)\) such that \(u=0\) at \(T\). Here \(\Omega\subseteq M\) is the control region, \({\mathbf 1}\) the characteristic function of a set. It is introduced the null controllability cost as the best \(C_{T,\Omega}\) in the estimate \[ \| g\| _{L^2({\mathbb R}\times M)}\leq C_{T,\Omega}\| u_0\| _{L^2(M)} \] where \(g\) solves the null controllability problem for initial data \(u_0\). The paper gives new results on the problem as to how the geometry of the control region influences \(C_{T,\Omega}\) for \(T>0\) small enough.

MSC:

93B05 Controllability
93B07 Observability
93C20 Control/observation systems governed by partial differential equations
80A20 Heat and mass transfer, heat flow (MSC2010)
35K20 Initial-boundary value problems for second-order parabolic equations
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