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Exact null internal controllability for the heat equation on unbounded convex domains. (English) Zbl 1282.93046

Summary: The liner parabolic equation \(\tfrac{\partial y}{\partial t}-\frac12\,\Delta y+F\cdot\nabla y={\vec{1}}_{\mathcal O_0}u\) with Neumann boundary condition on a convex open domain \(\mathcal O\subset \mathbb R^{d}\) with smooth boundary is exactly null controllable on each finite interval if \(\mathcal O_{0}\) is an open subset of \(\mathcal O\) which contains a suitable neighbourhood of the recession cone of \(\overline{\mathcal O}\). Here, \(F : \mathbb R^{d}\to \mathbb R^{d}\) is a bounded, \(C^{1}\)-continuous function, and \(F=\nabla g\), where \(g\) is convex and coercive.

MSC:

93B05 Controllability
35K20 Initial-boundary value problems for second-order parabolic equations
93C20 Control/observation systems governed by partial differential equations
47D07 Markov semigroups and applications to diffusion processes
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