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Interior controllability of a broad class of reaction diffusion equations. (English) Zbl 1206.93017
Summary: We prove the interior approximate controllability of the following broad class of reaction diffusion equation in the Hilbert spaces Z=L 2 (Ω) given by z ' =-Az+1 ω u(t), t[0,τ], where Ω is a domain in n , ω is an open nonempty subset of Ω, 1 ω denotes the characteristic function of the set ω, the distributed control uL(0,t 1 ;L 2 (Ω)) and A:D(A) is an unbounded linear operator with the following spectral decomposition: Az= j=1 λ j k=1 γ j z,ϕ j,k ϕ j,k . The eigenvalues 0<λ 1 <λ 2 <<λ n of A have finite multiplicity γ j equal to the dimension of the corresponding eigenspace, and {ϕ j,k } is a complete orthonormal set of eigenvectors of A. The operator -A generates a strongly continuous semigroup {T(t)} given by T(t)z= j=1 e -λ j t k=1 γ j z,ϕ j,k ϕ j,k . Our result can be applied to the nD heat equation, the Ornstein-Uhlenbeck equation, the Laguerre equation, and the Jacobi equation.
60J65Brownian motion
80A20Heat and mass transfer, heat flow
35K57Reaction-diffusion equations