Interior controllability of a broad class of reaction diffusion equations

Abstract

We prove the interior approximate controllability of the following broad class of reaction diffusion equation in the Hilbert spaces Z = L 2 () given by z ′ = - A z + 1 u (t), t [ 0, ], where is a domain in n, is an open nonempty subset of, 1 denotes the characteristic function of the set, the distributed control u L 2 (0, t 1; L 2 ()) and A: D (A) Z → Z is an unbounded linear operator with the following spectral decomposition: A z = j = 1 ∞ λ j k = 1 j z, j, k j, k. The eigenvalues 0 < λ 1 < λ 2 < < λ n → ∞ of A have finite multiplicity j equal to the dimension of the corresponding eigenspace, and { 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 j z, j, k j, k. Our result can be applied to the n D heat equation, the Ornstein-Uhlenbeck equation, the Laguerre equation, and the Jacobi equation. Copyright © 2009 H. Leiva and Y. Quintana.