Interior Controllability of a Broad Class of Reaction Diffusion Equations

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We prove the interior approximate controllability of the following broad class of reaction diffusion equation in the Hilbert spaces Z = L2(Ω) given by z′ = −Az + 1ωu(t), t ∈ [0, τ], where Ω is a domain in ℝn, ω is an open nonempty subset of Ω, 1ω denotes the characteristic function of the set ω, the distributed control u ∈ L2(0, t1; L2(Ω)) and A : D(A) ⊂ Z → Z is an unbounded linear operator with the following spectral decomposition: 〈z, ϕj,k〉ϕj,k. The eigenvalues 0 < λ1 < λ2 < ⋯<⋯λn → ∞ of A have finite multiplicity γj equal to the dimension of the corresponding eigenspace, and {ϕj,k} is a complete orthonormal set of eigenvectors of A. The operator −A generates a strongly continuous semigroup {T(t)} given by 〈z, ϕj,k〉ϕj,k. Our result can be applied to the nD heat equation, the Ornstein‐Uhlenbeck equation, the Laguerre equation, and the Jacobi equation.