Interior Controllability of a Broad Class of Reaction Diffusion Equations

We prove the interior approximate controllability of the following broad class of reaction diffusion equation in the Hilbert spacesZ=L2(Ω)given byz′=−Az+1ωu(t),t∈[0,τ], whereΩis a domain inℝn,ωis an open nonempty subset ofΩ,1ωdenotes the characteristic function of the setω, the distributed controlu∈L2(0,t1;L2(Ω))andA:D(A)⊂Z→Zis an unbounded linear operator with the following spectral decomposition:Az=∑j=1∞λj∑k=1γj〈z,ϕj,k〉ϕj,k. The eigenvalues0<λ1<λ2<⋯<⋯λn→∞ofAhave finite multiplicityγjequal to the dimension of the corresponding eigenspace, and{ϕj,k}is a complete orthonormal set of eigenvectors ofA. The operator−Agenerates a strongly continuous semigroup{T(t)}given byT(t)z=∑j=1∞e−λjt∑k=1γj〈z,ϕj,k〉ϕj,k. Our result can be applied to thenD heat equation, the Ornstein-Uhlenbeck equation, the Laguerre equation, and the Jacobi equation.