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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 \(Z=L^2(\Omega)\) given by \(z'=-Az+1_\omega u(t)\), \(t\in[0,\tau]\), where \(\Omega\) is a domain in \(\mathbb R^n\), \(\omega\) is an open nonempty subset of \(\Omega\), \(1_\omega\) denotes the characteristic function of the set \(\omega\), the distributed control \(u\in L(0,t_1;L^2(\Omega))\) and \(A:D(A)\subset\mathbb Z\to\mathbb Z\) is an unbounded linear operator with the following spectral decomposition: \(Az= \sum_{j=1}^\infty \lambda_j \sum_{k=1}^{\gamma_j}\langle z,\varphi_{j,k}\rangle\varphi_{j,k}\). The eigenvalues \(0<\lambda_1<\lambda_2<\cdots<\cdots\lambda_n\to\infty\) of \(A\) have finite multiplicity \(\gamma_j\) equal to the dimension of the corresponding eigenspace, and \(\{\varphi_{j,k}\}\) is a complete orthonormal set of eigenvectors of \(A\). The operator \(-A\) generates a strongly continuous semigroup \(\{T(t)\}\) given by \(T(t)z= \sum_{j=1}^\infty e^{-\lambda_jt} \sum_{k=1}^{\gamma_j}\langle z,\varphi_{j,k}\rangle\varphi_{j,k}\). Our result can be applied to the \(n\)D heat equation, the Ornstein-Uhlenbeck equation, the Laguerre equation, and the Jacobi equation.

MSC:

93B05 Controllability
60J65 Brownian motion
80A20 Heat and mass transfer, heat flow (MSC2010)
35K57 Reaction-diffusion equations
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References:

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