Resumen de Interior Controllability of a Broad Class of Reaction Diffusion Equations

Hugo Antonio Leiva A., Yamilet Quintana

• In this paper we prove the interior approximate controllability of the following broad class of reaction diffusion equation in the Hilbert spaces Z = L2( ) given by z�� = .Az + 1!u(t), t �¸ [0, �Ñ], where is a domain in Rn, �Ö is an open nonempty subset of , 1! denotes the characteristic function of the set �Ö, the distributed control u �¸ L2(0, t1;L2( )) and A : D(A) �¼ Z �¨ Z is an unbounded linear operator with the following spectral decomposition:

  Az = P�� j=1 �Éj Pj k=1 < z, �Ój,k > �Ój,k. The eigenvalues 0 < �É1 < �É2 < �E �E �E < �E �E �E �Én �¨ �� of A have finite multiplicity �Áj equal to the dimension of the corresponding eigenspace, and {�Ój,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 = ��X j=1 e.j t Xj k=1 < z, �Ój,k > �Ój,k.

  Our result can be apply to the nD heat equation, the equation modelling the damped flexible beam, the Ornstein-Uhlenbeck equation, the Laguerre equation and the Jacobi equation.