The controllability of systems governed by parabolic differential equations (Q1378597)

The control system is \[ y'(t)=Ay(t)+u(t) \] in a bounded domain \(\Omega\subset\mathbb{R}^m\) with boundary \(\Gamma\); \(A\) is the semigroup generator in \(L^2(\Omega)\) associated with the second order elliptic operator \[ {\mathcal A}=\sum_{j, k=1}^m {\partial \over \partial x_j} a_{jk}(x) {\partial \over \partial x_k} - c(x) \] and one has Dirichlet boundary conditions. The objective is to drive the initial condition \(y(0) = y_0\) to a target \(\bar y\) in time \(T,\) that is, \[ y(T) = \bar y \] by means of a control \(u \in L^2(0, T; L^2(\Omega)) = L^2((0, T) \times \Omega).\) Under suitable smoothness assumptions on the coefficients, the authors prove that this controllability problem has a solution if \(\bar y \in H^2(\Omega).\) The method used is a finite dimensional approximation. It should be pointed out that the novelty of this result lies in the fact that the target \(\bar y\) does not necessarily satisfy the boundary condition; if it does, that is, if \(y \in H^2_0(\Omega) = D(A),\) existence of the control in \(L^\infty(0, T; L^2(\Omega))\) follows from the equality \[ z = \int_0^T S(T - \sigma){1 \over T}(z - \sigma Az)d\sigma \] where \(S(t)\) is the semigroup generated by \(A\) and \(z \in D(A);\) it suffices to take \(z = \bar y - S(T)y_0\).