Controllability of Parabolic Equations

Abstract

Let Ω be a smooth bounded connected open set of \({\mathbb {R}}^d\) in the sense recalled in Sect. 1.7. We consider the second-order elliptic operator $$\displaystyle P_0 = \sum _{1\leq i,j\leq d} D_i (p^{ij}(x) D_j), \qquad \text{with}\ \ \sum _{1\leq i,j\leq d} p^{ij}(x) \xi _i \xi _j \geq C |\xi |{ }^2, $$ where \(p^{ij}\in {{\mathcal C}^\infty }(\overline {\varOmega };{\mathbb {R}})\) with all derivatives bounded and such that p ij = p ji, 1 ≤ i, j ≤ d. Let ω be an open subset of Ω. For T > 0, the controlled parabolic equation associated with P 0, in the time interval (0, T), with homogeneous Dirichlet boundary conditions, and for an initial condition y 0 in L 2(Ω), is given by $$\displaystyle \begin{aligned} \begin{cases} \partial_t y + P_0 y = \boldsymbol{1}_\omega v\quad & \text{in}\ (0,T)\times \varOmega,\\ y=0 & \text{on}\ (0,T)\times \partial\varOmega,\\ y(0) = y^0 & \text{in}\ \varOmega. \end{cases} \end{aligned} $$ The function v is the control. The goal is to drive the solution y to a prescribe state at time T > 0, yet only acting in the sub-domain ω. We shall make precise what can actually be achieved below.