LQR based $\omega -$stabilization of a heat equation with memory

We consider a heat equation with memory which is defined on a bounded domain in ${\mathbb{R}}^d$ and is driven by $m$ control inputs acting on the interior of the domain. Our objective is to numerically construct a state feedback controller for this equation such that, for each initial state, the solution of the closed-loop system decays exponentially to zero with a decay rate larger than a given rate $\omega >0$, i.e. we want to solve the $\omega$-stabilization problem for the heat equation with memory. We first show that the spectrum of the state operator $A$ associated with this equation has an accumulation point at $-{\omega }_0<0$. Given a $\omega \in (0,{\omega }_0)$, we show that the $\omega$-stabilization problem for the heat equation with memory is solvable provided certain verifiable conditions on the control operator $B$ associated with this equation hold. We then consider an appropriate LQR problem for the heat equation with memory. For each $n\in \mathbb{N}$, we construct finite-dimensional approximations $A_n$ and $B_n$ of $A$ and $B$, respectively, and then show that by solving a corresponding approximation of the LQR problem a feedback operator $K_n$ can be computed such that all the eigenvalues of $A_n+B_n{K}_n$ have real part less than $-\omega$. We prove that $K_n$ for $n$ sufficiently large solves the $\omega$-stabilization problem for the heat equation with memory. A crucial and nontrivial step in our proof is establishing the uniform (in $n$) stabilizability of the pair $(A_n+\omega I,B_n)$. We have validated our theoretical results numerically using two examples: an 1D example on a unit interval and a 2D example on a square domain.