LQR based ω-stabilization of a heat equation with memory

Abstract 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.