LQR based $ω-$stabilization of a heat equation with memory

TL;DR

The paper tackles ω-stabilization for a heat equation with memory on bounded domains by formulating an abstract evolution on X = L^2(Ω) × H^1_0(Ω) and proving spectral and semigroup properties of the state operator A. It then demonstrates ω-stabilizability via Hautus-type conditions on the control operator B for 0<ω<ω0, and designs a numerically tractable LQR controller using finite-dimensional Galerkin approximations A_n, B_n and Riccati equations to obtain Pi_n and K_n. The main theoretical result shows that for n large enough, K_n yields A_ω,n + B_n K_n stable, and K_n converges to the infinite-dimensional optimum K_∞ as n → ∞, yielding a practical ω-stabilizing controller K_ω = K_n P_n. The authors validate the approach with 1D and 2D numerical examples, illustrating eigenvalue placement and exponential decay in closed-loop dynamics. This work advances finite-dimensional, Riccati-based stabilization for PDEs with memory and suggests paths to robust, reduced-order output-feedback designs in future work.

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 $ω>0$, i.e. we want to solve the $ω$-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 $-ω_0<0$. Given a $ω\in(0,ω_0)$, we show that the $ω$-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 $-ω$. We prove that $K_n$ for $n$ sufficiently large solves the $ω$-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 $(A_n+ωI, B_n)$. 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.

Figures (1)

• Figure 1: The boundaries of the sectors $\Sigma$ in \ref{['eq:sectornu']} and $\Sigma_0$ in \ref{['eq:sigma0']} are shown using red and blue colored rays, respectively. The yellow dashed line represents the boundary of $\Sigma$ which lies inside $\overline{\Sigma_0}$. In Step 1 of the proof of Proposition \ref{['pr:unifstab']} we have shown that (for $n$ sufficiently large) the eigenvalues of $A^K_{\omega,n}$ are contained in $\Sigma_0$ and have real part less than $-\beta$, i.e. they lie in the green shaded region. Hence they are contained in $\Sigma$.

Theorems & Definitions (16)

• Theorem 3.1
• Theorem 3.2
• proof
• Remark 3.3
• Theorem 3.4
• proof
• Theorem 3.5
• Remark 3.6
• proof
• Proposition 4.2