Linear quadratic control of parabolic-like evolutions with memory of the inputs
Paolo Acquistapace, Francesca Bucci
TL;DR
The paper tackles a finite-horizon linear-quadratic control problem for abstract parabolic-like evolutions with memory, allowing an unbounded input operator and memory entered through a kernel $k$. Using a dynamic-programming/variational framework, it proves the existence and uniqueness of a closed-loop optimal control, expressed via a triple of operator gains $P_0(t)$, $P_1(t,p)$, $P_2(t,p,q)$ that solve a coupled system of quadratic differential equations, recovering the classical Riccati theory in the memoryless limit. The results extend LQ theory for PDEs to memoryful, unbounded-input settings, providing a rigorous synthesis of feedback gains $B^*P_0(t)$ and $P_1(t,p)$ and ensuring well-posedness of the associated Riccati-type system. This work thus advances the optimal control of memory-influenced PDEs, with potential impact on boundary-control problems in heat conduction and thermoelasticity where past input effects are significant.
Abstract
A study of the linear quadratic (LQ) control problem on a finite time interval for a model equation in Hilbert spaces which comprehends the memory of the inputs was performed recently by the authors. The outcome included a closed-loop representation of the unique optimal control, along with the derivation of a related coupled system of three quadratic (operator) equations which is shown to be well-posed. Notably, in the absence of memory the above elements -- namely, formula and system -- reduce to the known feedback formula and single differential Riccati equation, respectively. In this work we take the next natural step, and prove the said results for a class of evolutions where the control operator is no longer bounded. These findings appear to be the first ones of their kind; furthermore, they extend the classical theory of the LQ problem and Riccati equations for parabolic partial differential equations.