Linear quadratic control of parabolic-like evolutions with memory of the inputs

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 was 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.