The well-posedness of Riccati equations plays a central role in the study of the optimal control problem with quadratic functionals for linear partial differential equations (PDEs). Indeed, it allows the synthesis of the optimal control by solving the Riccati equation corresponding to the minimization problem, and then of the closed-loop equation. In this lecture I will first recall the principal steps of an established approach to the linear-quadratic problem for infinite-dimensional systems representing PDEs with distributed or boundary control. The failures or challenges stemming from the presence of unbounded control operators, combined with a hyperbolic or composite dynamics, will be highlighted. Finally, I will outline how the said approach proves successful even in the case of certain evolution equations with finite memory, thereby providing a first extension (to the realm of PDE’s) of the Riccati-based theory recently devised by L. Pandolfi in a finite dimensional context.

(Talk based on past work with P. Acquistapace (Pisa) and I. Lasiecka (Memphis), as well as on ongoing joint work with P. Acquistapace.)

Zoom meeting: Link