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arXiv · 2503.14046

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

Paolo Acquistapace·Francesca Bucci
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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.

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Paolo Acquistapace, Francesca Bucci. 2025-03-18. Linear quadratic control of parabolic-like evolutions with memory of the inputs. https://arxiv.org/abs/2503.14046

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