Riccati-based solution to the optimal control of linear evolution equations with finite memory

Paolo Acquistapace; Francesca Bucci

doi:10.3934/eect.2023035

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2024, Volume 13, Issue 1: 26-66. Doi: 10.3934/eect.2023035

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Riccati-based solution to the optimal control of linear evolution equations with finite memory

Paolo Acquistapace^1, and
Francesca Bucci^2, ,

1.
Dipartimento di Matematica, Università di Pisa (Ret.), Italy
2.
Dipartimento di Matematica e Informatica, Università degli Studi di Firenze, Italy

^*Corresponding author: Francesca Bucci
^*Corresponding author: Francesca Bucci

Received: April 2023

Revised: June 2023

Early access: July 4, 2023

Published online: July 4, 2023

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Abstract

In this article we study the optimal control problem with quadratic functionals for a linear Volterra integro-differential equation in Hilbert spaces. With the finite history seen as an (additional) initial datum for the evolution, following the variational approach utilized in the study of the linear-quadratic problem for memoryless infinite dimensional systems, we attain a closed-loop form of the unique optimal control via certain operators that are shown to solve a coupled system of quadratic differential equations. This result provides a first extension to the partial differential equations realm of the Riccati-based theory recently devised by L. Pandolfi in a finite dimensional context.

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Mathematics Subject Classification: Primary: 49N10, 35R09, 93C23, 49N35.

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References

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