$ L^p $-exact controllability of partial differential equations with nonlocal terms

Luisa Malaguti; Stefania Perrotta; Valentina Taddei

doi:10.3934/eect.2021053

Article Contents

2022, Volume 11, Issue 5: 1533-1564. Doi: 10.3934/eect.2021053

This issue Previous Article Exponential stabilization of a linear Korteweg-de Vries equation with input saturation Next Article Robustness of global attractors: Abstract framework and application to dissipative wave equations

$ L^p $-exact controllability of partial differential equations with nonlocal terms

1.
Department of Sciences and Methods for Engineering, University of Modena and Reggio Emilia, I-42122, Italy
2.
Department of Physics, Informatics and Mathematics, University of Modena and Reggio Emilia, I-41125, Italy

^* Corresponding author: luisa.malaguti@unimore.it
^* Corresponding author: luisa.malaguti@unimore.it

Received: February 2021

Revised: August 2021

Early access: October 2021

Published: October 2022

The authors are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) and acknowledge financial support from this institution

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Abstract

The paper deals with the exact controllability of partial differential equations by linear controls. The discussion takes place in infinite dimensional state spaces since these equations are considered in their abstract formulation as semilinear equations. The linear parts are densely defined and generate strongly continuous semigroups. The nonlinear terms may also include a nonlocal part. The solutions satisfy nonlocal properties, which are possibly nonlinear. The states belong to Banach spaces with a Schauder basis and the results exploit topological methods. The novelty of this investigation is in the use of an approximation solvability method which involves a sequence of controllability problems in finite-dimensional spaces. The exact controllability of nonlocal solutions can be proved, with controls in $ L^p $ spaces, $ 1<p<\infty $. The results apply to the study of the exact controllability for the transport equation in arbitrary Euclidean spaces and for the equation of the nonlinear wave equation.

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Mathematics Subject Classification: Primary: 93B05; Secondary: 34B10, 47H10.

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