# Controllability of evolution equations with memory

## TL;DR

This paper investigates the null controllability of evolution equations with memory, addressing the challenges posed by memory terms and establishing controllability conditions using duality and Carleman estimates.

## Contribution

It introduces new controllability criteria for evolution equations with memory, including finite-dimensional rank conditions and Carleman estimate-based results for parabolic equations.

## Key findings

- Rank conditions for controllability in finite-dimensional systems
- Null controllability established for certain parabolic equations with memory
- Use of Carleman estimates to prove controllability

## Abstract

This article is devoted to studying the null controllability of evolution equations with memory terms. The problem is challenging not only because the state equation contains memory terms but also because the classical controllability requirement at the final time has to be reinforced, involving the contribution of the memory term, to ensure that the solution reaches the equilibrium. Using duality arguments, the problem is reduced to the obtention of suitable observability estimates for the adjoint system. We first consider finite-dimensional dynamical systems involving memory terms and derive rank conditions for controllability. Then the null controllability property is established for some parabolic equations with memory terms, by means of Carleman estimates.