# Controllability properties for equations with memory of fractional type

## TL;DR

This paper investigates control systems with memory, including fractional derivatives, proving that approximate controllability is preserved while exact controllability to zero is a special case only for standard heat equations.

## Contribution

It establishes that approximate controllability extends to a broad class of memory systems, highlighting the uniqueness of the heat equation in controllability to zero.

## Key findings

- Approximate controllability is inherited by systems with memory.
- Controllability to zero is a singular property specific to the heat equation.
- The heat equation uniquely allows controllability to zero among memory systems.

## Abstract

We study a general class of control systems with memory, which in particular includes systems with fractional derivatives and integrals and also the standard heat equation. We prove that the approximate controllability property of the heat equation is inherited by every system with memory in this class while controllability to zero is a singular property, which holds solely in the special case that the system indeed reduces to the standard heat equation.