# Controllability of impulse controlled systems of heat equations coupled by constant matrices

## TL;DR

This paper investigates the controllability of heat equation systems with impulse controls coupled via constant matrices, establishing conditions for approximate controllability and demonstrating limitations for null controllability.

## Contribution

It provides a necessary and sufficient condition for approximate controllability based on Kalman's rank condition and characterizes the control instants needed for realization.

## Key findings

- Approximate controllability is characterized by Kalman's rank condition.
- Controllability over an interval can be achieved with controls at specific instants.
- Such systems are generally not null controllable.

## Abstract

This paper studies the approximate and null controllability for impulse controlled systems of heat equations coupled by a pair (A,B) of constant matrices. We present a necessary and sufficient condition for the approximate controllability, which is exactly Kalman's controllability rank condition of (A,B). We prove that when such a system is approximately controllable, the approximate controllability over an interval [0,T] can be realized by adding controls at arbitrary n different control instants 0<\tau_1<\tau_2<\cdots<\tau_n<T, provided that \tau_n-\tau_1<d_A, where d_A=\min\{\pi/|Im \lambda| : \lambda\in \sigma(A)\}. We also show that in general, such systems are not null controllable.