Greedy controllability of finite dimensional linear systems

TL;DR

This paper develops greedy algorithms for efficiently controlling parameter-dependent finite-dimensional linear systems, outperforming simple sampling methods, especially in heat-like equations due to dissipativity effects.

Contribution

It introduces optimal greedy-based procedures for controllability analysis of parameter-dependent systems, improving control approximation efficiency over traditional sampling methods.

Findings

Greedy algorithms outperform simple sampling in control approximation.

Heat-like equations require fewer samples due to dissipativity.

Numerical experiments confirm the efficiency of the proposed methods.

Abstract

We analyse the problem of controllability for parameter-dependent linear finite-dimensional systems. The goal is to identify the most distinguished realisations of those parameters so to better describe or approximate the whole range of controls. We adapt recent results on greedy and weak greedy algorithms for parameter depending PDEs or, more generally, abstract equations in Banach spaces. Our results lead to optimal approximation procedures that, in particular, perform better than simply sampling the parameter-space to compute the controls for each of the parameter values. We apply these results for the approximate control of finite-difference approximations of the heat and the wave equation. The numerical experiments confirm the efficiency of the methods and show that the number of weak-greedy samplings that are required is particularly low when dealing with heat-like equations, because of the intrinsic dissipativity that the model introduces for high frequencies.