Boundary approximate controllability under positivity constraints of linear systems

TL;DR

This paper develops frequency domain criteria for boundary approximate controllability of infinite-dimensional systems under positivity constraints, providing necessary and sufficient conditions and applying them to transportation and heat networks.

Contribution

It introduces new controllability conditions under positivity constraints, including a Kalman-type rank criterion for network systems, and analyzes controllability limitations for heat equations.

Findings

Controllability under positivity constraints characterized by a Kalman-type rank condition.

Approximate controllability achieved for heat networks with a single positive input.

Lack of controllability under unilateral control constraints for certain systems.

Abstract

This paper focuses on boundary approximate controllability under positivity constraints of a wide range of infinite-dimensional control systems. We develop frequency domain controllability criteria. Firstly, we derive a controllability result under positivity constraints on the control for such systems. Then, and more importantly, we provide a necessary and sufficient condition for controllability under positivity constraints on the control and the state. The obtained results are applied to the controllability of transportation and heat conduction networks. In particular, provided that the underlying graph is strongly connected, the controllability under positivity constraints on the control/state of transport network systems is fully characterized by a Kalman-type rank condition. For a system of heat equations with Robin boundary conditions on a path-like network, we establish approximate controllability under positivity state-constraint with a single positive input through the starting node. However, we prove the lack of controllability under unilateral control-constraint.