Minimal controllability problems on linear structural descriptor systems

TL;DR

This paper investigates minimal controllability problems in linear structural descriptor systems, revealing polynomial-time solutions for some cases and NP-hardness for others, thus extending controllability theory to descriptor systems.

Contribution

It demonstrates that MCP0 can be solved in polynomial time for descriptor systems and shows MCP1 is NP-hard, contrasting with traditional LTI systems.

Findings

MCP0 for descriptor systems is solvable in polynomial time.

MCP1 for descriptor systems is NP-hard.

Different derivation techniques are needed compared to LTI systems.

Abstract

We consider minimal controllability problems (MCPs) on linear structural descriptor systems. We address two problems of determining the minimum number of input nodes such that a descriptor system is structurally controllable. We show that MCP0 for structural descriptor systems can be solved in polynomial time. This is the same as the existing results on typical structural linear time invariant (LTI) systems. However, the derivation of the result is considerably different because the derivation technique of the existing result cannot be used for descriptor systems. Instead, we use the Dulmage--Mendelsohn decomposition. Moreover, we prove that the results for MCP1 are different from those for usual LTI systems. In fact, MCP1 for descriptor systems is an NP-hard problem, while MCP1 for LTI systems can be solved in polynomial time.