Minimal controllability problem on linear structural descriptor systems with forbidden nodes

TL;DR

This paper addresses the minimal controllability problem for descriptor systems with forbidden nodes, providing solvability conditions, optimal solutions, and an efficient algorithm based on graph theory and maximum matching.

Contribution

It introduces a new framework for MCP with forbidden nodes, deriving solvability conditions, optimal values, and an efficient algorithm, extending previous MCP work.

Findings

Solvability condition for MCP with forbidden nodes using bipartite graph theory.

Explicit formula for the optimal value of MCP with forbidden nodes.

An efficient alternating path algorithm for solving the MCP with forbidden nodes.

Abstract

We consider a minimal controllability problem (MCP), which determines the minimum number of input nodes for a descriptor system to be structurally controllable. We investigate the "forbidden nodes" in descriptor systems, denoting nodes that are unable to establish connections with input components. The three main results of this work are as follows. First, we show a solvability condition for the MCP with forbidden nodes using graph theory such as a bipartite graph and its Dulmage--Mendelsohn decomposition. Next, we derive the optimal value of the MCP with forbidden nodes. The optimal value is determined by an optimal solution for constrained maximum matching, and this result includes that of the standard MCP in the previous work. Finally, we provide an efficient algorithm for solving the MCP with forbidden nodes based on an alternating path algorithm.