Minimal Controllability Problem on Linear Structural Descriptor Systems With Forbidden Nodes

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.