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We consider minimal controllability problems (MCPs) on linear structuraldescriptor systems. We address two problems of determining the minimum numberof input nodes such that a descriptor system is structurally controllable. Weshow that MCP0 for structural descriptor systems can be solved in polynomialtime. This is the same as the existing results on typical structural lineartime invariant (LTI) systems. However, the derivation of the result isconsiderably different because the derivation technique of the existing resultcannot be used for descriptor systems. Instead, we use the Dulmage--Mendelsohndecomposition. Moreover, we prove that the results for MCP1 are different fromthose for usual LTI systems. In fact, MCP1 for descriptor systems is an NP-hardproblem, while MCP1 for LTI systems can be solved in polynomial time.

Minimal controllability problems on linear structural descriptor systems