Some complexity problems on single input double output controllers

We investigate the time complexity of constructing single input double output state feedback controller structures, given the directed structure graph G of a system. Such a controller structure defines a restricted type of P3-partition of the graph G. A necessary condition (∗) has been found and two classes of graphs have been identified where the search problem of finding a feasible P3partition is polynomially solvable and, in addition, (∗) is not only necessary but also sufficient for the existence of a P3-partition. It is shown further that the decision problem on two particular graph classes — defined in terms of forbidden subgraphs — remains NP complete, but is polynomially solvable on the intersection of those two classes. Moreover, for every natural number m, a stabilizing structure with Single Input m-Output controllers can be found in polynomial time for the system in question, if it admits one. ∗Basic Research in Computer Science, Centre of the Danish National Research Foundation.