A Controllability Problem of Finite-Field Networks

In this paper, the controllability of finite-field network (FFN) with single input is investigated through an algebra-theoretic perspective. An algebraic criterion on the matrix pair for controllability is derived, distinguishing FFNs from the classic real/complex-valued networks, and involving the algebraic structure caused by the system matrix pair. Basing on this, we further study a minimal controllability problem, i.e., finding a minimum number of agents to be affected by input, to make the system controllable. For FFN with single input, we present that the minimum number desired depends on the number of elementary divisors of the system matrix, if the given base satisfies certain condition. Meanwhile, we provide the corresponding method of constructing an optimal solution to the minimal controllability problem. In the end, we show that the set of all controllable pairs is dense in some sense by figuring the probability of the occurrence of a controllable pair.