Koopman operator approach for computing structure of solutions and Observability of non-linear finite state system

Abstract

Given a discrete dynamical system defined by a map in a vector space over a finite field called Finite State Systems (FSS), a dual linear system over the space of functions on the state space is constructed using the dual map. This system constitutes the well known Koopman linear system framework of dynamical systems, hence called the Koopman linear system (KLS). It is first shown that several structural properties of solutions of the FSS can be inferred from the solutions of the KLS. The problems of computation of structural parameters of solutions of non-linear FSS are computationally hard and hence become infeasible as the number of variables increases. In contrast, it has been well known that these problems can be solved by linear algebra for linear FSS in terms of elementary divisors of matrices and their orders. In the next step, the KLS is reduced to the smallest order (called RO-KLS) while still retaining all the information of the parameters of structure of solutions of the FSS. Hence when the order of the RO-KLS is sufficiently small, the above computational problems of non-linear FSS are practically feasible. Next, it is shown that the observability of the non-linear FSS with an output function is equivalent to that of the RO-KLS with an appropriate linear output map. Hence, the problem of non-linear observability is solved by an observer design for the equivalent RO-KLS. Such a construction should have striking applications to realistic FSS arising in Cryptology and Biological networks.