On characteristic cones of discrete autonomous nD systems: theory and algorithm

In this paper, we provide a complete answer to the question of characteristic cones for a general discrete autonomous nD system described by partial difference equations with real constant coefficients. A characteristic cone is a special subset (having the structure of a cone) of the domain, here Zn, such that the knowledge of the trajectories on this set uniquely determines them over the whole domain. The study of characteristic sets is relevant in many system-theoretic aspects. This importance of characteristic sets stems from the fact that they help quantify the ‘information’ required to solve a system of partial difference equations. In spite of its importance, the issue of characteristic sets for multidimensional systems have not been explored in its full generality except for Valcher’s seminal work for the special case of 2D systems in 2000. This apparent lack of progress in the last fifteen years is perhaps due to inapplicability of a crucial intermediate result by Valcher to cases with n > 2. We illustrate this inapplicability of the above-mentioned result in Section 3 with the help of an example. We then provide an answer to this open problem of characterizing characteristic cones for general n by proving a necessary and sufficient condition for a cone to be a characteristic cone for a given system of partial difference equations. In the second ∗The authors are with the Department of Electrical Engineering, Indian Institute of Technology Bombay, Powai, Mumbai, India. E-mail: {mousumi, debasattam}@ee.iitb.ac.in