Output reversibility in linear discrete-time dynamical systems

Output reversibility involves dynamical systems where for every initial condition and the corresponding output there exists another initial condition such that the output generated by this initial condition is a time-reversed image of the original output with the time running forward. Through a series of necessary and sufficient conditions, we characterize output reversibility in linear discrete-time dynamical systems in terms of the geometric symmetry of its eigenvalue set with respect to the unit circle in the complex plane. Furthermore, we establish that output reversibility of a linear continuous-time system implies output reversibility of its discretization. In addition, we present a control framework that allows to alter the system dynamics in such a way that a discrete-time system, otherwise not output reversible, can be made output reversible. Finally, we present numerical examples involving a discretization of a Hamiltonian system that exhibits output reversibility and an example of a controlled system that is rendered output reversible.