Linear output-reversible systems

In the present paper we consider the free response of a linear system involving dynamics and an output map. A second system is an output reversal of the original system if it can produce a time-reversed image every output of the original system. As a special case, a system is output reversible if it can produce time-reversed image of every one of its free output responses. Our main result is a spectral symmetry condition that provides a complete characterization of single-input, single-output, output-reversible systems. In particular, we show that a linear system is output-reversible if and only if its non-imaginary spectrum is symmetric with respect to the imaginary axis. As special cases, the class of output-reversible systems includes rigid body and Hamiltonian systems. This result suggests that stability and instability play a key role in the arrow of time, independently of dimensionality, continue, and sensitivity.