Linear Operator Differential Equations with Applications to Linear Time-Varying Systems

A classical result in linear system theory is that a causal linear time-varying (LTV) system has a smooth finite dimensional realization if and only if its impulse response exhibits a certain factorization. In which case, the standard four canonical realizations can be written explicitly in terms of the coefficient functions appearing in various forms of the inputoutput differential equation. On the other hand, little appears to be known about the non-factorizable case. Thus, the first goal of this paper is to examine in detail one specific instance of an LTV system whose impulse response is not factorizable and to show explicitly that its input-output map satisfies a type of linear functional differential equation, which will be referred to as a linear operator differential equation. This leads to the second goal of the paper which is to describe the general solutions of first-order and second-order linear operator differential equations. The treatment in the second-order case is focused primarily on singular equations, which are shown to describe the non-factorizable system presented in the first part of the paper.