Continuous LTI Input–Output Stable Systems on $${L^{p}(\mathbb {R})}$$ and $${\mathscr {D'}_{L^{p}}(\mathbb {R})}$$ Associated with Differential Equations: Existence, Invertibility Conditions and Inversion

A usual problem in analog signal processing is to ascertain the existence of a continuous single-input single-output linear time-invariant input–output stable system associated with a linear differential equation, i.e., of a continuous system such that, for every input signal in a given space of signals, yields an output, in the same space, which verifies the equation with known term the input, and to ascertain the existence of its inverse system. In this paper, we consider, as space of signals, the usual Banach space of $${L^{p}}$$ L p functions, or the space of distributions spanned by $${L^{p}}$$ L p functions and by their distributional derivatives, of any order (input spaces which include signals with not necessarily left-bounded support), we give a systematic theoretical analysis of the existence, uniqueness and invertibility of continuous linear time-invariant input–output stable systems (both causal and non-causal ones) associated with the differential equation and, in case of invertibility, we characterize the continuous inverse system. We also give necessary and sufficient conditions for causality. As an application, we consider the problem of finding a suitable almost inverse of a causal continuous linear time-invariant input–output stable non-invertible system, defined on the space of finite-energy functions, associated with a simple differential equation.