Characterization of input-to-output stability for infinite-dimensional systems

We prove a superposition theorem for input-to-output stability (IOS) of a broad class of nonlinear infinite-dimensional systems with outputs including both continuous-time and discrete-time systems. It contains, as a special case, the superposition theorem for input-to-state stability (ISS) of infinite-dimensional systems and the IOS superposition theorem for systems of ordinary differential equations known from the literature. To achieve this result, we introduce and examine several novel stability and attractivity concepts for infinite-dimensional systems with outputs: We prove criteria for the uniform limit property for systems with outputs, several of which are new already for systems with full-state output, we provide superposition theorems for systems which satisfy both the output Lagrange stability (OL) and IOS, give a sufficient condition for OL and characterize ISS in terms of IOS and input/output-to-state stability. Finally, by means of counterexamples, we illustrate the challenges appearing on the way of extension of the superposition theorems from the literature to infinite-dimensional systems with outputs.

💡 Research Summary

The paper “Characterization of input‑to‑output stability for infinite‑dimensional systems” develops a comprehensive theory of input‑to‑output stability (IOS) for a very broad class of nonlinear infinite‑dimensional control systems, covering both continuous‑time and discrete‑time dynamics. The authors start by observing that while input‑to‑state stability (ISS) has already been extended from ordinary differential equations (ODEs) to infinite‑dimensional settings such as time‑delay systems, PDEs, and abstract evolution equations, these extensions have always assumed that the output coincides with the state (full‑state output). In many practical situations, however, the output is a partial measurement, a tracking error, an observer error, or any other functional of the state and input. IOS is precisely the notion that captures robust stability of such output dynamics in the presence of external inputs.

To treat this general situation, the authors introduce a rigorous abstract framework: a control system is a sextuple (\Sigma=(I,X,U,\phi,Y,h)) where (I) is the time set ((\mathbb{R}_+) or (\mathbb{N}_0)), (X) and (Y) are Banach spaces for state and output, (U) is a space of admissible input functions, (\phi) is the transition map satisfying identity, causality and cocycle properties, and (h) is the output map. They assume forward completeness (every trajectory exists for all forward times), output continuity at the equilibrium point (OCEP), and boundedness of output reachability sets (BORS).

Within this setting a series of new stability concepts are defined:

• Output‑Lagrange stability (OL) – the output remains bounded for any bounded input and converges to zero when the input is zero.
• Output‑limit property (OLIM) – for any finite horizon there exists a small enough initial condition and input norm that guarantee the output stays below any prescribed (\varepsilon).
• Output‑uniform asymptotic gain (OUAG), output‑global uniform asymptotic gain (OGUAG) and output‑complete asymptotic gain (OCAG) – KL‑type gain estimates that are uniform with respect to initial conditions and inputs.
• Output‑uniform local stability (OULS), input‑output‑to‑state stability (IOSS), and several other notions that bridge ISS and IOS.

The central contribution is Theorem III.1 – the IOS Superposition Theorem. It states that if two systems each satisfy OL and IOS, then their interconnection (parallel or cascade) also satisfies IOS. The proof builds on the fact that OL provides a uniform bound on the output, while IOS supplies a KL‑type decay estimate; the combination yields a KL‑type bound for the composite system. This result generalizes the well‑known ISS superposition theorem for infinite‑dimensional systems (e.g.,