Input-to-state stability in integral norms for linear infinite-dimensional systems

We study integral-to-integral input-to-state stability for infinite-dimensional linear systems with inputs and trajectories in $L^p$-spaces. We start by developing the corresponding admissibility theory for linear systems with unbounded input operators. While input-to-state stability is typically characterised by exponential stability and finite-time admissibility, we show that this equivalence does not extend directly to integral norms. For analytic semigroups, we establish a precise characterisation using maximal regularity theory. Additionally, we provide direct Lyapunov theorems and construct Lyapunov functions for $L^p$-$L^q$-ISS and demonstrate the results with examples, including diagonal systems and diffusion equations.

💡 Research Summary

The paper investigates input‑to‑state stability (ISS) for infinite‑dimensional linear systems when both inputs and state trajectories are measured in integral norms, i.e., in (L^{p}) and (L^{q}) spaces. The authors begin by extending the classical admissibility theory, which links bounded control operators to well‑posedness, to the case of possibly unbounded input operators and to “(L^{p})–(L^{q}) admissibility”. They show that the familiar equivalence—exponential stability of the semigroup together with finite‑time (L^{p}) admissibility implying ISS—breaks down for integral‑to‑integral ISS. In particular, a counter‑example demonstrates that exponential stability does not guarantee infinite‑time (L^{p})–(L^{q}) admissibility when (p>q).

To overcome this difficulty, the authors focus on analytic semigroups and exploit the theory of maximal (L^{p}) regularity. They prove that for an analytic semigroup generated by (A), the system (\Sigma_{A,B}) is (L^{p})–(L^{p}) admissible for every control operator (B\in\mathcal L(U,X_{-1})) if and only if (A) possesses maximal (L^{p}) regularity on bounded intervals (and on (\mathbb R_{+}) for infinite‑time admissibility). This result bridges admissibility and maximal regularity, providing a precise characterisation of when integral‑norm ISS can be expected.

The paper then defines (L^{p})–(L^{q}) ISS: a system is said to be (L^{p})–(L^{q}) ISS if there exist constants (c,\gamma>0) such that for every horizon (T>0),

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