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Continuity of solutions to abstract linear control systems

Primarily AI-generated textHuman understanding: some partsmath.OC — Optimization and Controlmath.FA — Functional Analysis

Contributed by Frédéric Marbach ↗

Version 1 / Oct 03, 2026 / CC BY-SA 4.0

Abstract

It has long been known that the solutions to abstract linear control systems are continuous in time for controls in LpL^p with 1≤p<∞1 \le p < \infty. We prove that the same property remains valid for the endpoint case p=∞p = \infty, giving a positive answer to Weiss' 1989 Problem 2.4. The proof relies on a direct semigroup argument based on Phillips' lemma, does not require the input map to have an integral representation (which is not always the case for p=∞p = \infty), and actually entails that such systems are all of the zero-class (which fails for 1≤p<∞1 \le p < \infty).

Provenance statement

The key idea (using Phillips' lemma) was identified by GPT-6 Pro in its first attempt. The writing enhancement was done with the help of Opus 5.5. I have understood and checked (both informally, and by Lean formalization), the proofs of the main theorem and its corollary (Section 2). I have however neither fully understood nor checked the claims on the existence of systems which do not admit an integral representation (Section 3.2), which is nevertheless a classical fact in the field. The formalization was done by a combination of GPT-6 Pro and Opus 5.5.
FormalizationsPalomar

Tools used

OpenAI
ChatGPTVersion 6 Pro
Anthropic
Claude OpusVersion 5.5

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Version history

  1. v1Initial depositCurrentOct 03, 2026Submitted by Frédéric Marbach