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Continuity of solutions to abstract linear control systems

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

Contributed by Frédéric Marbach

Version 2 / 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 novel idea (identifying Phillips' lemma as the key ingredient) was produced 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 the proofs of all numbered statements of the paper (both informally, and by Lean formalization). The formalization was done by a combination of GPT-6 Pro and Opus 5.5, and registered in Palomar.
FormalizationsPalomar

Tools used

OpenAI
ChatGPTVersion 6 Pro
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Claude OpusVersion 5.5

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Version history

  1. v1Initial depositSupersededOct 03, 2026Submitted by Frédéric Marbach
  2. v2Submitted as a new versionCurrentOct 03, 2026Submitted by Frédéric Marbach