Continuity of states driven by bounded admissible inputs
Let $X$ and $U$ be Banach spaces, let $A$ generate a strongly continuous semigroup $S(t)$ on $X$, and let $X_{-1}$ be the completion of $X$ for the norm $\|x\|_{-1} = \|(\lambda - A)^{-1} x\|_X$, where $\lambda$ is in the resolvent set of $A$. Denote the extended semigroup on $X_{-1}$ by $S_{-1}$ and fix $B \in \mathcal{L}(U, X_{-1})$. Assume that $B$ is $L^\infty$-admissible: for every $T > 0$ there is $K_T < \infty$ such that for every strongly measurable $u \in L^\infty(0,T;U)$, $$\Phi_T u := \int_0^T S_{-1}(T-s) B u(s)\, ds \in X, \qquad \|\Phi_T u\|_X \le K_T \|u\|_{L^\infty}.$$
Weiss proved that if $B$ is $L^p$-admissible for some $1 \le p < \infty$, then $t \mapsto \int_0^t S_{-1}(t-s) B u(s)\, ds$ is continuous from $[0,T]$ to $X$ for every $u \in L^p(0,T;U)$. Preußler and Schwenninger proved the same for $L^\infty$-admissible $B$ and every $u \in L^\infty(0,T;U)$ when $X$ is reflexive, or $X$ has nontrivial type, or $X$ does not contain $c_0$ and $C([0,1];U)$ has property (V), or $X$ is weakly compactly generated and $L^\infty([0,1];U)$ is a Grothendieck space.
Problem. Must $t \mapsto \int_0^t S_{-1}(t-s) B u(s)\, ds$ be continuous from $[0,T]$ to $X$ for every such input $u$ and every $T > 0$?
Source: Weiss (1989), Problem 2.4; see Preußler and Schwenninger (2026), §4.4.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- OC — Optimization and Control
- Secondary
- FA
- Keywords
- admissible control operators, AIM Open Applied Problems, input-to-state stability, operator semigroups
References
@article{weiss1989,
author = {Weiss, George},
title = {Admissibility of unbounded control operators},
journal = {SIAM J. Control Optim.},
volume = {27},
number = {3},
year = {1989},
pages = {527--545},
doi = {10.1137/0327028},
}
@misc{arora2026,
author = {Arora, Sahiba and Preu{\ss}ler, Philip and Schwenninger, Felix L.},
title = {{ISS} and integral {ISS} coincide for linear systems with bounded inputs},
year = {2026},
eprint = {2609.40348},
archivePrefix = {arXiv},
}
@misc{preussler2026,
author = {Preu{\ss}ler, Philip and Schwenninger, Felix L.},
title = {Implications of structured continuous maximal regularity},
year = {2026},
eprint = {2605.12121},
archivePrefix = {arXiv},
}
@misc{aim-bounded-input-state-continuity,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Continuity of states driven by bounded admissible inputs}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/411-bounded-input-state-continuity.md},
note = {Entry 411 at commit a153f46, 2026-10-13},
}
Resolution claims
Let \(A\) generate a strongly continuous semigroup on an arbitrary Banach space \(X\), and let \(B\) be a bounded operator from an arbitrary Banach input space into the extrapolation space \(X_{-1}\). We prove that \(L^\infty\)-admissibility for all strongly measurable inputs implies that the finite-time control operator norms tend to zero with the time horizon. Consequently every state driven by a …
We show that input-to-state stability (ISS) and integral input-to-state stability (integral ISS) with respect to the space of essentially bounded functions are equivalent properties for linear systems. The proof combines recent results on Orlicz-admissibility and a duality result due to the authors. As a direct consequence, we show that, given an $L^\infty$-admissible control operator, the mild solutions of the linear …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: arXiv:2609.40348v1. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00490, "Continuity of states driven by bounded admissible inputs", clopen.solutions.