Almost Input-to-State Stability of the Damped Pendulum

Definitions and mathematical statement

Let $M=S^1\times{\mathbb{R}}$, where $S^1={\mathbb{R}}/(2\pi{\mathbb{Z}})$, and write $x=(\theta,\omega)$. Consider the controlled damped pendulum \[ \dot\theta=\omega,\qquad \dot\omega=-\sin\theta-\omega+d. \] For the forward-time formulation, an essentially bounded input is a measurable function $d:[0,\infty)\to{\mathbb{R}}$ with \[ \lVert d\rVert_\infty=\operatorname*{ess\,sup}_{t\geq0}|d(t)|<\infty. \] Write $x(t;x_0,d)$ for the trajectory starting from $x_0\in M$ at time zero; for measurable inputs, the equations are understood to hold almost everywhere along locally absolutely continuous trajectories. A function $\gamma:[0,\infty)\to[0,\infty)$ belongs to class $\mathcal K$ if it is continuous, strictly increasing, and satisfies $\gamma(0)=0$; unboundedness is not required.

The supplied question asks whether this system is almost input-to-state stable in the following stated sense: does there exist a single $\gamma\in\mathcal K$ such that, for every essentially bounded input $d$, there is an exceptional set $B_d\subseteq M$ of measure zero for which every $x_0\in M\setminus B_d$ satisfies \[ \limsup_{t\to\infty}|x(t;x_0,d)|\leq\gamma\left(\lVert d\rVert_\infty\right)? \] The intended assertion requires these trajectories to be defined for all $t\geq0$. The exceptional set may depend on the entire input $d$, while $\gamma$ must work for every input.

Unresolved geometric qualifications. The source does not define the state magnitude $|x|$ on the cylinder $M$, identify the reference state or target set relative to which that magnitude is measured, or specify the measure used for the exceptional set. To display the quantifiers without silently choosing these data, let $\rho:M\to[0,\infty)$ and $\mu$ stand for the intended, but unspecified, state-size function and measure. The proposed property then has the form \[ \exists\gamma\in\mathcal K\quad \forall d\in L^\infty([0,\infty);{\mathbb{R}})\quad \exists B_d\subseteq M\text{ measurable},\quad \mu(B_d)=0, \] \[ \forall x_0\in M\setminus B_d,\qquad \limsup_{t\to\infty}\rho\left(x(t;x_0,d)\right) \leq\gamma\left(\lVert d\rVert_\infty\right). \] Here $\rho$ and $\mu$ are placeholders for missing source data, not additional universally or existentially quantified choices. A fully specified mathematical question requires those data to be fixed.

Source qualification. Although the supplied statement fields present this pendulum question, the supplied literature assessment describes the extracted record as only a heading and a definition, insufficient for a standalone proposition. That discrepancy remains unresolved. The supplied open label is a curation status, not an independently established mathematical conclusion.

Short English statement

The proposed question asks for one input-dependent asymptotic bound that holds for every bounded forcing and for all initial states outside an exceptional null set that may depend on the forcing. The source leaves the state magnitude, target, and reference measure unspecified, and its literature assessment questions whether the extraction contains a complete proposition. This is a statement-only draft requiring human mathematical review.

Sources

• Supplied local problem registry: permanent UnsolvedMath ID $30001200$, exact source code catalogue entry, title Almost Input-to-State Stability of the Damped Pendulum. The supplied collection is Oberwolfach Reports Open Problems; the proposed year is 2009. No individual proposer is identified.
• Supplied source citation: Control Theory: On the Way to New Application Fields (2009), Oberwolfach Reports, pp. 593--680, DOI 10.4171/owr/2009/11. The source URL and the literature-source link labelled "original Oberwolfach report" are https://doi.org/10.4171/owr/2009/11. The supplied background describes Oberwolfach Reports as documenting workshops and their open-problem discussions.
• Supplied literature source labelled "Angeli 2004": https://doi.org/10.1109/TAC.2004.829594. No fuller bibliographic details or precise supporting passage are supplied.
• Literature assessment dated exactly 2026-08-21: the supplied status is "open," assigned under a triage rule because the extraction was assessed as a heading and definition rather than a complete standalone proposition. This is supplied dated curation derived from the OWR record, not independently verified source consultation or a verified current literature assessment. The supplied background requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and says its complete statement was preserved while malformed or undelimited notation was normalized. These are supplied editorial assertions; they do not resolve the discrepancy with the literature assessment or establish mathematical correctness. No cited source was independently consulted for this draft.