Computable Feasible-Horizon Sets

Determine computable lower boundaries of the admissible-horizon set for broad nonlinear systems with actuator magnitude and rate limits, uncertainty, safety constraints, and a common initial set, and characterize the associated deadline-dependent viable initial-state set.

Background

The review separates the plant’s minimum achievable time, the minimum horizon attainable by a controller family, the settling time of a particular tuning, and the bound certified by a Lyapunov proof. Existing results address selected constrained fixed-time or prescribed-time problems, but sharp nonlinear lower bounds with both magnitude and rate limits remain difficult.

The proposed characterization should use consistent quantifiers over the initial set, admissible inputs, feedback-law family, uncertainty, accuracy, and retention after arrival. It would replace gain escalation by an explicit computation of whether a requested deadline is physically and architecturally feasible.

References

The open problem is to make the lower boundary of $\mathcal T_{\mathrm{adm}$ computable for broad nonlinear classes and to quantify the gaps $T_{\mathrm{plant}{\min}(\mathcal X_0)\le T_{\mathcal C}{\min}(\mathcal X_0)\le T_{\kappa,\sup}(\mathcal X_0)\le T{\mathrm{cert}_\kappa(\mathcal X_0), \qquad \kappa\in\mathcal C$, where $T_{\kappa,\sup}(\mathcal X_0)=\sup_{z_0\in\mathcal X_0}T_\kappa(z_0)$ and $T{\mathrm{cert}_\kappa$ is the certified bound, all four for the same initial set, accuracy, uncertainty, and admissible inputs.

— From Asymptotic to Designer-Assigned-Time Control: A Review of Stability Notions, Design Mechanisms, and Controller Architectures  (2610.11964 - Bingöl, 8 Oct 2026) in Section 5.1, paragraph “Physical lower bounds, controller-family limits, and admissible horizons”

Open is a joint feasibility theory for $\text{safety} + \text{transient envelope} + \text{terminal accuracy} + \text{deadline} + \text{input magnitude/rate}$ that distinguishes remaining safe, returning to a safe set, and reaching a target without leaving it, and that returns the induced change of deadline or accuracy as an output of the analysis rather than as an unspecified slack variable.

— From Asymptotic to Designer-Assigned-Time Control: A Review of Stability Notions, Design Mechanisms, and Controller Architectures  (2610.11964 - Bingöl, 8 Oct 2026) in Section 5.1, paragraph “Zero-width envelopes, safety, and actuator feasibility”

The open direction is a unified specification and comparison framework in which the probability level, noise intensity, certified initial set, and control authority are explicit.

— From Asymptotic to Designer-Assigned-Time Control: A Review of Stability Notions, Design Mechanisms, and Controller Architectures  (2610.11964 - Bingöl, 8 Oct 2026) in Section 5.4, paragraph “Stochastic temporal guarantees”