Structural Possibility and Impossibility Results

Characterize which variables of nonlinear systems with internal dynamics, non-minimum-phase zero dynamics, underactuation, unknown control directions, and specified admissible input classes can satisfy assigned-time convergence properties, and establish corresponding possibility and impossibility results.

Background

Authority limitations alone do not determine whether an assigned-time guarantee is possible. Internal zero dynamics, underactuation, relative degree, actuation rank, and unknown control directions can restrict which outputs or states can converge within a selected horizon. Existing prescribed-time designs typically rely on normal-form or strict-feedback structures, while other results provide only practical guarantees for underactuated systems.

A general structural theory would identify the strength of the attainable guarantee for each regulated variable and specify the admissible input class required.

References

The open problem is to state, for a given structure, relative degree, stability and dimension of the internal dynamics, actuation rank, and sign uncertainty, which variables admit an assigned-time property, of which strength, and with which admissible input class.

— 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 “Structural obstructions: internal dynamics, underactuation, and control direction”

Missing is a comparison of fixed-, predefined-, and prescribed-time mechanisms under equal physical authority, reported as worst-case noise-to-terminal-error gains and horizon-accuracy curves that show where shorter horizons cease to improve practical accuracy.

— 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 “Horizon-dependent robustness and practical accuracy”

The open problem is to distribute a global deadline among estimation, reaching, filtering, adaptation, and plant dynamics so as to minimize peak error, input demand, noise sensitivity, or computation while preserving the final guarantee; quantitative peaking-deadline and transient-deadline bounds would turn this tuning heuristic into an optimization problem.

— 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 5.2, paragraph “Hidden transients and allocation of temporal budgets”

Appropriate temporal definitions for these systems, together with implementable memory-aware certificates, remain open.

— 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 “Memory, fractional dynamics, and delay”