Stability of Nonlinear Multicascade Systems

Definitions and mathematical statement

The source requests stability for nonlinear multicascade systems described in the report. The supplied verification says two reference-shaping-filter examples motivate the question, but no general class is specified precisely enough for a standalone system theorem. Their equations and hypotheses are not reproduced here.

For orientation only, a finite autonomous cascade may have blocks \[ \dot x_i=F_i(x_i,x_{i+1},\ldots,x_m)\quad(1\leq i<m), \qquad\dot x_m=F_m(x_m). \] This is an illustrative feedforward form, not an assertion that it is the report's exact multicascade model. A well-defined result must specify the block dimensions, coupling graph, regularity and growth conditions, equilibria or reference trajectories, inputs/feedback, and existence of forward solutions.

For an autonomous equilibrium at zero, the usual Lyapunov stability property is that every $\varepsilon>0$ admits $\delta>0$ such that solutions starting with $\|x(0)\|<\delta$ exist forward and satisfy $\|x(t)\|<\varepsilon$ for all $t\geq0$. Asymptotic, global asymptotic, exponential, input-to-state, and tracking stability are different targets. The source does not select one, and the filter motivation may require a reference-dependent error formulation instead of an autonomous equilibrium.

The intended research task is to give verifiable hypotheses yielding the appropriate stability property for the specified multicascade class once that class and target are recovered. No stability theorem for unrestricted nonlinear couplings is asserted, and no unstated subsystem or input-to-state assumptions are inserted. The two motivating filter examples cannot be reconstructed from their mention alone.

The supplied triage dated 21 August 2026 reports that the general class remained open in the report and that later stabilization results cover important subclasses. Stabilization through a chosen controller is not automatically stability of an unspecified existing interconnection. The assessment does not supply missing equations or a universal criterion; the record remains reconstructed_unverified.

Short English statement

The report-based multicascade class and intended stability notion are missing. The draft records the required system and target data, without assuming every cascade is stable or treating a later control-design result as an unconditional stability theorem.

Sources

• Local UnsolvedMath registry, permanent ID $30000193$, exact source code catalogue entry, Oberwolfach Reports Open Problems. Statement status reconstructed_unverified; supplied verification reports two motivating reference-shaping filters but an insufficiently specified general class.
• "Regelungstheorie (2005)," Oberwolfach Reports, pp. 565--645, DOI 10.4171/owr/2005/11: https://doi.org/10.4171/owr/2005/11. Proposed year 2005.
• Supplied "representative subclass result": https://doi.org/10.1016/j.automatica.2012.06.060. The other supplied literature source is the original problem report; no full system theorem or controller hypotheses are reproduced.
• Literature checked 21 August 2026; supplied status open. Registry created 25 August 2026 and updated 16 September 2026. This dated curation triage requires independent verification before citation.
• Registry curation attribution: Ulam AI / UnsolvedMath Contributors, CC BY 4.0. Underlying report and literature retain their supplied provenance; this statement-only draft requires human review.