Hyperexponential Stabilization

Updated 27 December 2025

Hyperexponential stabilization is a control strategy that guarantees decay rates faster than classical exponential stability by tuning nonlinear and time-varying feedback.
It employs nested and state-dependent Lyapunov functions to progressively accelerate convergence, achieving bounds as fast as e^(-ρ exp(ωt)).
This method is applied in infinite-dimensional, bilinear, and mean-field models to ensure robust stabilization even amid disturbances and uncertainties.

Hyperexponential stabilization refers to control and dynamical systems methodologies that guarantee a rate of convergence of the solution to equilibrium that is strictly faster than any exponential decay. In sharp contrast to classical exponential stability—where solutions satisfy bounds of the form $\|x(t)\|\le C\,e^{-\alpha t}$ for some $\alpha>0$ —hyperexponential stabilization yields convergence as $\|x(t)\|\le C\,e^{-\gamma t^\delta}$ for some $\delta>1$ or, in the extreme, as $\|x(t)\|\le C\,\exp(-\exp(\omega t))$ , commonly termed "superexponential" or "doubly-exponential" decay. This property is fundamentally appealing in control, infinite-dimensional systems, robust stabilization, and mean-field models, where extremely rapid convergence or mixing is required.

1. Definitions and Formal Characterization

Hyperexponential (or superexponential) stability generalizes exponential stability by requiring arbitrarily fast decay rates. Several non-equivalent formalizations exist:

Unrated Hyperexponential Stability: The zero equilibrium of $\dot{x}(t)=f(x(t))$ is hyperexponentially stable in domain $D$ if for all $r>0$ there exist $t'>0$ , $C>0$ , $\kappa>0$ such that $\|\Phi(t,x_0)\|\le C\,e^{-r\,t}$ for all $t>t'$ , $\|x_0\|\le\kappa$ (Zimenko et al., 2022).
Rated Hyperexponential Stability: For degree $r$ and parameters $\alpha=(\alpha_0,\dots,\alpha_r)>0$ , define

$\rho_{0,\alpha}(t)\!=\!\alpha_0 t,\quad \rho_{i,\alpha}(t)\!=\!\alpha_i\left(e^{\rho_{i-1,\alpha}(t)}-e^{\rho_{i-1,\alpha}(0)}\right),\,i=1,\dots,r$

and require $\|\Phi(t,x_0)\|\le C\,\exp\big(-\rho_{r,\alpha}(t)\big)$ (Zimenko et al., 2022). More generally, in infinite-dimensional Hilbert spaces the bound $\|X(t)\|\le M\,e^{-\alpha (t-t_0)^{1+\delta}}\|X_0\|$ with $\delta>0$ characterizes uniform hyperexponential stability (Labbadi et al., 20 Dec 2025), and superexponential convergence may be expressed as $\|u(t)-\psi_1(t)\|\le M e^{-\rho e^{\omega t}}$ (Alabau-Boussouira et al., 2019).

The key distinction from exponential stability is the possibility of tuning the decay rate arbitrarily fast by system design, typically via nonlinear, time-variant, or bilinear feedback.

2. Lyapunov Methods and Controller Design

Hyperexponential stabilization can be achieved by constructing Lyapunov functions whose decay rate accelerates as the system state nears equilibrium.

Explicit Nested Lyapunov Functions: Build a sequence $V_i(x)$ with nested sublevel sets $D_i$ and require $\dot{V}_i(x)\le -c_i V_i(x)$ , with $c_i\to\infty$ as $i\to\infty$ (Zimenko et al., 2022). The solution transitions through levels with progressively increasing exponential rate.
Single Lyapunov with State-Dependent Rate: Construct $V(x)$ so that $\dot{V}(x)\le -\beta(V(x)^{-1}) V(x)$ with $\beta(s)\to\infty$ as $s\to\infty$ , giving rapid decay near the origin (Zimenko et al., 2022).
Implicit Lyapunov Functions and LMI-Based Design: Design implicit $Q_1(V,x)$ , $Q_2(V,x)$ with matched zero-levels, and enforce $\dot{V}\le -c_1 V \prod_{i=1}^r \sigma_i^\alpha(V)$ for $V<1$ , and $\dot{V}\le -c_2 V$ for $V\ge 1$ . The feedback $u(V,x)$ includes a state-dependent gain $\varrho(V)^{\mu-1}$ yielding local rates tending to infinity; practical synthesis uses LMIs to compute stabilizing gains (Zimenko et al., 2022). The resulting controller ensures convergence rate $\|x(t)\|\le \sigma e^{-\rho_{1,\alpha}(t)}$ and robustness to noise, sample-and-hold implementation, and moderate input delays.

For finite-dimensional chains of integrators with unmatched perturbation, recursive time-varying state feedback using auxiliary variables and $\psi(t)$ -weighted gains achieves hyperexponential contraction for the first state and ISS for others. In discrete time, implicit Euler preserves the hyperexponential decay, with explicit rates $k^{-k}$ (Labbadi et al., 16 Nov 2025).

In infinite-dimensional Hilbert spaces with maximal monotone generator $A$ , employing a control law of the form $u=-K(1+t)^n X$ ensures Lyapunov drift $\dot{V}\le -\eta (1+t)^n V+\ldots$ , and time re-parametrization converts the decay into an exponential in the new time, yielding hyperexponential convergence of the original trajectory (Labbadi et al., 20 Dec 2025).

3. Fundamental Results in Infinite-Dimensional and Bilinear Systems

In the context of infinite-dimensional parabolic equations, bilinear (multiplicative) control laws enable superexponential stabilization. Under spectral gap and nondegeneracy conditions, explicit construction of a control $p(t)$ via the moment method ensures that for any initial data sufficiently close to the ground state, the solution satisfies $\|u(t)-\psi_1(t)\|\le M e^{-\rho e^{\omega t}}$ (Alabau-Boussouira et al., 2019). The control is obtained by solving an infinite moment problem using biorthogonal sequences. Applications include classical, variable-coefficient, and high-dimensional heat equations with Dirichlet or Neumann conditions, where explicit moment-based feedback delivers doubly-exponential decay of error to the ground state.

4. Quantitative Performance and Robustness

Hyperexponential stabilizers demonstrate rapid convergence and superior disturbance rejection:

In linear chains of integrators, hyperexponential control achieves settling times significantly smaller than finite-time analogues, is less sensitive to measurement noise, and tolerates input and sampling delays better (Zimenko et al., 2022).
For perturbed chains, after an initial contraction via a growing gain, a saturation mechanism “freezes” the gain in a neighborhood around the origin, ensuring bounded feedback and securing ISS for all remaining states (Labbadi et al., 16 Nov 2025).
For parabolic systems, the explicit moment-based control yields error decay at the scale of $\exp(-\exp(\omega t))$ or faster (Alabau-Boussouira et al., 2019).
In the Hilbert-space framework, noise/disturbance remainders decay as $O((1+t)^{-2n})$ , while the main mode decays with the desired superexponential profile (Labbadi et al., 20 Dec 2025).

5. Structural Aspects in Mean-Field, Probabilistic, and Stochastic Models

Beyond deterministic dynamical systems, hyperexponential stabilization is central to the global attraction of mean-field ODEs governing large-scale stochastic systems, notably queueing networks and load-balancing models. In systems with hyperexponential (Coxian) distributed job sizes, the evolution equations admit a unique global attractor under a monotonicity property with respect to a strong partial order and a telescoping “Lyapunov” integral decomposition (Houdt, 2018). Key elements include:

Coxian representation for modeling phase-type and hyperexponential distributions.
Structural drift conditions that ensure monotonicity and telescoping reduce the verification of global attraction to pairwise sandwiching and integral convergence.
Applicability to an extensive range of randomized and batch sampling load-balancing strategies.

6. Hyperexponential Stabilization in Mixture Models and Distribution Approximation

In probabilistic modeling, particularly of heavy-tailed phenomena, rapid stabilization of hyperexponential mixture components is critical for robust distributional approximation:

Hybrid Bernstein phase-type (BPH) and hyperexponential (HE) models leverage the rapid tail decay of HE but require robust parameter initialization for stability.
Stabilization is executed by penalized mean absolute error optimization, incorporating constraints for positivity and probability mass, solved using stochastic optimization (Adam) to find mixture parameters whose fitted tail matches heavy-tailed data precisely, leading to reliable performance across random initializations (Ziani et al., 30 Oct 2025).
The resultant fitting procedure outperforms pure BPH or HE approaches regarding tail metrics (e.g., mean, coefficient of variation, queueing-theoretic performance indices).

7. Extensions, Open Problems, and Critical Thresholds

Current theoretical work has characterized several critical thresholds for stabilizability:

In discrete-time uncertain systems with potentially hyperexponential nonlinearities, adaptive least-squares feedback can still achieve global stabilization provided the nonlinearity is polynomially bounded of degree $b<4$ on any subset of $\mathbb{R}$ of positive Lebesgue density—even if $f(x)$ grows exponentially almost everywhere (Liu et al., 2018).
For true hyperexponential or super-exponential growth $f(x)=O\left(e^{|x|^c}\right)$ with $c>1$ , necessary and sufficient density conditions for stabilization remain open, with the proposal that as long as the low-growth set is non-negligible, stabilization may persist.
Criticality of the $b=4$ threshold is established for polynomial growth; whether this threshold remains sharp in the hyperexponential regime is unresolved.

References

(Zimenko et al., 2022): "Stability analysis and stabilization of systems with hyperexponential rates"
(Labbadi et al., 20 Dec 2025): "On Hyperexponential Stabilization of Linear Infinite-Dimensional Systems"
(Alabau-Boussouira et al., 2019): "Superexponential stabilizability of evolution equations of parabolic type via bilinear control"
(Labbadi et al., 16 Nov 2025): "On hyperexponential stabilization of a chain of integrators in continuous and discrete time subject to unmatched perturbations"
(Ziani et al., 30 Oct 2025): "Approximating Heavy-Tailed Distributions with a Mixture of Bernstein Phase-Type and Hyperexponential Models"
(Houdt, 2018): "Global attraction of ODE-based mean field models with hyperexponential job sizes"
(Liu et al., 2018): "Is It Possible to Stabilize Discrete-time Parameterized Uncertain Systems Growing Exponentially Fast?"