Mean-Square Exponential Stabilization

• Mean-square exponential stabilization is defined as the property where the second moment of a stochastic system decays exponentially fast, regardless of initial conditions.
• Approaches like Lyapunov analysis, spectral factorization, LMIs, and generalized eigenvalue methods are used to design controllers that meet stability criteria under stochastic disturbances.
• Applications span LTI, MIMO, SPDEs, and nonlinear systems, demonstrating trade-offs between inherent instability, transmission uncertainties, and controller design complexity.

Mean-square exponential stabilization refers to the property (and corresponding synthesis problem) whereby the second moment of the state of a stochastic dynamical system decays exponentially fast in time (or discrete-time index), uniformly over all admissible initial conditions. This requirement emerges in the control of systems with process or channel uncertainty, including LTI and MIMO systems subject to stochastic multiplicative disturbances, time delays, or random erasures, as well as in stochastic PDEs and nonlinear systems over uncertain networks. Rigorous tests, trade-offs, and synthesis methodologies for mean-square exponential stabilization have been developed using operator theory, spectral factorization, Lyapunov approaches, Riccati and linear matrix inequalities (LMIs), and model-matching forms. The fundamental limitations are set by the interaction of the system’s open-loop instability, transmission channel statistics, and network-induced uncertainties.

1. Formal Definition and Core Criteria

A discrete-time stochastic system $x(k)$ is mean-square exponentially stable if

$$\mathbb{E}[\|x(k)\|^2] \leq M\, \rho^k\, \mathbb{E}[\|x(0)\|^2], \quad \forall\,k\geq 0, \ M>0,\, 0<\rho<1$$

for all initial states $x(0)$. This ensures uniform exponential decay of the second moment, i.e., the system “forgets” its initial condition exponentially fast in the mean-square sense (Li et al., 6 Feb 2024).

For deterministic LTI plants with stochastic perturbations (e.g., networked control with random delays and losses), the mean-square exponential stability is assessed via operator-theoretic small-gain-type criteria, where a weighted transfer matrix norm (often involving a spectral factor encapsulating uncertainty statistics) must have spectral radius strictly less than unity (Li et al., 6 Feb 2024, Lu et al., 2022, Pushpak et al., 2016).

2. Stochastic Feedback Interconnection and the Coefficient of Frequency Variation

In MIMO LTI feedback systems cascaded with linear stochastic (uncertain) systems, stochastic uncertainty blocks are factorized into a deterministic “mean” part $H(z)$ and a zero-mean uncertainty $\Omega$. The frequency-domain effect is captured by the coefficient of frequency variation

$W(z) = H^{-1}(z) \Phi(z)$

where $\Phi(z)$ is the spectral factor of the uncertainty’s energy spectrum $S_\Omega(z) = \Phi(z) \Phi^\top(z^{-1})$ (Li et al., 6 Feb 2024).

For each channel, the diagonal weight $W_j(z)$ quantifies the magnitude of uncertainty compared to the channel’s mean gain at each frequency. The effect of these $W_j$’s is embedded into a non-negative matrix

$$\widehat{T\,W} = \left[\, \|T_{ij}W_j\|_2^2 \,\right]$$

where $T_{ij}(z)$ is the $(i,j)$ entry of the closed-loop complementary sensitivity. Mean-square exponential stability is achieved if and only if

$\rho(\widehat{T\,W}) < 1$

where $\rho(\cdot)$ denotes the spectral radius (Li et al., 6 Feb 2024).

3. Necessary and Sufficient Conditions: Fundamental Limits

The mean-square stabilizability of a plant $P$ over a stochastic channel is equivalent to the existence of a controller $K$ such that $\rho(\widehat{T\,W}) < 1$. General synthesis is performed via Youla parameterization over all stabilizing controllers:

$T(z) = [U + MQ]\tilde{N}$

with $Q \in RH_\infty$, $P\,H = N\,M^{-1} = \tilde{M}^{-1}\tilde{N}$ being coprime factorizations.

Defining $J_{ij}(Q) = \|[(U + M Q)\tilde{N}]_{ij} W_j\|_2^2$, mean-square stabilizability holds if and only if

$$\rho_{\min} := \inf_{Q\in RH_\infty} \rho(J(Q)) < 1$$

(Li et al., 6 Feb 2024).

Fundamental limitations arise from:

• The number and magnitude of unstable open-loop poles
• Nonminimum-phase zeros’ proximity to the unit circle (or $s$-plane boundary)
• Input delays and the frequency content of uncertainty ($W$)
• Channel uncertainty statistics (e.g., loss rates, dropout probabilities)

For minimum-phase plants with input delays, one obtains a generalized eigenvalue condition involving the inner-outer factorization of $PH$ and the frequency-variation weights. Plants with nonminimum-phase zeros suffer stricter limitations since zeros aligned with unstable modes—or near the stability boundary—can drive $\rho_{\min} \geq 1$, rendering the system unstabilizable in the mean-square sense (Li et al., 6 Feb 2024).

4. Controller Synthesis and Design Approaches

Controller synthesis targets the minimization of $\rho(J(Q))$ subject to controller constraints, commonly formalized as a generalized eigenvalue or LMI feasibility problem. For structured plants, an explicit upper-triangular plant-factorization can yield direct diagonal (channel-wise) stabilizability tests:

$$B_i^*B_i \prec W_i^{-1}(i_{in}^*\,i_{in})\,W_i^{-*}$$

If such triangularization is possible, channel-wise synthesis decouples and the associated LMIs can be solved independently per channel (Li et al., 6 Feb 2024, Lu et al., 2022, Pushpak et al., 2016).

When $Q$ or $K$ is synthesized, explicit controller realization follows via $K(z) = (U + MQ)(V - NQ)^{-1}$.

5. Relationship to Mean-Square Stability in Infinite-Dimensional and Nonlinear Systems

For infinite-dimensional systems (SPDEs, PDEs), mean-square exponential stabilization is characterized via spectral radius criteria for suitable linear operators (e.g., discrete-time propagators or Lyapunov operators). Mean-square exponential decay rates can be directly identified from the deterministic drift’s spectrum and the statistics of the stochastic perturbation (Lang et al., 2017, Zhang et al., 2023).

In nonlinear settings, mean-square exponential stabilization relies on conditions derived from mean-square exponential contraction (dichotomy) of the linearized system, Lyapunov functionals, or the analysis of associated Riccati or Volterra equations. Regularity and dissipativity assumptions enable the extension of exponential mean-square stabilization results from the linear to the nonlinear regime (Zhu et al., 2019, Vaidya et al., 2016, Appleby et al., 2023, Yang et al., 26 Nov 2025, Lan et al., 2023).

6. Illustrative Applications and Examples

A representative example features a two-channel MIMO plant with distinct delay and dropout models per channel:

• Channel 1: random single-step delay with probability $p_0$, FIR weight $\alpha$
• Channel 2: memoryless dropout with probability $p_1$

Explicit forms are given for $H_1(z), S_{\Omega_1}(z), W_1(z)$, and the consolidated condition for mean-square stabilizability becomes an explicit inequality in terms of the plant's unstable pole, zeros, and channel parameters. Visualization in the $(p_0, p_1)$-plane directly delineates the region where mean-square stabilization is achievable (Li et al., 6 Feb 2024).

A second application in mixed-autonomy traffic PDEs with Markovian switching establishes mean-square exponential stabilization via boundary backstepping control, yielding explicit bounds on the tolerable stochastic deviation size (parameterized by $\epsilon^\star$) relative to the core stability margin of the deterministic design (Zhang et al., 2023).

7. Trade-offs, Limitations, and Physical Interpretation

The mean-square exponential stabilization problem inherently encodes the trade-off between system instability and statistical uncertainty. Key features include:

• Sensitivity to stochastic uncertainty: Higher uncertainty (quantified by channel loss, variance, or the $W$ weights) reduces the allowable level of plant instability or delays for which mean-square exponential stabilization is possible.
• Role of delays and NMP zeros: Large input delays or NMP zeros close to the unstable spectrum boundary sharply reduce the tolerable uncertainty margin.
• Conservatism of bounds: Explicitly computed uncertainty bounds (e.g., $\epsilon^\star$) are often conservative due to operator-norm and spectral-factor inequalities. Less conservative margins require system-specific small-gain, mode-dependent, or LMI analyses.
• Numerical and empirical validation: Monte Carlo and parameter-sweep studies confirm that fundamental region-of-stabilizability predictions from the theoretical criteria accurately govern the transition between mean-square exponential stability and instability under increasing uncertainty or delay (Li et al., 6 Feb 2024, Zhang et al., 2023).

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References

• Li, S., Lu, Y., & Su, Y. "Mean-Square Stability and Stabilizability for LTI and Stochastic Systems Connected in Feedback" (Li et al., 6 Feb 2024)
• Wang, X. et al. "Mean-Square Exponential Stabilization of Mixed-Autonomy Traffic PDE System" (Zhang et al., 2023)
• Zhu, Q., Chen, W., & He, S. "Nonuniform Mean-square Exponential Dichotomies and Mean-square Exponential Stability" (Zhu et al., 2019)
• Diwadkar, A., Pushpak, V., & Vaidya, U. "Mean Square Stability Analysis of Stochastic Continuous-time Linear Networked Systems" (Pushpak et al., 2016)
• Chen, N., Georgiou, T. T., & Stergiou, C. E. "Mean-square stability of linear systems over channels with random transmission delays" (Lu et al., 2022)
• Lawless, D. & Appleby, J. A. D. "Mean square asymptotic stability characterisation of perturbed linear stochastic functional differential equations" (Appleby et al., 2023)
• Xu, J. & Zhang, H. "Exponential Stabilization for Ito Stochastic Systems with Multiple Input Delays" (Xu et al., 2018)
• Wu, F. et al. "Mean-square exponential stability of exact and numerical solutions for neutral stochastic delay differential equations with Markovian switching" (Yang et al., 26 Nov 2025)
• Song, Z., He, Y., & Bao, J. "Mean square exponential stability of numerical methods for stochastic differential delay equations" (Lan et al., 2023)