Stochastic Passivity Framework

• Stochastic passivity framework is a generalization of classical passivity to stochastic systems using expectation-based energy dissipation criteria.
• It extends to controlled Itô stochastic port–Hamiltonian systems, with applications in power network control and data-driven neural surrogates.
• The framework provides a foundation for invariant measure, ergodic convergence, and robust feedback design in noisy environments.

The stochastic passivity framework generalizes classical passivity theory to stochastic systems, especially those described by Itô stochastic differential equations. The core concept is the replacement of strict, pathwise energy dissipation requirements with expectations or “weak” forms of dissipation, often localized to regions outside a neighborhood of a desired equilibrium. This framework not only extends to finite-dimensional and infinite-dimensional (e.g., Fokker–Planck) systems but also provides foundational underpinnings for feedback design, energy shaping, and structure-preserving neural surrogates for stochastic dynamical models.

1. Stochastic Port–Hamiltonian Systems and Passivity Notions

A canonical formulation considers controlled Itô stochastic port–Hamiltonian systems (SPHS) on $\mathbb{R}^n$: $$dx(t) = [J(x) - R(x)]\nabla H(x)\,dt + g(x)u(t)\,dt + \Sigma(x)\,dW_t, \quad y(t) = g(x)^\top\nabla H(x),$$ with $J(x) = -J(x)^\top$ (skew-symmetric), $R(x) = R(x)^\top \succeq 0$ (dissipation), and $\Sigma(x)$ the diffusion coefficient matrix for the Brownian driver $W_t$ (Cordoni et al., 2022, Persio et al., 8 Sep 2025).

Stochastic passivity relates the evolution of a storage (Lyapunov) function $H(x)$ to the supply rate $u^\top y$, replacing the deterministic dissipation inequality with an Itô generator condition: $$L H(x) = \langle [J(x)-R(x)]\nabla H(x) + g(x)u, \nabla H(x) \rangle + \frac{1}{2}\operatorname{Tr}[\Sigma(x)\Sigma(x)^\top \nabla^2 H(x)]$$ and requiring $L H(x) \leq u^\top y$. This can be interpreted at various levels of stochasticity, ranging from pathwise inequalities, supermartingale properties, to inequalities in expectation (Ackermann et al., 5 Dec 2025).

A refined notion, weak or ultimately stochastic passivity (Fang et al., 2016, Cordoni et al., 2022), demands the passivity inequality only outside a ball: for $\|x-x_e\|\geq C$,

$L H(x) \leq u^\top y,$

with possible strictness (including an additional negative-definite term $- \delta \|x - x_e\|^2$) for enhanced convergence.

2. Passivity-Driven Convergence: Invariant Measure and Ergodicity

For strictly ultimately stochastic passive closed-loop SPHS, with nondegenerate stochasticity ($\Sigma(x)\Sigma(x)^\top \succeq \nu I > 0$ for all $x$), one obtains the existence and uniqueness of a stationary (invariant) probability measure $\rho$. The process $x(t)$ converges in distribution to $\rho$, and time averages exhibit ergodic convergence: $$\lim_{t\to\infty}P[x(t)\in B|x(0)=x] = \rho(B), \quad \lim_{T\to\infty} \frac{1}{T}\int_0^T P[x(t)\in B]\,dt = \rho(B)$$ for every Borel set $B$ (Cordoni et al., 2022, Fang et al., 2016).

This framework thus replaces the Lyapunov-based notion of almost sure convergence with weak (distributional) convergence—a necessity in the presence of nonvanishing noise at the reference point.

3. Energy Shaping and Feedback Design

Feedback control in the stochastic passivity framework generalizes deterministic energy-shaping methods using Casimir-based matching: $$([J + J_a] - [R + R_a])K(x) = g(x)\varphi(x) - [J_a - R_a]\nabla H(x), \quad K(x) := \nabla H_a(x),$$ The shaped Hamiltonian is $H_d(x) = H(x) + H_a(x)$. Under suitable matching and strict ultimate passivity of both the base and auxiliary systems, the closed-loop SPHS remains strictly stochastically passive, therefore inheriting the unique invariant measure and ergodic stability properties (Cordoni et al., 2022).

In practice, feedback construction parallels deterministic interconnection-and-damping assignment, with additional attention to diffusion structure to ensure global or local weak passivity.

4. Generalizations: Passivity in Expectation, Indices, and Port-Hamiltonian Neural Networks

Stochastic passivity indices $(\nu, \rho)$ generalize the supply rate as $u^\top y - \nu y^\top y - \rho u^\top u$, enabling output-strict passivity in expectation: $\E[H(X_t)] - \E[H(X_0)] \leq \E\Bigl[\int_0^t \left(u^\top y - \nu \|y\|^2 - \rho \|u\|^2\right) ds\Bigr]$ (Persio et al., 8 Sep 2025). Passivity is enforceable even in data-driven models; stochastic port-Hamiltonian neural networks (pHNNs) are constructed to provably maintain weak passivity by design, via parameterizations and loss regularizations that enforce dissipativity and coisotropy of the diffusion (Persio et al., 8 Sep 2025).

5. Linear Stochastic (Port)-Hamiltonian Systems and Algebraic Characterizations

For linear stochastic input-state-output systems

$$dX_t = (A X_t + B u_t) dt + \sum_j (\mathfrak{A}_j X_t + \mathfrak{B}_j u_t) dW^j_t, \quad Y_t = C X_t + D u_t,$$

with quadratic storage $H(x) = \frac{1}{2}x^\top Q x$, passivity conditions are encoded as linear matrix inequalities (LMIs). Letting

$$\mathfrak{M}_Q = \begin{pmatrix} Q A + A^\top Q + \sum_j \mathfrak{A}_j^\top Q \mathfrak{A}_j & Q B - C^\top + \sum_j \mathfrak{A}_j^\top Q \mathfrak{B}_j \ B^\top Q - C + \sum_j \mathfrak{B}_j^\top Q \mathfrak{A}_j & - (D + D^\top) + \sum_j \mathfrak{B}_j^\top Q \mathfrak{B}_j \end{pmatrix},$$

the system is stochastically passive if and only if $\mathfrak{M}_Q \leq 0$; strong (pathwise) passivity further requires $Q \mathfrak{A}_j$ skew-symmetric, $Q \mathfrak{B}_j = 0$ for all $j$ (Ackermann et al., 5 Dec 2025). Canonical parameterizations mapping directly to stochastic PH structure exist.

6. Case Studies and Applications

The stochastic passivity and weak passivity frameworks have been applied in various engineering and control domains:

• Power Network Control: Passivity-based distributed controllers guarantee stochastic stability in DC microgrids with stochastic ZIP loads, ensuring current sharing and voltage regulation under Ornstein–Uhlenbeck load fluctuations (Silani et al., 2020), and regulate frequency and dispatch with wind generation modeled as stochastic processes (Silani et al., 2021).
• Physical Surrogates/Neural Models: Passivity-constrained neural surrogates accurately predict the behavior of stochastic interconnected mechanical and electrical systems, outperforming unconstrained alternatives and maintaining physical plausibility even under strong noise (Persio et al., 8 Sep 2025).
• General Nonlinear Stochastic Systems: The weak passivity framework enables the stabilization (in the sense of distributional convergence) of nonlinear and linear systems with persistent noise at the equilibrium, a setting where classical Lyapunov stability is unachievable (Fang et al., 2016).

7. Connections to Infinite-Dimensional Systems and the Fokker–Planck Formalism

The forward Kolmogorov (Fokker–Planck) equation associated with SPHS defines a PDE on the space of probability densities, possessing a formal infinite-dimensional port–Hamiltonian structure. Energy shaping in the original SPHS corresponds to Dirac-structure shaping in the Fokker–Planck equation, precisely aligning finite- and infinite-dimensional PH system viewpoints (Cordoni et al., 2022). This clarifies the relationship between ergodic convergence in distribution and energy dissipation at the PDE level.

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In summary, the stochastic passivity framework provides a rigorous foundation for stability and control of stochastic dynamical systems via modified Lyapunov methods, broadens the applicability of energy-based feedback to noisy environments, and underpins structure-preserving data-driven modeling approaches. Its mathematical development unifies stochastic analysis, control theory, and the port–Hamiltonian paradigm across both finite-dimensional and infinite-dimensional settings (Fang et al., 2016, Cordoni et al., 2022, Ackermann et al., 5 Dec 2025, Persio et al., 8 Sep 2025).