Generalized Zubov's Equation

• Generalized Zubov's Equation is a Lyapunov PDE that encodes the region of attraction through a bounded function that vanishes on the attractor and reaches 1 at the boundary.
• It generalizes classical theory by extending to closed invariant sets and perturbed systems, using viscosity solutions to handle nonlinear supremum-type formulations.
• Numerical approaches—such as augmented integration and deep learning with policy iteration—facilitate efficient, grid-free approximations of the exact attraction set.

Searching arXiv for the cited papers to ground the article in recent literature. arXiv search: (Wang et al., 26 Aug 2025) and (Kang et al., 2021) Generalized Zubov’s equation is a Lyapunov partial differential equation that encodes an attraction set through a bounded function normalized to vanish on the attractor and to approach $1$ at the attraction boundary. In the recent arXiv literature, the term appears in two closely related senses. One extends classical Zubov theory from an isolated equilibrium to a closed invariant set for an autonomous ODE, with the domain of attraction characterized by $\{x:V(x)<1\}$ (Kang et al., 2021). The other treats perturbed systems $$\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x},\boldsymbol{\delta})$$ and defines a viscosity solution of a fully nonlinear supremum-type PDE whose strict sublevel set yields the robust region of attraction (Wang et al., 26 Aug 2025). In both formulations, the equation provides a special Lyapunov function with exact boundary information rather than a conservative certificate chosen a priori.

1. Classical Zubov framework and bounded Lyapunov normalization

For a deterministic autonomous nonlinear system

$$\dot{\mathbf{x}}(t)=\mathbf{f}(\mathbf{x}(t)), \qquad \mathbf{x}\in\mathbb{R}^n,$$

with an asymptotically stable equilibrium at the origin, a maximal Lyapunov function can be written as

$$V(\mathbf{x})=\int_0^{+\infty}\|\boldsymbol{\phi}(t;\mathbf{x})\|^2\,dt,$$

where $\boldsymbol{\phi}(t;\mathbf{x})$ is the trajectory starting from $\mathbf{x}$. For suitable choices of the integrand, this function satisfies $V(\mathbf{0})=0$, is positive on the region of attraction away from the equilibrium, diverges at the boundary of the region of attraction or as $\|\mathbf{x}\|\to\infty$, and solves

$$\nabla V(\mathbf{x})^\top \mathbf{f}(\mathbf{x})=-\Phi(\mathbf{x}),$$

with $\{x:V(x)<1\}$0 positive definite (Wang et al., 26 Aug 2025).

The unboundedness of $\{x:V(x)<1\}$1 is numerically inconvenient. Zubov’s construction replaces it by a bounded Lyapunov function

$\{x:V(x)<1\}$2

so that $\{x:V(x)<1\}$3, $\{x:V(x)<1\}$4 in the attraction region away from the origin, and $\{x:V(x)<1\}$5 at the attraction boundary. The transformed function satisfies

$\{x:V(x)<1\}$6

and the region of attraction is recovered exactly as

$\{x:V(x)<1\}$7

This bounded normalization is the structural core that persists in generalized versions. What changes is the nature of the attractor, the admissible dynamics, and the precise Hamiltonian appearing in the PDE.

2. Two principal generalizations of Zubov’s equation

A first generalization replaces the isolated equilibrium by a closed invariant set $\{x:V(x)<1\}$8. For the autonomous ODE

$\{x:V(x)<1\}$9

an invariant set satisfies $$\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x},\boldsymbol{\delta})$$0 for all $$\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x},\boldsymbol{\delta})$$1 and all $$\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x},\boldsymbol{\delta})$$2. If $$\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x},\boldsymbol{\delta})$$3 is uniformly asymptotically stable and uniformly attracting, the domain of attraction is

$$\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x},\boldsymbol{\delta})$$4

and Zubov’s theorem yields continuous functions $$\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x},\boldsymbol{\delta})$$5 and $$\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x},\boldsymbol{\delta})$$6 such that $$\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x},\boldsymbol{\delta})$$7 on $$\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x},\boldsymbol{\delta})$$8, $$\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x},\boldsymbol{\delta})$$9 away from $$\dot{\mathbf{x}}(t)=\mathbf{f}(\mathbf{x}(t)), \qquad \mathbf{x}\in\mathbb{R}^n,$$0, $$\dot{\mathbf{x}}(t)=\mathbf{f}(\mathbf{x}(t)), \qquad \mathbf{x}\in\mathbb{R}^n,$$1 and $$\dot{\mathbf{x}}(t)=\mathbf{f}(\mathbf{x}(t)), \qquad \mathbf{x}\in\mathbb{R}^n,$$2 as $$\dot{\mathbf{x}}(t)=\mathbf{f}(\mathbf{x}(t)), \qquad \mathbf{x}\in\mathbb{R}^n,$$3, $$\dot{\mathbf{x}}(t)=\mathbf{f}(\mathbf{x}(t)), \qquad \mathbf{x}\in\mathbb{R}^n,$$4 as $$\dot{\mathbf{x}}(t)=\mathbf{f}(\mathbf{x}(t)), \qquad \mathbf{x}\in\mathbb{R}^n,$$5 approaches the boundary of $$\dot{\mathbf{x}}(t)=\mathbf{f}(\mathbf{x}(t)), \qquad \mathbf{x}\in\mathbb{R}^n,$$6, and

$$\dot{\mathbf{x}}(t)=\mathbf{f}(\mathbf{x}(t)), \qquad \mathbf{x}\in\mathbb{R}^n,$$7

Using the chain rule, the PDE form is

$$\dot{\mathbf{x}}(t)=\mathbf{f}(\mathbf{x}(t)), \qquad \mathbf{x}\in\mathbb{R}^n,$$8

In this setting, the constructed solution is extended by $$\dot{\mathbf{x}}(t)=\mathbf{f}(\mathbf{x}(t)), \qquad \mathbf{x}\in\mathbb{R}^n,$$9 outside the domain of attraction (Kang et al., 2021).

A second generalization concerns perturbed systems

$$V(\mathbf{x})=\int_0^{+\infty}\|\boldsymbol{\phi}(t;\mathbf{x})\|^2\,dt,$$0

where $$V(\mathbf{x})=\int_0^{+\infty}\|\boldsymbol{\phi}(t;\mathbf{x})\|^2\,dt,$$1 is bounded, $$V(\mathbf{x})=\int_0^{+\infty}\|\boldsymbol{\phi}(t;\mathbf{x})\|^2\,dt,$$2 is Lipschitz, and the equilibrium is singular in the sense that

$$V(\mathbf{x})=\int_0^{+\infty}\|\boldsymbol{\phi}(t;\mathbf{x})\|^2\,dt,$$3

The robust region of attraction is

$$V(\mathbf{x})=\int_0^{+\infty}\|\boldsymbol{\phi}(t;\mathbf{x})\|^2\,dt,$$4

that is, the set of initial conditions that converge under all admissible disturbances. The corresponding generalized Zubov equation is no longer linear in the disturbance argument; it is a supremum-type equation over worst-case disturbances (Wang et al., 26 Aug 2025).

A recurrent source of confusion is notation. In the invariant-set formulation, $$V(\mathbf{x})=\int_0^{+\infty}\|\boldsymbol{\phi}(t;\mathbf{x})\|^2\,dt,$$5 denotes the bounded Zubov function itself (Kang et al., 2021). In the robust formulation, $$V(\mathbf{x})=\int_0^{+\infty}\|\boldsymbol{\phi}(t;\mathbf{x})\|^2\,dt,$$6 denotes a generalized maximal Lyapunov function, and the bounded Zubov function is written as $$V(\mathbf{x})=\int_0^{+\infty}\|\boldsymbol{\phi}(t;\mathbf{x})\|^2\,dt,$$7 (Wang et al., 26 Aug 2025).

3. Robust generalized Zubov equation, worst-case disturbances, and viscosity solutions

For the perturbed system, the robust value function is defined by

$$V(\mathbf{x})=\int_0^{+\infty}\|\boldsymbol{\phi}(t;\mathbf{x})\|^2\,dt,$$8

with a nonnegative Lipschitz running cost $$V(\mathbf{x})=\int_0^{+\infty}\|\boldsymbol{\phi}(t;\mathbf{x})\|^2\,dt,$$9. A typical choice is

$\boldsymbol{\phi}(t;\mathbf{x})$0

This generalized maximal Lyapunov function is finite exactly on the robust region of attraction, vanishes at the origin, and diverges at the boundary of the robust region of attraction or as $\boldsymbol{\phi}(t;\mathbf{x})$1. Applying the exponential transform

$\boldsymbol{\phi}(t;\mathbf{x})$2

produces a bounded function whose strict sublevel set $\boldsymbol{\phi}(t;\mathbf{x})$3 is the robust region of attraction (Wang et al., 26 Aug 2025).

The generalized Zubov theorem recalled there states that if $\boldsymbol{\phi}(t;\mathbf{x})$4 satisfies $\boldsymbol{\phi}(t;\mathbf{x})$5, $\boldsymbol{\phi}(t;\mathbf{x})$6 on the robust region away from the origin, $\boldsymbol{\phi}(t;\mathbf{x})$7 at the boundary or at infinity, and is a viscosity solution of

$\boldsymbol{\phi}(t;\mathbf{x})$8

then

$\boldsymbol{\phi}(t;\mathbf{x})$9

is the robust region of attraction.

The supremum over disturbances yields the Hamiltonian

$\mathbf{x}$0

and the maximizing disturbance

$\mathbf{x}$1

is the optimal disturbance, or worst-case disturbance, at state $\mathbf{x}$2. This gives the robust equation a differential-game interpretation: nature selects disturbances to maximize accumulated cost, while the Lyapunov function is constructed for the worst environment.

Because the PDE is fully nonlinear and the solution need not be smooth, the appropriate solution concept is the viscosity solution. In the cited formulation, a viscosity subsolution satisfies the PDE in an inequality sense when tested with smooth functions touching from above, a viscosity supersolution satisfies it when tested with smooth functions touching from below, and a viscosity solution is both. This is the solution concept singled out for the robust generalized Zubov equation (Wang et al., 26 Aug 2025).

4. Integral representations and alternative normalizations

A central construction for the invariant-set version introduces a continuous function $\mathbf{x}$3 such that

$\mathbf{x}$4

with $\mathbf{x}$5 as $\mathbf{x}$6, and with $\mathbf{x}$7 bounded away from zero outside any neighborhood of $\mathbf{x}$8. The associated integral is

$\mathbf{x}$9

Under the assumptions stated in the source, $V(\mathbf{0})=0$0 is finite and continuous on the domain of attraction, equals $V(\mathbf{0})=0$1 on $V(\mathbf{0})=0$2, diverges as $V(\mathbf{0})=0$3, and satisfies

$V(\mathbf{0})=0$4

along trajectories (Kang et al., 2021).

The explicit integral-form solution is

$V(\mathbf{0})=0$5

with $V(\mathbf{0})=0$6. The differential identity

$V(\mathbf{0})=0$7

implies

$V(\mathbf{0})=0$8

The resulting characterization is exact: $V(\mathbf{0})=0$9 (Kang et al., 2021).

Two monotone bounded transforms therefore appear in the cited literature. The invariant-set construction uses $\|\mathbf{x}\|\to\infty$0 (Kang et al., 2021), whereas the robust construction uses $\|\mathbf{x}\|\to\infty$1 after first forming a worst-case integral value function (Wang et al., 26 Aug 2025). Both normalize the attraction problem into a bounded Lyapunov function with the boundary encoded by the limiting value $\|\mathbf{x}\|\to\infty$2.

5. Numerical methods: augmented integration, deep learning, and rollout-enhanced PINNs

For autonomous systems with a closed invariant set, one computational route is an augmented ODE system

$\|\mathbf{x}\|\to\infty$3

Here $\|\mathbf{x}\|\to\infty$4 approximates $\|\mathbf{x}\|\to\infty$5, and for large $\|\mathbf{x}\|\to\infty$6 it approximates $\|\mathbf{x}\|\to\infty$7 if the trajectory converges. The stopping logic described in the source is based either on the accumulated integral exceeding a large threshold $\|\mathbf{x}\|\to\infty$8, in which case $\|\mathbf{x}\|\to\infty$9 is set to $$\nabla V(\mathbf{x})^\top \mathbf{f}(\mathbf{x})=-\Phi(\mathbf{x}),$$0, or on the incremental growth of $$\nabla V(\mathbf{x})^\top \mathbf{f}(\mathbf{x})=-\Phi(\mathbf{x}),$$1 becoming very small, in which case $$\nabla V(\mathbf{x})^\top \mathbf{f}(\mathbf{x})=-\Phi(\mathbf{x}),$$2. The bounded Zubov function is then approximated by $$\nabla V(\mathbf{x})^\top \mathbf{f}(\mathbf{x})=-\Phi(\mathbf{x}),$$3. This computation is described as causality-free because the value at a point uses only the trajectory from that point, without a spatial grid (Kang et al., 2021).

The same source develops a deep learning method for high-dimensional systems. Training data are generated by sampling initial states, computing $$\nabla V(\mathbf{x})^\top \mathbf{f}(\mathbf{x})=-\Phi(\mathbf{x}),$$4 with the augmented-system procedure, and collecting additional points along stable trajectories. A feedforward neural network

$$\nabla V(\mathbf{x})^\top \mathbf{f}(\mathbf{x})=-\Phi(\mathbf{x}),$$5

is trained by minimizing the empirical mean-square error

$$\nabla V(\mathbf{x})^\top \mathbf{f}(\mathbf{x})=-\Phi(\mathbf{x}),$$6

with validation by RMSE (Kang et al., 2021).

For the robust equation, the cited 2025 paper introduces a rollout-enhanced physics-informed neural network with policy iteration. The bounded solution $$\nabla V(\mathbf{x})^\top \mathbf{f}(\mathbf{x})=-\Phi(\mathbf{x}),$$7 is approximated by a feedforward network $$\nabla V(\mathbf{x})^\top \mathbf{f}(\mathbf{x})=-\Phi(\mathbf{x}),$$8, with sigmoid hidden layers to enable accurate gradients and a linear output layer to regularize parameter growth. The empirical loss is

$$\nabla V(\mathbf{x})^\top \mathbf{f}(\mathbf{x})=-\Phi(\mathbf{x}),$$9

where the boundary term enforces $\{x:V(x)<1\}$00 and $\{x:V(x)<1\}$01 on the sampling boundary, the residual term enforces the generalized Zubov PDE at interior points using the current optimal disturbance, and the data term fits rollout-based anchor values $\{x:V(x)<1\}$02 (Wang et al., 26 Aug 2025).

Policy iteration alternates between policy evaluation and policy improvement. In the evaluation step, $\{x:V(x)<1\}$03 is updated to minimize the loss for a fixed disturbance policy and fixed anchor values. In the improvement step, the disturbance policy is updated through Hamiltonian maximization: $\{x:V(x)<1\}$04 Forward rollout under the current worst-case disturbance policy then produces truncated value estimates

$\{x:V(x)<1\}$05

which serve as anchor points. The stated function of these anchors is to prevent convergence to spurious singular solutions such as $\{x:V(x)<1\}$06 except near the origin and to provide global guidance in both low- and high-dimensional systems (Wang et al., 26 Aug 2025).

6. Applications, approximation results, limitations, and scope

The robust PINN-policy-iteration framework is validated on three benchmark systems. For a Van der Pol system with multiplicative disturbances,

$\{x:V(x)<1\}$07

with $\{x:V(x)<1\}$08, the paper reports that removing rollout by setting $\{x:V(x)<1\}$09 yields a singular degenerate solution that is essentially $\{x:V(x)<1\}$10 everywhere except near the origin, whereas rollout-enhanced training produces a smooth $\{x:V(x)<1\}$11 matching a finite-difference reference robust region of attraction. For an inverted pendulum with disturbed feedback gains over $\{x:V(x)<1\}$12, the learned value function agrees well with finite-difference estimates. For the $\{x:V(x)<1\}$13-dimensional system

$\{x:V(x)<1\}$14

with true robust region of attraction $\{x:V(x)<1\}$15, the method learns a $\{x:V(x)<1\}$16-dimensional value function, and the reported $\{x:V(x)<1\}$17-slice matches the known $\{x:V(x)<1\}$18 region (Wang et al., 26 Aug 2025).

The invariant-set and deep-learning framework is demonstrated on the New England $\{x:V(x)<1\}$19-generator, $\{x:V(x)<1\}$20-bus power system model. In that example, the state dimension is $\{x:V(x)<1\}$21, the weight is chosen as

$\{x:V(x)<1\}$22

the threshold is set to $\{x:V(x)<1\}$23, and the scaling is $\{x:V(x)<1\}$24. Each of the training and validation datasets uses $\{x:V(x)<1\}$25 trajectories of the augmented system, with $\{x:V(x)<1\}$26 additional points per stable trajectory, yielding $\{x:V(x)<1\}$27 and $\{x:V(x)<1\}$28. The network has $\{x:V(x)<1\}$29 hidden layers, $\{x:V(x)<1\}$30 neurons per hidden layer, $\{x:V(x)<1\}$31 activation, and a linear output, and it achieves

$\{x:V(x)<1\}$32

on the validation set, with maximum error about $\{x:V(x)<1\}$33 (Kang et al., 2021).

The same source proves an existence result for neural network approximation in power systems: for a system with $\{x:V(x)<1\}$34 generators and state dimension $\{x:V(x)<1\}$35, there exists a neural network $\{x:V(x)<1\}$36 with $\{x:V(x)<1\}$37 neurons such that

$\{x:V(x)<1\}$38

on a bounded set $\{x:V(x)<1\}$39, where $\{x:V(x)<1\}$40 and $\{x:V(x)<1\}$41 do not depend on $\{x:V(x)<1\}$42. This is presented as a polynomial dependence of the approximation error on the number of generators rather than an exponential grid dependence (Kang et al., 2021).

The cited limitations are also specific. For the robust PINN method, rollout and policy iteration increase computational cost; performance depends on $\{x:V(x)<1\}$43, $\{x:V(x)<1\}$44, $\{x:V(x)<1\}$45, network depth and width, the sampling region $\{x:V(x)<1\}$46, and rollout horizon; the method assumes known dynamics $\{x:V(x)<1\}$47 and known disturbance bounds $\{x:V(x)<1\}$48; and while $\{x:V(x)<1\}$49-dimensional examples are demonstrated, much higher dimensions may require more sophisticated architectures and sampling strategies (Wang et al., 26 Aug 2025). For the augmented-integration and data-generation route, computation near the attraction boundary is expensive because the integral diverges there, and the choices of $\{x:V(x)<1\}$50, $\{x:V(x)<1\}$51, and $\{x:V(x)<1\}$52 are data-driven and heuristic (Kang et al., 2021).

Across these formulations, generalized Zubov’s equation occupies a precise position between converse Lyapunov theory, domain-of-attraction analysis, Hamilton–Jacobi structure, and data-driven computation. Its defining feature is the exact set characterization

$\{x:V(x)<1\}$53

with the main variations arising from whether the attractor is a general invariant set or a robust equilibrium under worst-case disturbances, and whether the solution is treated through explicit integral constructions, grid-free trajectory integration, or neural approximators trained from PDE residuals and rollout data.