Adversarial Physics-Informed Machine Learning for Robust Optimal Safe Predefined-Time Stabilization: A Game-Theoretic Approach

TL;DR

A physics-informed learning algorithm is introduced that robustly learns the Nash safely predefined-time stabilizing control strategy and demonstrates the efficacy and resilience of the proposed method in ensuring robust optimal safe predefined-time stabilization under adversarial disturbances.

Abstract

We develop a game-theoretic framework for adversarially robust optimal safe predefined-time stabilization of parameter-dependent nonlinear dynamical systems with nonquadratic cost functionals. Our approach ensures that all system trajectories remain within a specified admissible set and converge to equilibrium in a predefined time despite adversarial disturbances. The control problem is formulated as a two-player zero-sum differential game, where the controller is a minimizing player and the adversary a maximizing player. We derive sufficient conditions for the existence of a saddle-point solution and safe predefined-time stability using a barrier Lyapunov function that satisfies a differential inequality and the steady-state Hamilton-Jacobi-Isaacs (HJI) equation. To address the analytical intractability of solving the HJI equation, we introduce a physics-informed learning algorithm that robustly learns the Nash safely predefined-time stabilizing control strategy. Simulation results demonstrate the efficacy and resilience of the proposed method in ensuring robust optimal safe predefined-time stabilization under adversarial disturbances.

Figures (6)

• Figure 1: Block diagram of the adversarial PINN control framework for robust optimal safe predefined-time stabilization. The adversarial PINN is trained offline using a set of collocation points to solve the steady-state HJI equation subject to predefined-time stability and safety constraints. The learned Nash control strategy $\hat{u}^\star(\cdot)$ is then applied in feedback to ensure robust optimal safe predefined-time stabilization under the adversarial disturbance $\hat{a}^\star(\cdot)$.
• Figure 1: Adversarial physics-informed learning architecture to solve the robust optimal safe predefined-time stabilization problem. A set of collocation points $\left\{x_i\right\}_{i=1}^M$ is randomly sampled in $\mathcal{S}$. The training is formulated as a constrained optimization problem: at each collocation point, the loss function penalizes deviations from the HJI equation \ref{['eqn: HJI']}, while the constraint function enforces the differential inequality \ref{['eqn: lyap4']} to ensure safe predefined-time stability. Constraints \ref{['eqn: lyap1']}-\ref{['eqn: lyap3']} are satisfied by the design of the PINN architecture. This constrained optimization problem is solved iteratively using the augmented Lagrangian method, with the Lagrange multipliers updated at each iteration according to \ref{['update_multiplier']} and \ref{['update_multiplier2']}.
• Figure 1: Nash value, Nash feedback control strategy, and Nash feedback adversary strategy for the bounded safe set. (Left) Exact Nash value $V$, exact Nash control strategy $u^\star$, and exact Nash adversary strategy $a^\star$. (Middle) Approximate Nash value $\hat{V}$, approximate Nash control strategy $\hat{u}$, and approximate Nash adversary strategy $\hat{a}$. (Right) Symmetric absolute error for learning the Nash value, the Nash control strategy, and the Nash adversary strategy.
• Figure 2: Robust optimal safe predefined-time stabilization for the bounded safe set. (Left) Closed-loop state trajectories $x(t), \ t \geq 0$, starting from different initial conditions near the boundary $\partial \mathcal{S}$ of the safe set. (Right) Time evolution of the Euclidean norm $\|x(t)\|_2, \ t \geq 0,$ of each trajectory shown in the left plot. In both plots, trajectories starting from the same initial condition are marked with the same color.
• Figure 3: Nash value, Nash feedback control strategy, and Nash feedback adversary strategy for the unbounded safe set. (Left) Exact Nash value $V$, exact Nash control strategy $u^\star$, and exact Nash adversary strategy $a^\star$. (Middle) Approximate Nash value $\hat{V}$, approximate Nash control strategy $\hat{u}$, and approximate Nash adversary strategy $\hat{a}$. (Right) Symmetric absolute error for learning the Nash value, the Nash control strategy, and the Nash adversary strategy.

Theorems & Definitions (18)

• Definition 1: bhat2000finite
• Definition 2: polyakov2011nonlinear
• Definition 3: jimenez2020lyapunov
• Remark 1
• Definition 4: kokolakis2024safe
• Remark 2
• Definition 5: bertsekasnonlinear
• Theorem 2.1: kokolakis2025safe
• Remark 3
• Definition 6