arXiv subjects

Kaijing Lyu

Publications and source records attributed to Kaijing Lyu.

3 recordsLinked to original sources

arXiv2026

Robust stabilization of hyperbolic PDE-ODE systems via Neural Operator-approximated gain kernels

This paper investigates the mean square exponential stabilization problem for a class of coupled PDE-ODE systems with Markov jump parameters. The considered system consists of multiple coupled hyperbolic PDEs and a finite-dimensional ODE, where all system parameters evolve according to a homogeneous continuous-time Markov process. The control design is based on a backstepping approach. To address the computational complexity of solving kernel equations, a DeepONet framework is proposed to learn the mapping from system parameters to the backstepping kernels. By employing Lyapunov-based analysis, we further prove that the controller obtained from the neural operator ensures stability of the closed-loop stochastic system. Numerical simulations demonstrate that the proposed approach achieves more than two orders of magnitude speedup compared to traditional numerical solvers, while maintaining high accuracy and ensuring robust closed-loop stability under stochastic switching.

math.OC↗

arXiv2026

Operator learning for prescribed-time stabilization of reaction-diffusion systems

This paper addresses boundary prescribed-time stabilization of a one-dimensional heat equation with spatially and temporally varying coefficients. In contrast to asymptotic or exponential stabilization, prescribed-time stabilization ensures convergence to equilibrium within a user-defined time that is independent of the initial condition, a property that is particularly attractive in applications with stringent transient performance requirements. The backstepping design for this problem requires solving, at each time instant, a two-dimensional time-dependent kernel Partial Differential Equation (PDE) whose solution continuously varies with the plant coefficients. The repeated numerical solution of this parabolic kernel PDE results in a prohibitive computational burden, thereby limiting real-time applicability. To overcome this limitation, we propose a neural-operator-based approximation of the mapping from the time-varying system coefficient to the corresponding backstepping kernel. The operator is trained offline using representative solutions of the kernel PDE and subsequently deployed online to generate the required time-varying kernels in real time. We establish, via Lyapunov analysis, that the resulting neural-operator-based controller preserves prescribed-time stability provided that the operator approximation error satisfies an explicit bound. Furthermore, we investigate a direct approximation of the full feedback law mapping the plant parameter functions and state measurements to the boundary control input. For this setting, we prove semiglobal practical prescribed-time stability of the closed-loop system. Numerical experiments demonstrate that the proposed approach reduces the computational cost of kernel generation by several orders of magnitude, thereby enabling real-time prescribed-time stabilization for heat equations with spatially and temporally varying coefficients.

math.OC↗

arXiv2025

Neural Operators for Adaptive Control of Traffic Flow Models

The uncertainty in human driving behaviors leads to stop-and-go instabilities in freeway traffic. The traffic dynamics are typically modeled by the Aw-Rascle-Zhang (ARZ) Partial Differential Equation (PDE) models, in which the relaxation time parameter is usually unknown or hard to calibrate. This paper proposes an adaptive boundary control design based on neural operators (NO) for the ARZ PDE systems. In adaptive control, solving the backstepping kernel PDEs online requires significant computational resources at each timestep to update estimates of the unknown system parameters. To address this, we employ DeepONet to efficiently map model parameters to kernel functions. Simulations show that DeepONet generates kernel solutions nearly two orders of magnitude faster than traditional solvers while maintaining a loss on the order of \(10^{-2}\). Lyapunov analysis further validates the stability of the system when using DeepONet-approximated kernels in the adaptive controller. This result suggests that neural operators can significantly accelerate the acquisition of adaptive controllers for traffic control.

math.OC↗