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

Kaijing Lyu, Umberto Biccari, Jun-Min Wang

Abstract

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.

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

Abstract

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.
Paper Structure (12 sections, 6 theorems, 83 equations, 5 figures, 2 tables)

This paper contains 12 sections, 6 theorems, 83 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Neural-operator approximation for prescribed-time backstepping
  3. Contributions and organization
  4. Problem formulation
  5. DeepONet approximation of the time-varying kernel
  6. Lyapunov analysis under the NO-approximated control law $\lambda\mapsto k$
  7. Approximating the full feedback law $(\lambda, v)\mapsto U$
  8. Simulations
  9. Simulations’ configuration, dataset generation and NO training
  10. Simulations results: exponential stabilization with NO-approximated kernel $\lambda \mapsto k$
  11. Simulations results: practical stabilization with NO-approximated full feedback law $(\lambda,v) \mapsto U$
  12. Conclusions and open problems

Key Result

Lemma 1

Assume that $\lambda \in G_\alpha(0,T; C^0(0,1))$ for some parameter $\alpha\in[1,2]$. Let with $c \in C^\infty(0,T)$. Then the kernel equation eq:kernel_1 admits a unique strong solution where Moreover, there exits a constant $C>0$, such that

Figures (5)

  • Figure 1: Open-loop evolution of the state $v$
  • Figure 2: Backstepping kernels $k$ and $\hat{k}$ at $t = 5$s, together with the $L^2$ approximation error.
  • Figure 3: Backstepping kernels $k$ and $\hat{k}$ at $t = 7$s, together with the $L^2$ approximation error.
  • Figure 4: Closed-loop evolution of the true state $v$ and the state $\hat{v}$ obtained using the NO–approximated backstepping kernel $k$. The NO is trained to learn the mapping $\lambda(\cdot,\cdot)\mapsto k$, and the resulting approximate kernel is used to construct the backstepping feedback law.
  • Figure 5: Closed-loop state responses under two control inputs: the "perfect control" $U$ computed from the exact backstepping kernel and the "approximate control" $\hat{U}$ generated directly by a NO trained to learn the mapping $(\lambda(\cdot,\cdot),\, v) \mapsto U$.

Theorems & Definitions (13)

  • Definition 1
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Lemma 2
  • proof
  • Theorem 3
  • ...and 3 more