On the Optimal Control Problem of Stochastic Semilinear Partial Differential Equations with Non-Globally Lipschitz Coefficients

Technical Summary: Optimal Control of Stochastic Semilinear PDEs with Non-Globally Lipschitz Coefficients

Problem Formulation
The paper addresses the optimal control problem for stochastic semilinear partial differential equations (SPDEs) of the form:
$$dy = (Ay + f(y) + u(t))dt + \sigma(y)dW(t)$$
defined on a bounded domain $D \subset \mathbb{R}^d$ with Dirichlet boundary conditions. The system is driven by an infinite-dimensional $Q$-Wiener process $W(t)$. The study considers two distinct time horizons: a finite interval $[0, T]$ and an infinite horizon $[0, +\infty)$.

The objective is to minimize cost functionals of the form:
$$J(u) = \mathbb{E}\int_0^T \int_D (y^2 + u^2) dx dt$$
(for finite time) and
$$J(u) = \mathbb{E}\int_0^\infty \int_D (e^{-\gamma t}y^2 + u^2) dx dt$$
(for infinite time), where $y$ is the state and $u$ is the control variable.

The primary mathematical challenge lies in the nonlinearity $f(y)$. Unlike standard literature which often assumes global Lipschitz continuity or polynomial growth, this work considers coefficients where $f$ is only $C^1$ with a one-sided Lipschitz condition ($f'(s) \le \lambda_f$) and potentially exhibits faster-than-polynomial growth (e.g., exponential terms found in the Frank–Kamenetskii equation). The diffusion coefficient $\sigma$ is assumed to be globally Lipschitz. A specific coupling condition, $|f'(s)\sigma^2(s)| \le C(1+|s|^2)$, is imposed to manage the interaction between the reaction and noise terms.

Methodology
The authors propose an approximation scheme to overcome the lack of global Lipschitz continuity in $f$, which prevents the direct application of the stochastic maximum principle (a tool requiring bounded derivatives).

1. Cut-off Approximation: The authors construct a sequence of auxiliary problems where the nonlinearity $f$ is replaced by a truncated version $f_k(s) = f(\min(\max(-k, s), k))$. These truncated coefficients are globally Lipschitz and possess bounded derivatives (after suitable smoothing), thereby satisfying the regularity requirements for the stochastic maximum principle.
2. Existence of Optimal Controls: For both the original problem and the sequence of approximating problems, the existence of weak martingale solutions (optimal pairs of state and control) is established. This relies on the lower semicontinuity of the cost functional and the weak compactness of admissible controls in $L^2$ spaces.
3. Convergence Analysis: The core of the methodology involves proving that the solutions to the approximating problems converge to the solution of the original problem. This is achieved through:
   • Energy Estimates: Deriving uniform bounds on the solutions using Itô's formula and Gronwall's inequality.
   • Compactness Arguments: Utilizing the Jakubowski-Skorokhod theorem to handle the weak convergence of probability measures and establishing tightness of the solution sequences.
   • Weak Convergence: Demonstrating that weak limits of the approximating controls and states satisfy the original SPDE and minimize the original cost functional.

Key Contributions and Results

• Existence for Non-Globally Lipschitz Coefficients: The paper establishes the existence of optimal controls for SPDEs where the reaction term $f$ does not satisfy global Lipschitz or standard polynomial growth conditions. This extends previous results that were restricted to monotone or polynomially growing nonlinearities.
• Approximation Scheme for Finite Time Intervals:
  • The authors prove that the optimal cost values of the approximating problems ($J_k^*$) converge to the optimal cost of the original problem ($J^*$) as the truncation parameter $k \to \infty$.
  • They demonstrate that the sequence of optimal controls for the approximating problems is strongly compact in $L^2(\Omega_T; H)$. Any weak subsequential limit of these controls constitutes an optimal control for the original problem.
  • The corresponding state trajectories converge strongly in $L^2(\Omega_T; H)$.
• Approximation Scheme for Infinite Time Intervals:
  • For the infinite horizon problem, the authors propose a two-step approximation: first truncating the nonlinearity (as in the finite case) and second, truncating the time horizon to $[0, n]$.
  • They prove that the optimal costs for the finite-horizon problems converge to the infinite-horizon optimal cost as $n \to \infty$.
  • The sequence of controls extended by zero beyond time $n$ forms a minimizing sequence for the infinite-horizon problem. Weak limits of these sequences yield optimal controls for the infinite-horizon problem.
• Weak Martingale Solutions: The results are formulated in terms of weak martingale solutions, acknowledging that the probability space and Wiener process may need to be constructed as part of the solution (rather than being fixed a priori), which is standard for SPDEs with non-Lipschitz coefficients.

Significance and Claims
The paper claims to fill a gap in the literature regarding optimal control for SPDEs with coefficients that lack bounded Fréchet derivatives. While the stochastic maximum principle is a powerful tool for constructing feedback controls, its applicability is severely limited by the requirement of bounded derivatives. By introducing a cut-off approximation scheme, the authors provide a rigorous method to:

1. Prove the existence of optimal controls in settings where standard maximum principle techniques fail.
2. Construct these controls via a convergent sequence of regularized problems where the maximum principle is applicable.

The authors note that their conditions cover important physical models, such as the Frank–Kamenetskii equation, where the reaction term grows exponentially, a scenario where standard polynomial growth assumptions are violated. The work provides a theoretical foundation for solving such control problems without requiring the coefficients to be globally Lipschitz.