Convergence analysis for an implementable scheme to solve the linear-quadratic stochastic optimal control problem with stochastic wave equation

TL;DR

The paper develops a rigorously analyzed implementable numerical scheme for a stochastic linear-quadratic control problem governed by a stochastic wave equation with affine noise. It derives a coupled FBSPDE optimality system via a stochastic Pontryagin maximum principle, and then discretizes in space with conforming finite elements and in time with an implicit midpoint rule, proving strong convergence rates without resorting to Malliavin calculus. A gradient-descent framework is introduced, augmented by artificial iterates that enable exact computation of conditional expectations and eliminate costly Monte Carlo sampling, yielding a scalable algorithm with provable error bounds. An explicit rate of convergence in the space-time discretization is established (order $\tau + h^2$), and the overall method combines theoretical guarantees with practical efficiency, as confirmed by numerical experiments. The approach provides a robust, implementable pathway for SLQ problems constrained by SPDEs, with potential applicability to stochastic wave-structure control and related uncertain-media scenarios.

Abstract

We study an optimal control problem for the stochastic wave equation driven by affine multiplicative noise, formulated as a stochastic linear-quadratic (SLQ) problem. By applying a stochastic Pontryagin's maximum principle, we characterize the optimal state-control pair via a coupled forward-backward SPDE system. We propose an implementable discretization using conforming finite elements in space and an implicit midpoint rule in time. By a new technical approach we obtain strong convergence rates for the discrete state-control pair without relying on Malliavin calculus. For the practical computation we develop a gradient-descent algorithm based on artificial iterates that employs an exact computation for the arising conditional expectations, thereby eliminating costly Monte Carlo sampling. Consequently, each iteration has a computational cost that is proportional to the number of spatial degrees of freedom, producing a scalable method that preserves the established strong convergence rates. Numerical results validate its efficiency.

Figures (5)

• Figure 1: A flowchart outlining the error analysis and algorithmic approach for the SLQ problem. Here, PMP denotes Pontryagin's maximum principle.
• Figure 2: Surface plots for a path of the $\ell$‑th iterate over the space–time domain: (A) control iterate $(t, x)\mapsto U_{h\tau}^{(\ell)}(\omega, t, x)$; (B) displacement state iterate $(t, x)\to X_{1,h\tau}^{(\ell)}(\omega,t, x)$.
• Figure 3: (A) Histogram (empirical density) of $\{U_{h \tau}^{(\ell)}(t_{N-1}, 0.5;\omega_{i})\}_{i=1}^{\rm M}$, and (B) decay of the (approximated) cost functional $\ell\mapsto J_{h\tau}^{\rm M}(X_{1,h\tau}^{(\ell)},U_{h\tau}^{(\ell)})$ for ${\rm M}=1000$.
• Figure 4: Comparison of the iterates under three noise levels (columns). Rows show various profiles of a single path of a displacement iterate $X_{1, h\tau}^{(\ell)}(\cdot; \omega)$, and velocity iterate $X_{2,h\tau}^{(\ell)}(\cdot; \omega)$. In Row 1,2,3: Displacement iterate $x\mapsto X_{1, h\tau}^{(\ell)}(t, x, \omega)$ and velocity iterate $x\mapsto X_{2, h\tau}^{(\ell)}(t, x, \omega)$, respectively, for different times $t=0.25, 0.50, 0.75$.
• Figure 5: Comparison of the iterates under three noise levels (columns). Rows show various profiles of the single path of the displacement iterate $X_{1,h\tau}^{(\ell)}(\cdot; \omega)$, and velocity iterate $X_{2,h\tau}^{(\ell)}(\cdot; \omega)$. In Row 1,2,3: Displacement iterate $t\mapsto X_{1, h\tau}^{(\ell)}(t, x, \omega)$ and velocity iterate $t\mapsto X_{2, h\tau}^{(\ell)}(t, x, \omega)$, respectively, for different spatial points $x=0.25, 0.50, 0.75$.

Theorems & Definitions (75)

• Remark 1.1: Computational time
• Definition 2.1
• Lemma 2.1
• proof
• Proposition 2.2: Existence of a unique optimal tuple
• proof
• Lemma 2.3: Existence and uniqueness of a solution to BSPDE \ref{['1.5']}
• proof
• Remark 2.1
• Theorem 2.4: Pontryagin's maximum principle