Optimal control with stochastic PDE constraints and uncertain controls

The optimal control of problems that are constrained by partial differential equations with uncertainties and with uncertain controls is addressed. The Lagrangian that defines the problem is postulated in terms of stochastic functions, with the control function possibly decomposed into an unknown deterministic component and a known zero-mean stochastic component. The extra freedom provided by the stochastic dimension in defining cost functionals is explored, demonstrating the scope for controlling statistical aspects of the system response. One-shot stochastic finite element methods are used to find approximate solutions to control problems. It is shown that applying the stochastic collocation finite element to the formulated problem leads to a coupling between stochastic collocation points when a deterministic optimal control is considered or when moments are included in the cost functional, thereby obviating the primary advantage of the collocation method over the stochastic Galerkin method for the considered problem. The application of the presented methods is demonstrated through a number of numerical examples. The presented framework is sufficiently general to also consider a class of inverse problems, and numerical examples of this type are also presented.

💡 Research Summary

This paper addresses optimal control problems where the governing partial differential equations (PDEs) contain stochastic coefficients and the control itself is uncertain. The authors formulate the state variable z(x,ω) and the control u(x,ω) as random fields defined on the tensor‑product Hilbert space L²(D)⊗L²(Ω). Crucially, the control is decomposed additively into an unknown deterministic mean component ū(x) (the signal to be implemented) and a known zero‑mean stochastic component u₀(x,ω) that models actuator uncertainty. An analogous decomposition is applied to boundary controls.

Two families of cost functionals are introduced. The first, J₁, penalises the L²‑distance between the stochastic state and a target field, adds a term involving the L²‑norm of the state’s standard deviation, and includes regularisation terms for the distributed and boundary controls. By adjusting the weight β on the variance term, one can directly influence the variability of the system response. The second functional, J₂, replaces the stochastic‑state term with the distance between the mean state \bar{z} and the deterministic target, thereby focusing solely on the expected response while still allowing variance penalisation.

To make the problem tractable, a finite‑dimensional noise assumption is adopted: all random fields (diffusion coefficient κ, Dirichlet data, Neumann data, and controls) are expressed as functions of a finite set of independent random variables ξ₁,…,ξ_L. This enables a parametric representation of the stochastic PDE, either via a truncated Karhunen–Loève expansion or a generalized polynomial chaos basis.

The authors employ a “one‑shot” approach: they introduce Lagrange multipliers λ and χ, construct a Lagrangian that combines the cost functional and the PDE constraints, and derive first‑order optimality conditions by variational differentiation. The resulting optimality system consists of the original stochastic state equation, an adjoint equation (the backward stochastic PDE associated with the cost functional), and an algebraic optimality condition linking the control to the adjoint (γ u+λ=0). Solving this coupled system yields the optimal deterministic control mean ū directly, without the need for iterative optimisation loops.

Two stochastic finite‑element discretisation strategies are examined. The stochastic Galerkin method leads to a large, coupled linear system whose block structure reflects the interaction of spatial and stochastic degrees of freedom. The stochastic collocation method, which normally decouples the problem into independent deterministic solves at selected collocation points, loses this non‑intrusivity when the cost functional contains statistical moments or when the control is deterministic. In such cases, collocation points become coupled, negating its primary advantage. Consequently, the Galerkin approach is shown to be more suitable for the class of problems considered.

Because the Galerkin system can be very large, the paper proposes two preconditioners tailored to its block structure: a block‑diagonal preconditioner that approximates each stochastic block independently, and a scaling preconditioner that balances the contributions of the stochastic and spatial components. Both preconditioners dramatically improve the convergence of Krylov subspace solvers such as GMRES.

Extensive numerical experiments are presented. In two‑ and three‑dimensional diffusion problems, the authors demonstrate how varying the weights (α, β, γ, δ) trades off between mean performance and variance reduction. They also treat an inverse‑type problem where the control is purely stochastic (no deterministic mean) and the goal is to recover the mean component from noisy observations of the state. All code is released under the LGPL, enabling reproducibility.

In summary, the paper contributes (1) a clear decomposition of stochastic controls into deterministic and random parts, (2) cost functionals that explicitly incorporate statistical moments, (3) a rigorous comparison of stochastic collocation versus Galerkin for PDE‑constrained optimisation, (4) efficient preconditioned solvers for the resulting one‑shot Galerkin systems, and (5) an extension of the framework to stochastic inverse problems. The methodology is broadly applicable to engineering domains where uncertainty in material properties, loads, or actuators must be accounted for in optimal design and control.