[Paper Review] Stochastic Maximum Principle for a PDEs with noise and control on the boundary

Giuseppina Guatteri|arXiv (Cornell University)|Jul 19, 2008

Stochastic processes and financial applications12 references3 citations

TL;DR

This paper establishes a stochastic maximum principle for stochastic partial differential equations (SPDEs) with noise and control acting on the boundary, using a novel approach that avoids second-order approximations by leveraging regularity of the adjoint equation. The key contribution is a necessary optimality condition for boundary-controlled SPDEs with non-convex control spaces, valid under fractional power regularity assumptions and a convergence rate of $\varepsilon^{\alpha}$ with $\frac{1}{2} < \alpha < 1$. The framework applies to both Neumann and Dirichlet boundary conditions via weighted $L^2$ spaces and unbounded operators derived from resolvent theory.

ABSTRACT

In this paper we prove necessary conditions for optimality of a stochastic control problem for a class of stochastic partial differential equations that is controlled through the boundary. This kind of problems can be interpreted as a stochastic control problem for an evolution system in an Hilbert space. The regularity of the solution of the adjoint equation, that is a backward stochastic equation in infinite dimension, plays a crucial role in the formulation of the maximum principle.

Motivation & Objective

• To develop a stochastic maximum principle for SPDEs where both noise and control are applied on the boundary, a setting less explored than diffused noise and control.
• To address the challenge of non-convex control spaces, which typically require second-order expansions, by proving a first-order optimality condition with $\varepsilon^{\alpha}$ convergence rate.
• To establish the regularity of the adjoint equation in the domain of the formal adjoint of the unbounded operator arising from boundary control, enabling the formulation of the maximum principle.
• To extend the maximum principle to SPDEs with Dirichlet boundary conditions by introducing a weighted $L^2$ space framework.
• To provide two concrete examples—Neumann and Dirichlet boundary control problems—demonstrating the applicability of the theoretical framework.

Proposed method

• Formulate the SPDE as an infinite-dimensional stochastic evolution equation in a Hilbert space $H = L^2(0,1)$ or $L^2(0,\infty; (\rho^{\theta+1} \wedge 1)\,d\rho)$, using the generator $A$ of the Laplacian with appropriate boundary conditions.
• Introduce unbounded operators $(\lambda - A)D$ and $(\lambda - A)D_1$ to represent the boundary control and noise terms, where $D$ and $D_1$ map controls and Wiener processes into the domain of fractional powers of $\lambda - A$, ensuring mild solution existence.
• Use resolvent theory and the existence of solutions $d^i(x)$ to Neumann or Dirichlet eigenvalue problems to define the boundary operators $D$ and $D_1$, ensuring the required regularity for the unbounded terms.
• Derive the adjoint equation as a backward stochastic differential equation in the infinite-dimensional Hilbert space $H$, proving that its solution takes values in the domain of $[(\lambda - A)D]^*$, which is essential for the maximum principle.
• Establish the maximum principle by analyzing the first-order variation of the cost functional, showing that the Hamiltonian condition holds under $\varepsilon^{\alpha}$-order approximation with $\alpha > \frac{1}{2}$, avoiding second-order terms.
• Verify the conditions of the maximum principle in two examples: one with Neumann boundary conditions and a bounded domain, and another with Dirichlet conditions on an unbounded domain, using weighted $L^2$ spaces to ensure operator regularity.

Experimental results

Research questions

• RQ1 Can a stochastic maximum principle be formulated for SPDEs where both noise and control are localized on the boundary, rather than distributed in space?
• RQ2 How can the maximum principle be established when the control space is non-convex, which typically necessitates second-order expansions?
• RQ3 What regularity conditions are required for the adjoint equation to ensure the validity of the maximum principle in the presence of unbounded boundary operators?
• RQ4 How can boundary noise and control be embedded into a Hilbert space framework when standard $L^2$ spaces fail to support the required operator regularity?
• RQ5 To what extent can the framework be extended to Dirichlet boundary conditions, which are known to pose difficulties in semigroup and evolution equation formulations?

Key findings

• The paper establishes a stochastic maximum principle for SPDEs with boundary control and noise, valid even when the control space is non-convex, by proving a first-order optimality condition with $\varepsilon^{\alpha}$ convergence rate for $\frac{1}{2} < \alpha < 1$.
• The adjoint equation solution is shown to take values in the domain of $[(\lambda - A)D]^*$, which is essential for the Hamiltonian formulation and ensures the validity of the maximum principle.
• For Neumann boundary conditions on $[0,1]$, the existence of $d^i \in H^{2\alpha}(0,1)$ solving the eigenvalue problem ensures the required regularity of the boundary operators.
• For Dirichlet conditions on $[0,\infty)$, the use of a weighted $L^2$ space $L^2(0,\infty; (\rho^{\theta+1} \wedge 1)\,d\rho)$ allows the Dirichlet map to take values in $D((\lambda - A)^\alpha)$ for $\alpha > \frac{1}{2}$, enabling the formulation of the evolution equation.
• The cost functional is well-defined and satisfies the required smoothness and growth conditions in both examples, with $l$ and $h$ having bounded or sublinearly growing derivatives.
• The theoretical results are validated through two concrete examples: one with Neumann boundary control and space-time white noise, and another with Dirichlet boundary control and a boundary Wiener process, both showing that the hypotheses of the main theorem are fulfilled.