A class of stochastic control problems with state constraints

2026-03-05

AI Summary

This study addresses a linear-quadratic stochastic optimal control problem subject to state constraints, aiming to steer the system trajectory away from prescribed forbidden regions in space-time while minimizing the expected cost of state and control. By modeling the dynamics via diffusion processes and leveraging stochastic control theory together with probabilistic representation techniques, the authors establish a probabilistic representation of the value function under regularity conditions on the constraint set and derive its explicit solution. The resulting optimal control policy is strongly adapted and implementable via the filtration generated by the underlying Brownian motion. In addition to providing several analytical examples, this work offers a systematic framework for solving stochastic control problems with state constraints.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Intelligent Robots: State Estimation

Application Category

• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Web Mining and Content Analysis: Content-based information diffusion
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs

Abstract

We obtain a probabilistic solution to linear-quadratic optimal control problems with state constraints. Given a closed set $\mathcal{D}\subseteq [0,T]\times\mathbb{R}^d$, a diffusion $X$ in $\mathbb{R}^d$ must be linearly controlled in order to keep the time-space process $(t,X_t)$ inside the set $\mathcal{C}:=([0,T]\times\mathbb{R}^d)\setminus\mathcal{D}$, while at the same time minimising an expected cost that depends on the state $(t,X_t)$ and is quadratic in the speed of the control exerted. We find a probabilistic representation for the value function and an optimal control under a set of mild sufficient conditions concerning the coefficients of the underlying dynamics and the regularity of the set $\mathcal{D}$. The optimally controlled dynamics is in strong form, in the sense that it is adapted to the filtration generated by the driving Brownian motion. Fully explicit formulae are presented in some relevant examples.

Problem

• stochastic control
• state constraints
• linear-quadratic control
• diffusion processes
• optimal control

Innovation

• stochastic control
• state constraints
• probabilistic representation
• linear-quadratic optimization
• strong solution