Lagrangian dual method for solving stochastic linear quadratic optimal control problems with terminal state constraints

TL;DR

This paper develops a Lagrangian duality framework for solving stochastic linear quadratic control problems with terminal state constraints, providing theoretical solutions and an iterative augmented Lagrangian method with proven convergence.

Contribution

It introduces a strong duality theorem under specific conditions and proposes two novel approaches, including a closed-form solution and an ALM algorithm, for constrained stochastic LQ problems.

Findings

Strong duality theorem established under convexity and surjectivity conditions.

Closed-form solution construction based on duality principles.

Convergence of the augmented Lagrangian method proved.

Abstract

A stochastic linear quadratic (LQ) optimal control problem with a pointwise linear equality constraint on the terminal state is considered. A strong Lagrangian duality theorem is proved under a uniform convexity condition on the cost functional and a surjectivity condition on the linear constraint mapping. Based on the Lagrangian duality, two approaches are proposed to solve the constrained stochastic LQ problem. First, a theoretical method is given to construct the closed-form solution by the strong duality. Second, an iterative algorithm, called augmented Lagrangian method (ALM), is proposed. The strong convergence of the iterative sequence generated by ALM is proved. In addition, some sufficient conditions for the surjectivity of the constraint mapping are obtained.