Stochastic Linear-quadratic Control Problems with Affine Constraints

TL;DR

This paper addresses stochastic linear-quadratic control problems with affine constraints, deriving dual formulations and conditions for solution uniqueness using Pontryagin's principle and Lagrangian duality.

Contribution

It establishes the dual problem, proves solution equivalence under Slater's condition, and introduces a new condition for solution uniqueness in affine constrained stochastic control.

Findings

Dual problem formulation for constrained stochastic control

Equivalence of primal and dual solutions under Slater's condition

New sufficient condition for solution invertibility and uniqueness

Abstract

This paper investigates the stochastic linear-quadratic control problems with affine constraints, in which both equality and inequality constraints are involved. With the help of the Pontryagin maximum principle and Lagrangian duality theory, the dual problem of original problem is established and the state feedback form of the solution to the optimal control problem is obtained. Under the Slater condition, the equivalence is proved between the solutions to the original problem and the ones of the dual problem, and the KKT condition is also provided for solving original problem. Especially, a new sufficient condition is given for the invertibility assumption, which ensures the uniqueness of the solutions to the dual problem.