Algorithmic Optimality Guarantees for Nonsmooth $H_\infty$ Output-Feedback Policy Search

Stabilizing a dynamical system is a fundamental problem that serves as a cornerstone for many complex tasks in the field of control systems. The problem becomes challenging when the system model is unknown. Among the Reinforcement Learning (RL) algorithms that have been successfully applied to solve problems pertaining to unknown linear dynamical systems, the policy gradient (PG) method stands out due to its ease of implementation and can solve the problem in a model-free manner. However, most of the existing works on PG methods for unknown linear dynamical systems assume full-state feedback. In this paper, we take a step towards model-free learning for partially observed linear dynamical systems with output feedback and focus on the fundamental stabilization problem of the system. We propose an algorithmic framework that stretches the boundary of PG methods to the problem without global convergence guarantees. We show that by leveraging zeroth-order PG update based on system trajectories and its convergence to stationary points, the proposed algorithms return a stabilizing output feedback policy for discrete-time linear dynamical systems. We also explicitly characterize the sample complexity of our algorithm and verify the effectiveness of the algorithm using numerical examples.

We study continuous-time stochastic linear--quadratic control with a scalar state and nonnegative control constraints. With quadratic running costs, randomized exploration can change the optimal mean feedback. We introduce an auxiliary problem without entropy in which the conditional policy mean is nonnegative and the covariance is prescribed in proportion to the squared state. Moment reduction and state separation yield two regional Riccati equations and characterize the finite-horizon value function and optimal conditional mean. We give a sufficient control--cost compatibility condition under which this mean coincides with the optimal feedback of the original constrained problem. For time-invariant coefficients, we characterize the undiscounted infinite-horizon optimum and establish mean square asymptotic stability under the stated assumptions. For unknown dynamics, we develop model-free actor--critic schemes for finite and infinite horizons. Numerical experiments compare the learned policies with their analytical targets and examine the effect of exploration on the optimal mean.

Finite-horizon optimal-control computations repeatedly solve two-point Pontryagin boundary value problems whose conditioning can deteriorate as the horizon grows. We give a verifiable data-level certificate under which it does not. Hyperbolicity of the reduced state--costate transition matrix, together with scaled stable--unstable boundary transversality, yields an endpoint-corrected Green inverse with horizon-independent constants and weighted contractions transfer this inverse to the nonlinear problem, so the original Pontryagin endpoint rows $x_0=x_{\rm in}$ and $p_T=r_x(x_T,y)$ carry a unique local stationary branch whose first-order expansion and Lipschitz constants are uniform in the horizon. Consequently the finite-horizon feedback map is horizon-uniformly Lipschitz, first-order expandable, and satisfies an exact shrinking-horizon consistency identity. Symplectic and Riccati criteria certify the hypotheses from matrix data: every stabilizable definite linear-quadratic system with invertible dynamics and a locally concave terminal Hessian at the reference qualifies. Reproducible computations illustrate both certificates.