Sparse Ergodic Control with Control-Dependent Noise via Physics-Informed Neural Networks

Sparse ergodic control provides a natural framework for long-run stochastic decision-making under resource constraints. Existing formulations, however, are typically restricted to control-affine systems with control-independent diffusion. When the diffusion coefficient depends explicitly on the control input, the associated ergodic Hamilton–Jacobi–Bellman (HJB) equation becomes non-separable through the term tr[a(x, u)∇2V], so classical arguments based on control-affine separability no longer apply directly. In this work, we study sparse ergodic control of stochastic systems with control-dependent diffusion and nonlinear dynamics within a viscosity-solution and learning-based framework. To address the discontinuous l0-type sparsity penalty, we introduce smooth non-convex sparsity approximations that preserve differentiability while retaining sparse threshold behavior. Within a viscosity-solution framework, we analyze the existence and uniqueness properties of the associated ergodic pair and establish localized approximation error estimates for the smooth approximation. We further characterize a quasi-threshold sparse structure of the resulting optimal feedback policies in non-affine stochastic systems with control-dependent noise. On the computational side, we develop a Physics-Informed Neural Network (PINN)-based solver with adaptive residual-driven sampling for high-dimensional sparse ergodic HJB equations, together with a distributed monotone-inspired iterative scheme for weakly coupled multi-agent systems. Numerical experiments on multi-robot swarm navigation and renewable-integrated smart-grid control demonstrate that the proposed methods produce sparse control policies while preserving stable long-run performance under stochastic disturbances.