Sparse Stochastic Optimal Control under Control-Dependent Diffusion and Time-Varying Dynamics

We propose a unified framework for sparse stochastic optimal control in systems governed by time-varying dynamics and control-dependent diffusion—key features in robotic manipulators, energy systems, and other real-world applications. In contrast to conventional models assuming control-independent noise, our formulation explicitly captures the feedback between control effort and stochastic disturbances. This leads to a modified Hamilton–Jacobi–Bellman (HJB) equation that is nonconvex, nonsmooth, and nonmonotone. To address these challenges, we develop a relaxed viscosity solution framework that ensures analytical well-posedness. We also establish structural conditions under which sparse $L_0$-optimal control can be exactly recovered via its convex $L_1$ relaxation, leading to provable bang–off–bang strategies. Numerical simulations on robotic arms and photovoltaic–battery systems demonstrate the effectiveness of the proposed approach, showing superior control sparsity, robustness to noise, and safety guarantees under uncertainty. The results suggest our method offers a principled and scalable solution for real-time decision-making in noisy, resource-constrained environments.