Approximate Sparse Stochastic Control for Time-Varying Systems with Control-Dependent Diffusion

This paper proposes a scalable framework for sparse stochastic optimal control in time-varying systems with control-dependent diffusion. By relaxing the classical stochastic normality condition, we establish an approximate equivalence between ℓ ₀ and ℓ ₁ optimal control formulations under local or probabilistic regularity. The framework unifies sparsity, stochasticity, and safety by incorporating variational inequalities to handle state and control constraints. To overcome the curse of dimensionality, we develop efficient numerical solvers based on sparse grids, Tensor-Train decomposition, and neural residual networks. Numerical experiments on robotic and energy systems demonstrate substantial gains in control sparsity, safety compliance, and computational scalability, confirming the practicality and robustness of the proposed approach in high-dimensional, safety-critical settings.