2025/04/25 by Yurii Averboukh, Averboukh, Yurii, A. N. Volkov +1
Computer Science · Mathematics · #46G05 #49J52 #49K20 #49L25 #49N35 #82C22 #93C20 #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Optimization and Control (math.OC) #Optimization and Variational Analysis #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2504.18232
openalex publication_date 2025/04/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper explores the application of nonsmooth analysis in the Wasserstein space to finite-horizon optimal control problems for nonlocal continuity equations. We characterize the value function as a strict viscosity solution of the corresponding Bellman equation using the notions of ε-subdifferentials and ε-superdifferentials. The main paper's result is the fact that continuous subsolutions and supersolutions of this Bellman equation yield lower and upper bounds for the value function. These estimates rely on proximal calculus in the space of probability measures and the Moreau-Yosida regularization. Furthermore, the upper estimates provide a family of approximately optimal feedback strategies that realize the concept of proximal aiming.