Second-Order $Λ$-Sets and Extensions to Non-Smooth, Hybrid, and Stochastic Optimal Control

This paper develops a comprehensive extension of the $Λ$-set framework for optimal control, introducing second-order $Λ$-sets and generalizing the theory to non-smooth, hybrid, and stochastic hybrid systems. We first establish second-order necessary conditions that incorporate curvature information of the reachable set, providing refined optimality criteria that bridge classical second-variation methods with the geometric $Λ$-set approach. The framework is then extended to Filippov systems with discontinuous dynamics and to hybrid dynamical systems with state-dependent switching, yielding new necessary conditions for optimality in these settings. Furthermore, we introduce stochastic $Λ$-sets for systems subject to both continuous diffusion and discrete random switching, connecting the framework to Peng's stochastic maximum principle. Throughout the paper, detailed examples -- including nonholonomic systems, mechanical systems with friction, and stochastic temperature control -- illustrate the theoretical developments and demonstrate the practical applicability of the extended $Λ$-set theory. The results unify and generalize existing maximum principles, offering a powerful geometric tool for analyzing optimal control problems across a broad spectrum of system classes, from classical smooth systems to modern stochastic hybrid systems.