Second order optimality conditions for piecewise regular extremals in Optimal Control

What the math gives to ML

The paper develops a stability and second-order-analysis toolkit for trajectories generated by the maximum of finitely many smooth Hamiltonian branches. Its transferable asset is the combination of explicit switching surfaces, Lipschitz continuity of the resulting hybrid flow, and monotone evolution of a Lagrangian tangent subspace even when the vector field is only piecewise smooth. This suggests a hybrid neural ODE or state-space block with explicit branch switching and a positive-semidefinite sensitivity regularizer. The first implementation should test whether this improves gradient stability and loss descent relative to an unconstrained hard-switch neural ODE.