Second-order conditions for optimal control problems with mixed control-state constraints and control appearing linearly

We study optimal control problems with a two-sided mixed control-state constraint and assume that the control variable appears linearly in both the system dynamics and constraints. By defining the control-state constraint as a new control variable, the optimal control problem is transformed into an optimal control problem with simple bounds on the new control variable. In view of Pontryagin's Minimum Principle, optimal controls of the transformed problem are concatenations of bang-bang or singular arcs. Second-order sufficient conditions (SSC) for such bang-singular controls have recently been given in the literature. We summarize results on SSC and illustrate their numerical verification on the optimal control of the Rayleigh equation for various bounds in a two-sided control-state constraint.