Pontryagin’s maximum principle [@branicky98; @sussmann99; @riedinger03] and subsequently, various computational techniques have been developed to solve this problem [@shaikh07; @soler12; @pakniyat14]. Based on the Pontryagin’s maximum principle, the optimal solution should satisfy a two-point boundary value problem (BVP). However, similar to the classical control problem, the difficulties related to numerical solution of the necessary condition of optimality limits the application of this approach. In [@riedinger99], it has been shown that for a Linear-Quadratic (LQ) problem it is sufficient to solve a sequence of Riccati equations with proper transversality conditions at the switching times in order to optimize the continuous inputs but the mode switches should be still calculated based on the enumerations of all the possible switches at each time step. In order to ease the computational burden of finding the optimal switching behavior, in [@bengea05], the switched system is embedded in larger family of systems defined by the convex hull of the switched subsystems. It has been shown that if the sufficient and necessary conditions for optimality in the embedded systems exists, the bang-bang optimal solution of the embedded problem is also the optimal solution of the switched system; otherwise a sub-optimal solution can be derived.

[@borrelli05] propose an 