off-line method to synthesize an optimal control law for a discrete linear hybrid system with linear inequality constraints. The proposed method is a combination of dynamic programming and multi-parametric quadratic programming which designs a feedback law for continuous and discrete inputs in the feasible regions. A simpler approach in [@bemporad99] uses a mixed integer linear/ quadratic program to solve the optimal control problem for mixed logical dynamical systems.

Optimizing the cost function with respect to the switching times has been studied for autonomous systems by [@egerstedt03; @johnson11; @wardi12] and for non-autonomous systems by [@kamgarpour12]. By using the derivative of the cost function with respect to the switching time, these methods use nonlinear programming techniques to optimize the cost function. However, in general these methods do not consider the sensitivity of the continuous inputs’ control law to the switching times.

While many of the aforementioned approaches are computationally demanding for real-time robotic applications, there is a class of efficient optimal control algorithms known as Sequential Linear Quadratic (SLQ) methods which can be applied to real-time, complex robotic applications [@neunert16]. An SLQ algorithm sequentially solves the extremal problem around the latest estimation of the optimal trajectories and improves these optimal trajectories using 