problem based on the parameterization of switching times. We assume that the switched system dynamics consist of $I$ subsystems where the system dynamics for the $i$th subsystem (${ i \in \{1,2,\dots,I\} }$) is as follows $$\label{eq:system_dynamics}
\dot{\vx}(t) = \vf_i \left( \vx(t),\vu(t) \right) \qquad \text{\textit{for }} \, t_{i-1} \leq t < t_{i},$$ where $\vx(t) \in \mathbb{R}^{n_x}$ is the continuous state, $\vu(t) \in \mathbb{R}^{n_u}$ is the piecewise continuous control input, and $\vf_i: \mathbb{R}^{n_x} \times \mathbb{R}^{n_u} \rightarrow \mathbb{R}$ is the vector field of subsystem $i$ which is continuously differentiable and Lipschitz up to the first order derivatives. $t_i$ is the switching time between subsystem $i$ and $i+1$. $t_0$ and $t_I$ are respectively the given initial time and the final time. The initial state is $\vx_0$ and $\vx(t_{i}^-) = \vx(t_{i}^+)$ at the switching moments because of the state continuity condition. The optimal control problem for the switched system in Equation  is defined as $$\label{eq:general_opt}
\min\limits_{\scriptstyle \begin{matrix}
\vt\!\in\!\mathbb{T}, \vu(\cdot)
\end{matrix}
} \Phi(\vx(t_I))+ \sum_{i=1}^I{\int_{t_{i-1}}^{t_{i}} { L_i(\vx,\vu)dt}},$$ where $\Phi(\cdot)$ and $L_i(\cdot,\cdot)$ are the final cost and the running cost (for subsystem $i$) which are continuously differentiable and Lipschitz up to the second order derivatives. $\mathbb{T}$ is a polytope in $\mathbb{R}^{^{I-1}}$ defined as $\mathbb{T} = \left\{ (t_1, \dots, 