%% formulation
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\begin{align*}
\min_{p} \quad & \int_0^{t_f} \bm{x}^T \bm{I} \bm{x}\ \mathrm{d}t \\
\text{subject to:} \quad & \dot{\bm{x}} = \begin{cases} \bm{A}_1 \bm{x} + \bm{G} p & 0 \leq t \leq t_1 \\ \bm{A}_2 \bm{x} + \bm{G} p & t_1 \leq t \leq t_2 \\ \bm{A}_3 \bm{x} & t_2 \leq t \leq t_f \end{cases} \\
& \bm{x}(0) = \bm{x}_0 \\
& \bm{x}^{L}(t_1) - \bm{x}^{R}(t_1) = \bm{E}_1 \\
& \bm{x}^{L}(t_2) - \bm{x}^{R}(t_2) = \bm{E}_2 \\
\text{where:} \quad & 0 < t_1 < t_2 < t_f
\end{align*}
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\end{document}

%% solution