\documentclass{book}
\usepackage{amsthm,amssymb,amsmath}
\begin{document}

\begin{align}
-\dot{g}^{(k)}(t) =
\begin{cases}
 \begin{eqnarray}
 \label{eq:j}
(A-SP)^Tg^{(k)}(t)+PA_1x^{(k-1)}(t-\tau)+Pf(x^{(k-1)})(t) \nonumber\\
&&+f^T_x(x^{(k-1)}(t))\lambda^{(k-1)}(t)+A_1\lambda^{(k-1)}(t+\tau)
 \end{eqnarray}	   &t_0\leq t\leq t_f-\tau\\
 \begin{eqnarray}
 \label{eq:j}
 (A-SP)^Tg^{(k)}(t)+PA_1x^{(k-1)}(t-\tau)+Pf(x^{(k-1)})(t) \nonumber\\
&&+f^T_x(x^{(k-1)}(t))\lambda^{(k-1)}(t)
 \end{eqnarray}    & t_f-\tau\leq t\leq t_f\\
\end{cases}
\end{align}%

\end{document}