Consider the optimal control problem with an optimal trajectory $x^*(t)$ and an initial point $(x_1^0,x_2^0)$ \begin{equation} \begin{split} \dot{x}=A x + Bu \\\ J=\int^\infty_0(x_2^2+\epsilon u^2)dt \end{split} \end{equation} whera $(A,B)$ is minimal. How do I analyze $\lim_{\epsilon\rightarrow 0}x^*(t)$?

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You can find an analysis of your problem in the following articles

your problem deals with bang-bang or singular controls. It is also possible that the optimal control chatters.