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Inconsistent solutions to linear optimal control problem

One of the necessary conditions is $L_u = 0$. If you assume that $\mu(t) = 0$, it follows from $L_u = \mu_x(t) + \mu(t) = 0$ that $\mu_x(t) = 0$. Because $u(t)$ can take any value when $\mu_x(t) = 0$, ...
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