= Solution
<Pontryagin maximum principle> leaves the initial <costate> unknown. Terminal conditions determine these unknown constants and, when terminal time is free, the time. In a maximization convention with terminal reward $\Phi$ and running cost $L$, use $H=p\cdot f-L$. A free terminal coordinate gives $p_i(T)=\partial_i\Phi$; a fixed terminal coordinate instead allows a constraint multiplier. More generally $p(T)-\nabla\Phi$ is normal to the permitted terminal manifold. The <free-terminal-time transversality condition> is $H(T)+\Phi_T=0$. These conditions convert the adjoint and state equations into a boundary-value problem. Reversing the <costate> sign gives the equivalent minimization convention.
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