Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 4 / 28I / 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.