Optimal movement control models of Langevin and Hamiltonian types

We study a class of optimal stochastic control problems arising from the control of movements. Exact solutions are first presented for linear cases for both the during- and post-movement control problem, depending on a parameter α>0α>0. It is found that for the Langevin type equation and for the post-movement control case, a non-degenerate solution exists only when α>1/2α>1/2. For the Langevin type equation and for the during-movement control, a non-degenerate solution is found when α>1α>1. For the post-movement control and the Hamiltonian type equation, an optimal control signal is obtained and is non-degenerate when α>1/2α>1/2. Again for the during-movement control, we find an optimal non-degenerate control signal when α>1α>1. All results are then generalized to nonlinear control cases (the first order perturbation of linear cases). Numerical examples are included to illustrate the applications of our results.