Optimal control for nonlinear systems driven by a known exogenous signal

We consider optimal control problems for continuous-time systems with time-dependent dynamics, in which the time-dependence arises from the presence of a known exogenous signal. The problem has been elegantly solved in the case of linear input-affine systems, for which it has been shown that the solution has a remarkable structure: it is given by the sum of two contributions; a state feedback, which coincides with the unperturbed optimal control law, and a purely feedforward term in charge of compensating the effect of the exogenous signal. The objective of this note is to extend the above result to nonlinear input-affine systems. It is shown that, while some of the relevant features of the linear case indeed rely heavily on linearity and are not preserved in the nonlinear setting, several structural claims can be proved also in the nonlinear case.