Optimal Control for Nonlinear Systems Driven by a Known Exogenous Signal

Abstract

We consider optimal control problems for continuous-time systems with time-dependent dynamics, in which the time-dependence arises from the presence of a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">known</i> exogenous signal. The problem has been elegantly solved in the case of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">linear</i> 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 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">unperturbed</i> optimal control law, and a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">purely feedforward</i> term in charge of compensating the effect of the exogenous signal. The objective of this article is to extend the above result to <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">nonlinear</i> input-affine systems. It is shown that, while some of the relevant features of the linear case indeed rely heavily on <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">linearity</i> and are not preserved in the nonlinear setting, several structural claims can be proved also in the nonlinear case.