Singular-perturbation of the time-optimal soft-constrained cheap-control problem

The solution of a time-optimal soft-constrained control problem with linear dynamics is considered. The cost function has no penalty on the integral of the state. The solution is formulated in terms of the controllability Grammian and is obtained as the solution of a system of linear algebraic equations and a nonlinear scalar algebraic equation. As the state approaches the origin, or equivalently, as the control becomes cheap, the optimal final time becomes small. This introduces a highly degenerate hierarchy of amplitude scales. An approach based solely on expanding the controllability Grammian is developed to obtain an asymptotic solution of the problem without resorting to boundary-layer theory