  We address the problem of computing a control for a time-dependent nonlinear
system to reach a target set in a minimal time. To solve this minimal time
control problem, we introduce a hierarchy of linear semi-infinite programs, the
values of which converge to the value of the control problem. These
semi-infinite programs are increasing restrictions of the dual of the nonlinear
control problem, which is a maximization problem over the subsolutions of the
Hamilton-Jacobi-Bellman (HJB) equation. Our approach is compatible with generic
dynamical systems and state constraints. Specifically, we use an oracle that,
for a given differentiable function, returns a point at which the function
violates the HJB inequality. We solve the semi-infinite programs using a
classical convex optimization algorithm with a convergence rate of O(1/k),
where k is the number of calls to the oracle. This algorithm yields
subsolutions of the HJB equation that approximate the value function and
provide a lower bound on the optimal time. We study the closed-loop control
built on the obtained approximate value functions, and we give theoretical
guarantees on its performance depending on the approximation error for the
value function. We show promising numerical results for three non-polynomial
systems with up to 6 state variables and 5 control variables.
