Asymptotic approach on conjugate points for minimal time bang-bang controls

We focus on the minimal time control problem for single-input control-affine systems ẋ = X(x) + u1Y1(x) in IR with fixed initial and final time conditions x(0) = x̂0, x(t f ) = x̂1, and where the scalar control u1 satisfies the constraint |u1(·)| 6 1. For these systems a concept of conjugate time tc has been defined in e.g. [3, 30, 33] in the bang-bang case. Besides, theoretical and practical issues for conjugate time theory are well known in the smooth case (see e.g. [5, 32]), and efficient implementation tools are available (see [11]). The first conjugate time along an extremal is the time at which the extremal loses its local optimality. In this work, we use the asymptotic approach developed in [44] and investigate the convergence properties of conjugate times. More precisely, for ε > 0 small and arbitrary vector fields Y1, ..., Ym, we consider the minimal time problem for the control system ẋ = X(xε)+uε1Y1(x )+ε ∑m i=2 u ε i Yi(x ), under the constraint ∑m i=1(u ε i ) 2 6 1, with the fixed boundary conditions x(0) = x̂0, x(t f ) = x̂1 of the initial problem. Under appropriate assumptions, the optimal controls of the latter regularized optimal control problem are smooth, and the computation of associated conjugate times t c falls into the standard theory; our main result asserts the convergence, as ε tends to 0, of t c towards the conjugate time tc of the initial bang-bang optimal control problem, as well as the convergence of the associated extremals. As a byproduct, we obtain an efficient algorithmic way to compute conjugate times in the bang-bang case.