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 RnRn with fixed initial and final time conditions x(0)=xˆ0, x(tf)=xˆ1, and where the scalar control u1u1 satisfies the constraint |u1(⋅)|⩽1|u1(⋅)|⩽1. For these systems a concept of conjugate time tctc has been defined in e.g. Agrachev et al. (2002) [23], Maurer and Osmolovskii (2004) [21], and Noble and Schättler (2002) [28] in the bang–bang case. Besides, theoretical and practical issues for conjugate time theory are well known in the smooth case (see e.g. Agrachev and Sachkov (2004) [43] and Milyutin and Osmolovskii (1998) [15]), and efficient implementation tools are available (see Bonnard et al. (2007) [35]). 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 Silva and Trélat (in press) [36] and investigate the convergence properties of conjugate times. More precisely, for ε>0ε>0 small and arbitrary vector fields Y1,…,YmY1,…,Ym, we consider the minimal time problem for the control system ẋε=X(xε)+u1εY1(xε)+ε∑i=2muiεYi(xε), under the constraint ∑i=1m(uiε)2⩽1, with the fixed boundary conditions xε(0)=xˆ0, xε(tf)=xˆ1 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 tcε falls into the standard theory; our main result asserts the convergence, as εε tends to 0, of tcε towards the conjugate time tctc 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.