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

Abstract

Abstract We focus on the minimal time control problem for single-input control-affine systems x = X ( x ) + u 1 Y 1 ( x ) in R n with fixed initial and final time conditions x ( 0 ) = x ˆ 0 , x ( t f ) = x ˆ 1 , and where the scalar control u 1 satisfies the constraint | u 1 ( ⋅ ) | ⩽ 1 . For these systems a concept of conjugate time t c has been defined in e.g. Agrachev et al. (2002) [23] , Maurer and Osmolovskii (2004) [21] , and Noble and Schattler (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 Trelat (in press) [36] and investigate the convergence properties of conjugate times. More precisely, for e > 0 small and arbitrary vector fields Y 1 , … , Y m , we consider the minimal time problem for the control system x e = X ( x e ) + u 1 e Y 1 ( x e ) + e ∑ i = 2 m u i e Y i ( x e ) , under the constraint ∑ i = 1 m ( u i e ) 2 ⩽ 1 , with the fixed boundary conditions x e ( 0 ) = x ˆ 0 , x e ( t f ) = x ˆ 1 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 e falls into the standard theory; our main result asserts the convergence, as e tends to 0, of t c e towards the conjugate time t c 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.