Bang-singular-bang extremals: sufficient optimality conditions

Abstract

In this paper, we give second-order sufficient conditions for strong local optimality of a bang-singular-bang extremal in the minimum time problem. The conditions are given in terms of some regularity assumptions on the extremal and of the coercivity of the extended second variation associated with the minimum time problem with fixed endpoints on the singular arc. The conditions are close to the necessary conditions in the usual sense; namely, we require strict inequalities where the necessary conditions have nonstrict inequalities.

Discretization of semilinear bang-singular-bang control problems

“…Optimality conditions for (CP) in case of a bang-singular-bang control switching structure have been derived in [5], see also [7][8][9][10]26,36,39,44]. Allover the paper, we use the following assumptions: (H1) (strict structural assumption)…”

“…In order to make usage of the strong regularity approach for the variational inequalities (VI h δ ), the space R d will be equipped with a norm reflecting the use of Goh's transformation in deriving (39) and the estimates in Lemma 7: Define Ξ = (R d , | · |) with the norm of an element ξ = ( p, x, u, μ) given by…”

“…Concerning the theoretical analysis of numerical methods in bang-singular control case, important advances have been achieved in deriving optimality conditions [5,9,20,21,[37][38][39]44], but also in stability investigation of solutions [14,15,40]. For the latter, the shooting method has been utilized e.g.…”

Minimum-time strong optimality of a singular arc: Extended Dubins problem

“…Thus, it can be asked to be coercive with respect to the L 2 -norm of the new control variable. This approach, both for classical and intrinsically-defined second variations, has been widely used in the analysis of sufficient optimality conditions (see for instance [3], [6], [11], [12], and references therein).…”

“…In the standard theory, this lift is obtained using the Hamiltonian flow of the maximized Hamiltonian H max , see for example [1], [2]. When the extremal is singular the standard method does not work and H max cannot be used (see [11]), then, to construct the lifted trajectories, we consider the solutions of a system governed by a Hamiltonian greater than or equal to H max , as suggested by the approach used in [11], [12]. This Hamiltonian method is particularly effective for studying strong optimality, since it allows us to compare trajectories that belong to a C 0 -neighborhood of the reference trajectory, independently on the value of the control.…”