Second order optimality conditions in the smooth case and applications in optimal control

Abstract

Abstract. The aim of this article is to present algorithms to compute the first conjugate time along a smooth extremal curve, where the trajectory ceases to be optimal. It is based on recent theoretical developments of geometric optimal control, and the article contains a review of second order optimality conditions. The computations are related to a test of positivity of the intrinsic second order derivative or a test of singularity of the extremal flow. We derive an algorithm called COTCOT (Conditions of Order Two and COnjugate Times), available on the web, and apply it to the minimal time problem of orbit transfer, and to the attitude control problem of a rigid spacecraft. This algorithm involves both normal and abnormal cases.Mathematics Subject Classification. 49K15, 49-04, 70Q05.

Optimal Control and Applications to Aerospace: Some Results and Challenges

“…In that case, the notion of conjugate point must be replaced with the notion of focal point. The theory and the resulting algorithms remain however similar (see [23,2]). …”

“…These cases may appear to be quite trivial, but actually in practice this issue is far from being obvious because a priori, given some extremities, we are not able to say if the resulting problem can be solved with a normal extremal (that is, with a p 0 = −1). It could happen that it is not: this is the case for instance for certain initial and final conditions in the well-known minimal-time attitude control problem (see [23], where such abnormals are referred to as exceptional singular trajectories).…”

“…Under the strict Legendre assumption, there holds t c > 0, and this first conjugate time characterizes the (local) optimality status of the trajectory (see [17,23,19,50,51,52]). is not an immersion at p 0 (that is, its differential is not injective).…”

“…Its proof can be found in [17,23,2]. Essentially it states that computing a first conjugate time reduces to compute the vanishing of some determinant along the extremal.…”

“…Note however that the domain of definition of the exponential mapping requires a particular attention in order to define properly these Jacobi fields according to the context: normal or abnormal extremal, final time fixed or not. A more complete exposition can be found in the survey article [23], which provides also some algorithms to compute first conjugate times in various contexts (however, always in the case where the control can be expressed as a smooth function of x and p) and some practical hints for algorithmic purposes (see also section 2.4.2).…”