Solving optimal control problems for delayed control-affine systems with quadratic cost by numerical continuation

Abstract

In this paper we introduce a new method to solve fixed-delay optimal control problems which exploits numerical homotopy procedures. It is known that solving this kind of problems via indirect methods is complex and computationally demanding because their implementation is faced with two difficulties: the extremal equations are of mixed type, and besides, the shooting method has to be carefully initialized. Here, starting from the solution of the non-delayed version of the optimal control problem, the delay is introduced by numerical homotopy methods. Convergence results, which ensure the effectiveness of the whole procedure, are provided. The numerical efficiency is illustrated on an example.

Continuity of Pontryagin Extremals with Respect to Delays in Nonlinear Optimal Control

“…where C2 0 is another constant. Coupling the two last results with the Grönwall's inequality, (23) follows. The conclusion comes from (23) and the fact that t is a Lebesgue point of u τ (•) and of u τ (• − τ 2 ).…”

“…Assumptions (B 1 ) and (C 3 ) play a complementary role in proving continuity for the adjoint vectors (see Section 3.3.4). They are related to the strict Legendre-Clebsch condition and the uniqueness of solutions for (5), and, even if these assumptions seem to be restrictive, many problems in applications satisfy them and examples can be found in [9,10,11,23,24,25,26]. In particular, Assumption (C 3 ) is instrumentally used also to derive strong continuity of controls from weak continuity when there are delays on the control, because of the following general fact.…”

“…Implementing homotopy methods combined with the Maximum Principle with delays, as shortly described above, however requires a number of considerations which go beyond the scope of the present paper. In a forthcoming work, we will show that this approach indeed happens to be competitive with respect to other approaches (in particular, direct methods that have been classically applied in this context, see, e.g., [9,10]) and we will illustrate it on a nontrivial and nonacademic optimal control problem with launch vehicles, a case of application where delays are the main obstacle to efficiently embark numerical approaches (the authors already showed the numerical efficiency of this approach on a simpler problem, see [23]).…”

“…Let t j < t t τ f be a Lebesgue point of u τ (•), u τ (• − τ 2 ). First, let us show that (23) xπ τ (t) − xτ (t) = η1 wτ t 1 ,u 1 (t) + o(η1) = η1 ṽτ t 1 ,ω − u 1 (t 1 ) (t) + ṽτ t 1 +τ 2 ,ω + u 1 (t 1 ) (t) + o(η1).…”

Optimal Control of Endoatmospheric Launch Vehicle Systems: Geometric and Computational Issues

“…Motivated by the convergence result established in [54], [55], we could add one further homotopic step on the delay. For computational times, even if many simulations on different missions for (OIP) show that Algorithm 1 can run between 0.5 Hz and 1 Hz, we cannot ensure a real-time processing yet.…”