Optimal Control and Applications to Aerospace: Some Results and Challenges

Abstract

This article surveys the classical techniques of nonlinear optimal control such as the Pontryagin Maximum Principle and the conjugate point theory, and how they can be implemented numerically, with a special focus on applications to aerospace problems. In practice the knowledge resulting from the maximum principle is often insufficient for solving the problem in particular because of the well-known problem of initializing adequately the shooting method. In this survey article it is explained how the classical tools of optimal control can be combined with other mathematical techniques to improve significantly their performances and widen their domain of application. The focus is put onto three important issues. The first is geometric optimal control, which is a theory that has emerged in the 80's and is combining optimal control with various concepts of differential geometry, the ultimate objective being to derive optimal synthesis results for general classes of control systems. Its applicability and relevance is demonstrated on the problem of atmospheric re-entry of a space shuttle. The second is the powerful continuation or homotopy method, consisting of deforming continuously a problem towards a simpler one, and then of solving a series of parametrized problems to end up with the solution of the initial problem. After having recalled its mathematical foundations, it is shown how to combine successfully this method with the shooting method on several aerospace problems such as the orbit transfer problem. The third one consists of concepts of dynamical system theory, providing evidence of nice properties of the celestial dynamics that are of great interest for future mission design such as low cost interplanetary space missions. The article ends with open problems and perspectives.

Optimal control of the Lotka–Volterra system: turnpike property and numerical simulations

“…A more detailed explanation about the initialization problems of that method can be found in Section 2.4 of Trélat [34], or Chapter 9 of Trélat [33].…”

Optimal location of controllers for the one-dimensional wave equation

“…The relaxed first problem of minimizing (17) over U L is then equivalent to the optimal control problem of determining a control a ∈ U L steering the infinite dimensional control system (26) from the initial conditions (27) to the final condition…”

“…We used the code COTCOT (see [2]). The convergence is then obtained instantaneously and can permit, when combined with a continuation method, to reach very large values of N (see [27] for a survey on these numerical approaches).…”

Galerkin Optimal Control

“…These optimal control methods take two main forms, indirect and direct. Recent surveys of these techniques are provided by Betts [29,30], Trélat [31] and Ross [32] and a historical perspective by Stryk et al [33]. Indirect methods (such as the shooting and multiple shooting methods) solve Pontryagin's necessary conditions for optimality.…”

“…From Equation (2.29), the modes, {a j } N j=0 , and the nodes, {x N j } N j=0 , are related by the generalized Vandermonde matrix (2.30) by the relationships [47] 31) wherex N j = x N (t j ) for j = 0, 1, . .…”