Static Solutions of the Hamilton-Jacobi-BellmanEquation for Circular Orbit Transfers

Abstract

A LYAPUNOV control law was proposed by Chang et al. [1] for transfer between Keplerian orbits. For circular orbits, a Lyapunov function was chosen based on the energy E and the angularmomentumL, and a control lawwas derived in order tomake the derivative of the Lyapunov function negative-definite. However, the optimality of the control law was not analyzed. In this Note, the optimality of these transfers is shown by deriving a cost function for which these transfers are optimal. In particular, the link between the Lyapunov approach and the static solution of the Hamilton–Jacobi– Bellman equation (HJBE) for the two-body orbit transfer is shown. Moreover, it is shown that for long times of flight, constantpropulsive-acceleration transfers can be designed in feedback form with almost the same fuel cost in comparison with fuel-optimal transfers. Bonnard et al. [2] considered the geometric structure of timeoptimal control for spacecraft transfer in elliptic Keplerian orbits. A time-optimal control problem was proposed for constant specific impulse engines and the controllability of the system was analyzed. The methodology to compute the optimal control was based on the traditional indirect method, which was solved numerically by means of a shooting method. Freeman and Kokotovic [3] introduced the concept of a robust control Lyapunov function (rclf) and showed that every rclf satisfies the steady-state Hamilton–Jacobi–Isaacs equation associated with a meaningful game. Indeed, having a control Lyapunov function, it can be used for determination a stabilizing feedback control that is optimal in some sense. This technique is known as inverse optimality method. As an example, Bharadwaj et al. [4] used this method to design globally stabilizing attitude control laws. Primbs et al. [5] investigated the relation between optimal control problems and control Lyapunov functions from a receding-horizon perspective. In that work, it was shown that a broad class of Lyapunov controllers admit natural extensions to recedinghorizon formulations, which are locally optimal. However, these ideas were of a general character and applied only to a simple nonlinear system. In this Note, these ideas are applied specifically for mediumto low-thrust spacecraft orbital transfer in a two-body setting. In particular, a closed-form solution of an infinite-horizon HJBE for two-body, circular-orbit transfers is derived and is compared with fuel-optimal finite time transfers. The solution of the HJBE provides the optimal control at every point in phase space in feedback form. However, the nonlinearity of this partial differential equation (PDE)makes it extremely difficult to solve analytically. For linear systems subject to quadratic cost functions, it is possible to obtain an analytic form for the optimal cost [6], but for general nonlinear systems that are of practical interest, this is not the case in general. The numerical approach to solve this PDE is still challenging, as the solutions are in general not smooth and differentiable at every point. However, the great drawback of dynamic programming is, as Bellman himself calls it, “the curse of dimensionality.” Even recording the solution to a moderately sized problem involves an enormous amount of storage. If only an optimal path from a known initial point is desired, it is wasteful and tedious to find a whole field of extremals [7]. After a review of the main result obtained by Chang et al. [1], the static form of the HJBE is described and the optimality of the Lyapunov control law for a regulized fuel-optimal cost function is shown. The application of this control law is illustrated with a threedimensional transfer from a low Earth orbit (LEO) to the geostationary Earth orbit (GEO). The results are shown to be applicable to both variable specific impulse (VSI) and engines with constantpropulsive-acceleration capability. The examples illustrate the relation between the (infinite-horizon) transfers and actual finite time, fuel-optimal solutions. Finally, the application of thismethod to provide good initial guesses of the costates (relevant to the indirect formulation and solution of the optimal control problem) is explored.