## The as a restricted 3 bodyproblem involves the

The optimization of an interplanetary trajectory considering it as a restricted 3 bodyproblem involves the motion of a massless spacecraft under the gravitational eld gen-erated by two primaries of masses m1 and m2, respectively, which move in a circularmotion due to their mutual interaction, using continuous low thrust propulsion2, 3,where m1>m2>>m. Non-linear and time varying Equations of motion are then gov-erned on the basis of the restricted 3 body problem.The optimization of the interplanetary trajectory using continuous low thrust isdi erent from that of a general impulsive thrust trajectory optimization. It involvesthe application of Euler-Lagrange optimality conditions. Preliminarily a suitableperformance index J along with constraints are formulated. This is followed by theformulation of an Hamiltonian4, 12.Optimization theory is then applied to the formulated Hamiltonian equation whichyields a Two-Point Boundary Value Problem (TPBVP). Such TPBVPs are generallyvery dicult to solve. These TPBVPs are then reduced to an Initial Value Problem(IVP) to arrive at the solution using numerical integration techniques.There are two basic approaches:1. Direct Approach2. Indirect Approach3.1 Direct ApproachThe direct approach is to discretize the original problem and transform it into aparameter optimization problem. Explicit integration of the system di erential equa-tions is avoided. Instead, algebraic expressions approximate the di erential equationsDept. of ICE, MIT 2Objectivelocally, and the resulting system of nonlinear simultaneous equations is then solvedby mathematical programming.3.2 Indirect ApproachIndirect solutions for the optimal control explicitly use the necessary conditions ofoptimality i,e the Euler-Lagrange equations and the system co-state (the Lagrangemultiplier) variables. Euler-Lagrange optimality conditions are then applied to Hamil-tonian H to obtain State and Co-state equations from which Two Point BoundaryValue Problem is obtained (TBVP).Among the rst and best known of the indirect solution methods are initial value,or shooting methods5, 6, 13, in which one guesses either the initial or terminalboundary conditions, then integrates numerically forward or backward and nallyreadjusts this initial guess iteratively so that the boundary conditions at the other endare also satis ed. These methods, however, are extremely sensitive to the accuracy ofthe initial guess, thus making the numerical computation for an improved guess quitedicult. Determining the solution for a given problem using direct method is muchsimpler when compared to that of an indirect method and it does not guarantee ahigh level of accuracy. However, indirect method is complex and guarantees a higherlevel of accuracy.

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