Automatica, Vol. 8, pp. 581-588. Pergamon Press, 1972. Primed in Great Britain. A Manifold Imbedding Algorithm for Optimization Problems" Un algorithme d'eneastrement de eoUeeteur pour probl mes d'optimisation Ein vielseitiger Einbettungsalgorithmus ftir Optimierungsproblerne A.riropwrM BKaIOqeHHa Mam@om, a ,rta npo6aeM onTa anm Y. HONTOIR~" and J. B. CRUZ, JR.~" The shooting method for numerical computation of controls satisfying the necessary conditions of the Maximum Principle can be improved by imbedding of the boundary conditions and use of a continuation method. Sammm'y--A manifold imbedding algorithm is described for the solution of two-point boundary value problems arising in optimization problems. A one-parameter imheddina of the initial and terminal manifolds is introduced so that a member of the resulting imbedded class of problems has an immediate solution. problem is'the first of a finite ~qus~ce of problems including the original ¢me as last element. By a continuation procedure, these problems are mccc~ively solved. In contrast to a previous imbedding technique for solving two-point boundary value problems, the system equations are left unchanged. Intermediate problems therefore keep their phy~cal meaning and the choice of a proper imbedding is easier. Moreover, a reduction of computation time is achieved. The manifold imbedding method may he regarded as a variation of the Nzwton-Raphson iterative procedure chamcte~zed by an extended region of convergence. The resulting reduction of guesswork and the nnifu~ approach to a wide range of optimization problems are amongst the favc~rablefeature. The minimization of the transfer time of a low-thrmt ion ~ between the oebits of Earth and Mars illustrates the application of the method. I. INTRODUCTION Tmn~ are basically three types of numerical methods for solving control optimization problems. Direct methods start with an initial guess of the control and by an iterative procedure minimiTe the cost function. Indirect methods reduce the optimization problem to a two-point boundary value problem (TPBVP) using the maximum prin- ciple. This problem is solved by a shooting pro- cedure which determines a set of missing terminal * Received 15 July 1971; revised 15 March 1972. The original version of this paper was presented at the 5th IFAC Congress which was held in Paris, France during June 1972. It was recommended for publication in revised form by Associate Editor A. Wierzbicki. t Coordinated Science Laboratory and Department of Electrical Engineering, University of Illinois, Urbana, Illinois 61801, U.S.A. conditions such that all boundary conditions are satisfied. Finally, dynamic programming con- verts the problem to solving the Hamilton-Jacobi- Bellman partial differential equation. References [1-3] constitute a brief sample of the literature on the subject. Each method possesses some unfavorable fea- tures. The prohibitively slow rate of convergence of a first order direct method as well as the di~- culty of choosing a proper initial guess which ensures the convergence of a second order method are well-known. Starting an indirect method is not easy due to the instability of the adjoint system. Dynamic programming requires a large amount of computation and an impractical memory storage. The purpose of this paper is to describe an algorithm which enhances the capability of the shooting method. An imbedding of the initial and final boundary conditions of the problem and the continuation technique provide the basis for the algorithm. This continuation method for functional equa- tions is not new [4-7]. Let T be a transformation of a Banach space into itself and T(x)=0 an equa- tion to be solved. Suppose T is imbedded in the family {T(k, x):ks[0, 1]} such that T(1, x)=T(x) and the equation T(0, x)=0 has a known unique solution x(0). Under suitable feasibility conditions, iterative methods can be used to successively find the solutions of T[k, x(k)]-----0 for the finite sequence of imbedding parameters 0<kl< ... <ks=l. Another practical method for continuing the solution is the integration of the differential equation underl.vmg x(k). 581