The Use of Legendre Polynomial Expansions in Solving Optimal Control Problems in the Spacel2

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Abstract
A method for solving a class of general deterministic optimal control problems is presented here. The method consists of relating the functions involved in the problem to sequences and then converting the problem to one which deals with these sequences alone. The function-sequence correspondence is defined by representing each function by its Legendre polynomial expansion and then relating the function to the sequence of coefficients in this expansion. After this is done, the problem is converted to one in l 2; by determining the equivalents in l 2; of differentiation, inner multiplication, and multiplication. The resulting problem in l 2; is a non-linear programming problem which consists of an infinite array of equations, inequalities, and expressions, each of which involves infinite polynomial expressions. To solve a problem of this type it must be approximated by a finite non-linear programming problem. After this is done various methods can bo used for solving the final problem.

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