Method of Conjugate Gradients for Optimal Control Problems with State Variable Constraint

Abstract

This chapter discusses the method of conjugate gradients for optimal control problems with state variable constraint. The method of conjugate gradients is a useful computational tool in solving optimal control problems with state variable constraint. The method is basically simple and relatively easy to program. Although the search directions are only locally conjugate with respect to the Hessian of the performance functional, they still provide satisfactory convergence. The conjugate gradient method has a higher rate of convergence in comparison with the method of steepest descent; however, the difference in the rate of convergence is less pronounced for this constrained problem as compared with the cases of unconstrained problems. The convergence is along the expanding sequence of sets, the intersections of the linear spaces spanned by the search directions, and the set of admissible controls instead of expanding sequence of subspaces. In converting the constrained control problem to an equivalent unconstrained one by introducing a penalty function, the computational process involves more time in contrast to the approach that considers the constraints directly; however, it requires less programming work.