Shooting Methods for Two-Point Boundary Value Problems of Discrete Control Systems

Abstract

Two-point boundary value problems (TPBVP) are an important class of problems which appear frequently in optimal control. These may be well conditioned or ill conditioned. A wellconditioned TPBVP will have a system matrix with linearly independent columns due to closeness of its eigenvalues. On the other hand an ill conditioned TPBVP will have a system matrix with almost linearly dependent columns due to wide variation of its eigenvalues. In other words, a wellconditioned system is a onetime scale system whereas an ill conditioned system is a multi-time scale system. Ill conditioned systems are computationally stiff systems with widely separated eigenvalues. The stiffness increases with increase in time scales. The solution of TPBVP of discrete control systems is obtained by shooting method, that is, a number of initial value problems (IVP) will be shot to get the solution of TPBVP. The solution of a wellconditioned TPBVP is easier compared to an ill-conditioned TPBVP. An ill-conditioned TPBVP requires orthonormalization process to make the columns of the system matrix linearly independent. More the stiffness more the number of orthonormalization processes. Here the method of complimentary functions is used for well-conditioned systems and Conte's method for ill-conditioned systems. First we develop shooting methods for well-conditioned and illconditioned TPBVP of discrete control systems. Later the methods are supported with two illustrative examples one for each case.