Use of symbolic computation to gain insight into a difficult eigenvalue problem

摘要

The solution of a nonlinear dynamic optimal control problem often requires an algorithm which updates a trial solution from iteration to iteration. In such algorithms, it is important to ensure stability so that the iterations are stable and converge in a satisfactory manner. Particular emphasis is given to an algorithm, known as dynamic integrated system optimisation and parameter estimation, DISOPE, (P.D. Roberts, 1993), for the solution of continuous nonlinear optimal control problems subject to model-reality differences. The DISOPE technique is described as applied to the nonlinear optimal control situation. Then, for the purpose of local convergence and stability analysis, the linear situation is considered, where it is shown that the method can be represented as a unit memory linear repetitive process. A stability theorem is then stated, providing a sufficient and necessary condition for asymptotic stability if the solutions of a given equation all lie within a unit circle in the complex plane. The computational solution of this equation is then considered. Finally, taking a simple scalar case, symbolic computation is employed to achieve insight into the nature of the solutions of the stability equation.