A Study of the Weak Non-linear Optimal Control Problem Using the Fixed Point Theorem

Yasunari SHIDAMA
Toyomi OHTA

IEICE TRANSACTIONS (1976-1990)   Vol.E72    No.12    pp.1317-1325
Publication Date: 1989/12/25
Online ISSN: 
Print ISSN: 0000-0000
Type of Manuscript: Special Section PAPER (Special Issue on the 2nd Karuizawa Workshop on Circuits and Systems)
Category: Nonlinear Problems

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It is generally thought to be difficult to construct the optimum control law for the non-linear systems. The number of research papers in this field is rather small, compared with those of the neighboring fields. Among them, Garrad's study of ε-perturbation to approximate the law, Nishikawa's study about quasi-optimal control and Ohkubo's proposal to approximate system's non-linearity by the tensor products should be put more importance. Yet it is still unknown whether the optimum feedback control law for any non-linear system exists or not. In this paper, the existence conditions of this control law for weak non-linear system, which is composed of linear quadratic part and weak non-linear one, are studied using the fixed point theorem. The non-linear part is contained in εg, where ε is a small positive real number and g implies non-linear function. The existence conditions of control law for this composite system is examined, assuming the initial condition to be restrained in some bounded domain. Thus, the existence conditions for the optimum control law are resulted in terms of those for certain kind of implicit function: this implicit function relates Lagrange multipliers to the state variables of the system, using the variation principle.