Uncertain Systems and Fuzzy Differential Equations

V. Lakshmikantham

Florida Institute of Technology

Department of Mathematical Sciences

Melbourne, FL 32901 USA

The study of fuzzy differential equations is initiated and some typical results are given in order to understand the intricacies involved in incorporating fuzziness in the terms of differential equations. Here new notion of stability is introduced.

The mathematical modeling of various physical phenomena involves two inconveniences. The first is caused by excessive complexity of the situation being modeled and consequent uncertainty of the systems. This leads to two further consequences:

(1) We are not able to formulate the model, and

(2) the model constructed is too complicated with uncertainties to be used in practice.

Despite such imperfect knowledge, attempts have been made in some selected mathematical model, to devise controllers that will steer the system in a certain required fashion. The second inconvenience consists of indeterminancy caused by our subjective inability to differentiate events exactly and its main property is vaguness. Thus there is a need for a mathematical apparatus to describe the vague notions to overcome the obstacles in modeling and such an apparatus is the concept of fuzzy set theory.

Recently, the theory of fuzzy differential equations has been initiated and the basic results have been systematically investigated, including stability theory. In this paper, we shall introduce some typical results in the theory of fuzzy differential equations which help us understand the intracacies involved in incorporating fuzziness in the theory of differential equations.