Fuzzy Laplace Transform Based On Henstock Integral And Its Applications

Laplace transform is widely used in the research of transformation and calculation involving real variable functions and complex variable functions,linear differential equations and algebraic equations,and the analysis and synthesis of control sys-tems(such as signal flow,dynamic structure,Nyquist stability criterion,root locus method,control system correction method).However,in the mathematical model-ing of the control system,the uncertainties of the parameters are inevitably involved,and this uncertainty can be often expressed as a fuzzy number.Therefore,there are many researches on the fuzzy Laplace transform.Based on the theory of fuzzy Henstock integral,the fuzzy Laplace transform and fuzzy convolution formula are proposed and discussed,and the characterization theorem is given.The results can be directly applied to the discussion of solving problems in discontinuous fuzzy sys-tems.Firstly,considering the existing results on fuzzy Laplace transform and their applications were based on Zaheh's decomposition theorem and even formally char-acterized by the integrals of real-valued functions directly.That is to say,it has not been solved the problem of the existence of fuzzy Laplace transform in essence.In this paper,the fuzzy Laplace transform was incorporated into the framework of Henstock integral and proposed by using the fuzzy Henstock integrals on infinite intervals.In addition,as a theoretical basis,the existence and operation principles of the fuzzy Laplace transform are investigated.Further,the convolution of fuzzy number-valued functions and real-valued func-tions is defined and its basic properties are studied.At the same time,the convolu-tion theorem of Laplace transform based on the fuzzy Henstock integral is obtained.Finally,as an application,the initial value problem of a class of discontinuous fuzzy differential equations is discussed by using the fuzzy Laplace transform;the solution of two kinds of fuzzy Volterra integral equations is discussed by using the convolution theorem;several examples are given.