Fuzzy continuous dynamical system: A multivariate optimization technique

Abstract

This paper presents a multivariate optimization technique for the numerical simulation of continuous dynamical systems whose parameters, functional forms and/or initial conditions are modelled by fuzzy distributions. Fuzzy differential equation (FDE) is interpreted by using the strongly generalized differentiability concept and is shown that by this concept any FDE can be transformed to a system of ordinary differential equations (ODEs). Next by solving the associate ODEs one can find solutions for FDE. This approach has an inherited drawback of increasing the volume of uncertainty at each instance of time generally with nonlinear functional forms. We attempted to solve the corresponding ODEs with fuzzy (interval valued) initial condition and tried to evaluate the region of uncertainty at every time instance by solving the ODEs with initial conditions on the surface of region of uncertainty. Solving differential systems with fuzzy continuous distribution is an infinite problem as the fuzzy distributions are generally subsets of R <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</sub> and hence of infinite cardinality. We took the initial region of uncertainty and simulate the uncertainty region in the next state. We used connection matrix and formed an augmented system with taking account of the partial differentials of the state variables at every instance of time, where Partial derivatives tell about locally linearized behaviour of the system. Property of sufficiency of vertices (psv) is used as an optimization criterion to get the extreme values of the uncertainty region and a rough sketch of the volume of uncertainty could be estimated. Here we present a methodology to numerically simulate interval calculus and implements a new approach to the numerical integration of fuzzy dynamical systems, where the propagation of imprecision as a fuzzy distribution in the phase space is solved by a constrained multivariate optimization technique. Numerical simulations of some fuzzy dynamical systems are also reported. Finally ecological degradation in wetlands in India is modelled by fuzzy initial value problem and some sustainable solution is proposed.