Theory Of Fuzzy Differential Equations Research Articles

Calculus for fuzzy functions with strongly linearly independent fuzzy coefficients

This paper establishes a theory of calculus for fuzzy-number-valued functions whose range is embedded in a specific Banach space of fuzzy numbers. These spaces are generated by strongly linearly independent (SLI) fuzzy numbers whose operations are induced by an isomorphism with Rn. We use the notion of Fréchet differentiability and Riemann integrability for these functions and present a fundamental theorem of calculus. Lastly, we develop a theory of fuzzy differential equations and provide an existence and uniqueness theorem.

Stabilizability And Solvability Of Fuzzy Differential Equations Using Backstepping Method

Solution and stability of systems of fuzzy ordinary differential equations are of great important and may be sometimes difficult to carry out in the theory of fuzzy differential equations. Therefore, need for efficient approaches and techniques seem to be necessary for solving and stabilizing such type of problems. In this paper the backstepping method will be modified and used for analyzing systems of fuzzy ordinary differential equations. Such type of problems that might be encountered in studying the mathematical problems resulting from modeling complex physical or engineering real life phenomena’s. The topic of this paper is considered as a part of the control theory of fuzzy differential equations, in which the adaptive backstepping method based on constructing Lyapunove function in quadratic form will be combined with the so called the α-level sets in fuzzy set theory. The reliability of the proposed approach is presented by solving three examples, linear and nonlinear, in which the orginal system may be unstable.

A Peano theorem for fuzzy differential equations with evolving membership grade

The Peano theorem on the existence without possible uniqueness of solutions has been a perplexing problem in the theory of fuzzy differential equations. The difficulty appears to be due to the standard use of the supremum metric ▪∞ defined by the supremum over the Hausdorff metric between the level sets of the fuzzy sets. Another may have been the classical formulation of fuzzy differential equations in terms of the Hukuhara derivative of the level sets. Here a Peano theorem is established for fuzzy differential equations formulated in a recent paper by the authors by combining Hüllermeier's suggestion of defining fuzzy differential equations at each level set via differential inclusions with Aubin's morphological equations, which allow non-local set evolution. A major difference from previous publications is the use of the endograph metric ▪end, essentially the Hausdorff metric between the endographs in Rn×[0,1] of fuzzy sets, instead of the supremum metric ▪∞. Another is that the membership grades of the fuzzy sets are also allowed to evolve under the fuzzy differential equations. The result applies for a very general class of fuzzy sets without additional assumptions of fuzzy convexity, compact supports or even normality.

Properties of Reflected Liu Process at the Origin

Liu process is a special fuzzy process which plays an important role in the theory of fuzzy differential equations. This paper proposes the concept of reflected Liu process at the origin. Then some properties of this type of Liu process are deduced.

A class of linear differential dynamical systems with fuzzy matrices

This paper investigates the first order linear fuzzy differential dynamical systems with fuzzy matrices. We use a complex number representation of the α-level sets of the fuzzy system, and obtain the solution by employing such representation. It is applicable to practical computations and has also some implications for the theory of fuzzy differential equations. We then present some properties of the 2-dimensional dynamical systems and their phase portraits. Some examples are considered to show the richness of the theory and we can clearly see that new behaviors appear. We finally present some conclusions and new directions for further research in the area of fuzzy dynamical systems.

Set differential equations versus fuzzy differential equations

The study of fuzzy differential equations (FDEs) forms a suitable setting for a mathematical modeling of real world problems in which uncertainties or vagueness pervades. In recent years, the theory of FDEs has been investigated extensively in the original formulation as well as in an alternative framework, which leads to ordinary multivalued differential inclusions. It has recently been realized that initiating the study of set differential equations in a metric space has several advantages, in addition to providing a natural setting for considering FDEs. In this paper, we present some interesting results in this direction with the necessary background material.

Revisiting fuzzy differential equations

We revisit the original formulation of fuzzy differential equations (FDEs) and provide examples to show that the theory of FDEs is richer than the corresponding theory of ordinary differential equations (ODEs) and therefore need not be rejected. The alternative formulation in terms of differential inclusions does not always have advantage over ODEs, as claimed. We also discuss a stability result via Lyapunov functions in this setup.

Towards the theory of fuzzy differential equations

Comparative analysis of two ideologically distinct approaches in the theory of fuzzy differential equations is given. Domains of applications, advantages and shortcomings of those approaches are described. New definitions of solutions to fuzzy differential equations are proposed.

Existence of fixed points of fuzzy mappings via theory of fuzzy differential equations

Recently, the study of fuzzy differential equations has gained importance due to its application. In this paper we prove the existence of a unique solution for fuzzy initial value problems. Further, employing stability theory and contraction mapping principle, we obtain fixed points of fuzzy nonlinear mappings.

Stability theory of fuzzy differential equations via differential inequalities

Lyapunov theory of stability is developed for fuzzy differential equations employing suitable theory of differential inequalities. Method of perturbing Lyapunov function is utilized which provides weaker assumptions for discussing stability theory. Mathematics subject classification (1991): 34D20, 34A40, 34L99.