Fuzzy Differential Inclusion Research Articles

On fuzzy fractional differential inclusion driven by variational–hemivariational inequality in Banach spaces

The aim of this paper is to examine an evolution problem (FFDIVHVI) involving a fuzzy fractional differential inclusion and a variational–hemivariational inequality (VHVI) in Banach spaces. First, we show a uniqueness and existence theorem for VHVI under the theory of monotone operators and the surjectivity theorem. Then, by utilizing fixed point theorem for multivalued contraction mapping and fuzzy set theory, we establish the existence result for FFDIVHVI. In addition, it is proven that the collection of all mild trajectories of FFDIVHVI exhibits compactness. Finally, we illustrate the applicability of the abstract theory by a nonlinear quasistatic thermoelastic frictional contact problem for which we provide existence results.

Robust Adaptive Control for a Class of T-S Fuzzy Nonlinear Systems with Discontinuous Multiple Uncertainties and Abruptly Changing Actuator Faults

In the complex environment, the suddenly changing structural parameters and abrupt actuator failures are often encountered, and the negligence or unproper handling method may induce undesired or unacceptable results. In this paper, taking the suddenly changing structural parameters and abrupt actuator failures into consideration, we focus on the robust adaptive control design for a class of heterogeneous Takagi–Sugeno (T-S) fuzzy nonlinear systems subjected to discontinuous multiple uncertainties. The key point is that the switch modes not only vary with the system time but also vary with the system states, and the intrinsic heterogeneous characteristics make it difficult to design stable controllers. Firstly, the concepts of differential inclusion are introduced to describe the heterogeneous fuzzy systems. Meanwhile, a fundamental lemma is provided to demonstrate the criteria of the boundness for a Filippov solution. Then, by using the set-valued Lie derivative of the Lyapunov function and introducing a vector of specific continuous functions, the closed-loop T-S fuzzy differential inclusion systems are proved to be ultimately bounded. The sufficient conditions for system stability are derived in term of linear matrix inequalities (LMIs), which can be solved directly. Finally, a numerical example is provided to illustrate the effectiveness of the proposed control algorithm.

A new class of fuzzy fractional differential inclusions driven by variational inequalities

The goal of this paper is to consider a new fuzzy differential system consisting of a fuzzy fractional differential inclusion combined with a variational inequality, which is called fuzzy fractional differential variational inequality (FFDVI). Such a model captures the desired features of both fuzzy fractional differential inclusions and fractional differential variational inequalities within the same framework. By employing the set-valued version of the Krasnoselskii fixed point theorem, an existence of solutions is obtained for the FFDVI under some mild conditions. Moreover, an approximating algorithm is provided to find a solution of the FFDVI. Finally, two numerical examples are given to illustrate our main results.

Parameter Estimations of Fuzzy Forced Duffing Equation: Numerical Performances by the Extended Runge-Kutta Method

In this work, the forced Duffing equation with secondary resonance will be considered our subject by assuming that the initial values has uncertainty in terms of a fuzzy number. The resulted fuzzy models will be studied by three fuzzy differential approaches, namely, Hukuhara differential and its generalization and fuzzy differential inclusion. Applications of fuzzy arithmetics to the models lead to a set of alpha-cut deterministic systems with some additional equations. These systems are then solved by the extended Runge-Kutta method. The extended Runge-Kutta method is chosen as our numerical approach in order to enhance the order of accuracy of the solutions by including both function and its first derivative values in calculations. Among our fuzzy approaches, our simulations show that the fuzzy differential inclusion is the most appropriate approach to capture oscillation behaviors of the model. Using the aforementioned fuzzy approach, we then demonstrate how to estimate parameters to our generated fuzzy simulation data.

On Darboux‐Type Differential Inclusions with Uncertainty

In this article we prove the existence results for solutions of the Darboux‐type problems in fuzzy partial differential inclusions with local conditions of integral types. We present two problems involving open and closed level sets of a given fuzzy mapping. In the first case fuzzy differential inclusion has been transformed into an equivalent Darboux‐type problem for partial differential equations and then using the Tychonoff fixed point theorem we prove the existence result for this crisp case. For the second case we use Nadler’s fixed point theorem and selection theorem of Kuratowski‐Ryll‐Nardzewski to find the solution of given differential inclusions problem. We furnish an example to validate our results.

Different Solution Strategies for Solving Epidemic Model in Imprecise Environment

We study the different solution strategy for solving epidemic model in different imprecise environment, that is, a Susceptible‐Infected‐Susceptible (SIS) model in imprecise environment. The imprecise parameter is also taken as fuzzy and interval environment. Three different solution procedures for solving governing fuzzy differential equation, that is, fuzzy differential inclusion method, extension principle method, and fuzzy derivative approaches, are considered. The interval differential equation is also solved. The numerical results are discussed for all approaches in different imprecise environment.

Boundary value problems for semilinear fuzzy impulsive differential inclusions based on semigroups in Banach spaces

A first-order semilinear fuzzy differential inclusion with fuzzy impulse characteristics and linear boundary conditions is considered in separable Banach spaces. By means of semigroup properties, stacking approach and the fixed point theorem for multivalued map due to Dhage, the existence results for fuzzy solution are established. Some general criteria are presented for a class of semilinear fuzzy differential inclusions including higher dimensional and infinite dimensional uncertain systems, and the conditions for existence of solutions in the results are concise and mild. Also, several examples are given to illustrate the utility and applicability.

Existence of local and global solutions of fuzzy delay differential inclusions

Fuzzy delay differential inclusions are introduced and studied in this paper. The local and global existence theorems under different conditions are proved by using selection theorems and Kakutani’s fixed point theorem. Under the tangential condition, a global viable solution for a fuzzy delay differential inclusion is proved to exist. The property of the solution sets is achieved. Some known results of fuzzy differential inclusions and fuzzy differential equations are extended, which might be helpful in the analysis of dynamic systems with uncertainties.

Averaging of Fuzzy Controlled Differential Inclusions with Terminal Quality Criterion

We analyze the possibility of application of an averaging scheme to the problems of control with terminal quality criterion in the case where the behavior of the system is described by a controlled fuzzy differential inclusion containing a small parameter. Translated from Neliniini Kolyvannya, Vol. 16, No. 1, pp. 105–110, January–March, 2013.

Quasisolutions of fuzzy differential inclusions

We introduce the notion of a quasisolution of a fuzzy differential inclusion and study the relationship between the sets of ordinary solutions and quasisolutions.

Different initial conditions in fuzzy Tumor model

One of the best ways for better understanding of biological experiments is mathematical modeling. Modeling cancer is one of the complicated biological modeling that has uncertainty. Therefore, fuzzy models have studied because of their application in achievement uncertainty in modeling. Overall, the main purpose of this modeling is creating a new view of complex phenomena. In this paper, fuzzy differential equation model consisting of tumor, the immune system and normal cells has been studied. Model derived from a classical model DePillis in 2003, which some parameters from a clinical point of view can be described in the region. In this model, by considering fuzzy parameters from clinical point of view, the three-dimensional fuzzy tumor cells in terms of time and membership function are pictured and region of uncertainties are determined. To access the uncertainty area we use fuzzy differential inclusion method that is one of the including methods of solving differential equations. Also, different initial conditions on the model are inserted and the results of them are analyzed because tumor has different treatment in different initial conditions. Results show that fuzzy models in the best way justify what happens in the reality.

Some studies on uncertainty management in dynamical systems using cybernetic approaches and fuzzy techniques with applications

Uncertain information processing by fuzzy if–then rules has received much attention. Here we have taken a different path to model a system, about which we do not have precise information, namely modelling the system by fuzzy-valued functions without resorting to fuzzy if–then rules. As a result, the phase (state) space of the system becomes a fuzzy set and the underlying fuzzy mapping becomes a fuzzy attainability set mapping. Uncertain or fuzzy dynamical systems have been defined in terms of fuzzy attainability set mappings. Fuzzy differentiable dynamical systems have been discussed with a particular emphasis on fuzzy differential inclusion (FDI) relations. An evolutionary algorithm for solving one-dimensional FDIs has been developed. A model of the creation of a tropical cyclone in the form of a vortex, created by winds coming from different directions and colliding under certain conditions, has been proposed in terms of FDIs. A model has been considered for the highly uncertain system of evolution of a tumour in a human body and an FDI relation model of the whole system has been proposed and simulated. The model of an evolution of turbulence, as a random occurrence of vortices in a three-dimensional dynamic fluid, has been proposed and simulated, where each vortex is modelled by a fuzzy-valued function, where uncertain parameter and variable values are fuzzy numbers. All the systems represented by FDI relations have been simulated with the help of the evolutionary algorithm mentioned above.

Complexity analysis, uncertainty management and fuzzy dynamical systems

In this paper, we present a brief study on various paradigms to tackle complexity or in other words manage uncertainty in the context of understanding science, society and nature. Fuzzy real numbers, fuzzy logic, possibility theory, probability theory, Dempster‐Shafer theory, artificial neural nets, neuro‐fuzzy, fractals and multifractals, etc. are some of the paradigms to help us to understand complex systems. We present a very detailed discussion on the mathematical theory of fuzzy dynamical system (FDS), which is the most fundamental theory from the point of view of evolution of any fuzzy system. We have made considerable extension of FDS in this paper, which has great practical value in studying some of the very complex systems in society and nature. The theories of fuzzy controllers, fuzzy pattern recognition and fuzzy computer vision are but some of the most prominent subclasses of FDS. We enunciate the concept of fuzzy differential inclusion (not equation) and fuzzy attractor. We attempt to present this theoretical framework to give an interpretation of cyclogenesis in atmospheric cybernetics as a case study. We also have presented a Dempster‐Shafer's evidence theoretic analysis and a classical probability theoretic analysis (from general system theoretic outlook) of carcinogenesis as other interesting case studies of bio‐cybernetics.

Fuzzy differential inclusions as substitutes for stochastic differential equations in population biology

The familiar procedure of adding white noise to deterministic systems of equations may not be appropriate or even possible in some modelling problems arising in the biological sciences. Although some mathematical handle on indeterminant factors (i.e. noise) may be necessary, sometimes the probabilistic requirements involved cannot be rigorously verified for the data set in hand. As an alternative, we discuss here the modelling utility of fuzzy differential inclusions associated with given systems of nonlinear ODE's. We give concrete examples and give account of the conservative stochastic mechanics of E. Nelson applied to growth of a dimorphic clone, and its fuzzy differential inclusion analogue.

A Survey of Viability Theory

Some theorems of viability theory which are relevant to nonlinear control problems with state constraints and state-dependent control constraints are motivated and surveyed. They all deal with viable solutions to nonlinear control problems, i.e., solutions satisfying at each instant given state constraints of a general and diverse nature. Some classical results on controlled invariance of smooth nonlinear systems are adopted to the nonsmooth case, including inequality constraints bearing on the state and state-dependent constraints on the controls. For instance, existence of a viability kernel of a closed set (corresponding to the largest controlled invariant manifold) is provided under general conditions, even when the zero-dynamics algorithm does not converge. The concepts of slow and heavy viable solutions are introduced, providing concrete ways of regulating viable solutions, by closed-loop feedbacks and closed-loop dynamical feedbacks. Viability theorems also allow the extension of Lyapunov’s second method to nonsmooth observation functions and the construction of “best” Lyapunov functions. As an application, “fuzzy differential inclusion” is presented. Proofs and complements can be found in [Viability Theory, to appear, 1991]. They rely on properties of differential inclusion (see [Differential Inclusions, Springer-Verlag, Berlin, New York, 1984]) and set-valued analysis, (see [Set-Valued Analysis, Birkhäuser, Basel, 1990]).