Şahin Emrah Amrahov

Nizami Gasilov, Şahin Emrah Amrahov, Afet Golayoğlu Fatullayev

In this study, we consider a linear differential equation with fuzzy boundary values. We express the solution of the problem in terms of a fuzzy set of crisp real functions. Each real function from the solution set satisfies differential equation, and its boundary values belong to intervals, determined by the corresponding fuzzy numbers. The least possibility among possibilities of boundary values in corresponding fuzzy sets is defined as the possibility of the real function in the fuzzy solution. In order to find the fuzzy solution we propose a method based on the properties of linear transformations. We show that, if the corresponding crisp problem has a unique solution then the fuzzy problem has unique solution too. We also prove that if the boundary values are triangular fuzzy numbers, then the value of the solution at any time is also a triangular fuzzy number. We find that the fuzzy solution determined by our method is the same as the one that is obtained from solution of crisp problem by the application of the extension principle. We present two examples describing the proposed method.

Şahin Emrah Amrahov, Iman N. Askerzade

We consider a fuzzy linear system with crisp coefficient matrix and with an arbitrary fuzzy number in parametric form on the right-hand side. It is known that the well-known existence and uniqueness theorem of a strong fuzzy solution is equivalent to the following: The coefficient matrix is the product of a permutation matrix and a diagonal matrix. This means that this theorem can be applicable only for a special form of linear systems, namely, only when the system consists of equations, each of which has exactly one variable. We prove an existence and uniqueness theorem, which can be use on more general systems. The necessary and sufficient conditions of the theorem are dependent on both the coefficient matrix and the right-hand side. This theorem is a generalization of the well-known existence and uniqueness theorem for the strong solution.

Şahin Emrah Amrahov, Nizami Gasilov, Afet Golayoglu Fatullayev

In this paper, we consider a time-optimal control problem with uncertainties. Dynamics of controlled object is expressed by crisp linear system of differential equations with fuzzy initial and final states. We introduce a notion of fuzzy optimal time and reduce its calculation to two crisp optimal control problems. We examine the proposed approach on an example.

Nizami Gasilov, Afet Golayoğlu Fatullayev, Şahin Emrah Amrahov

In this paper, a linear system of equations with crisp coefficients and fuzzy right-hand sides is investigated. All possible cases pertaining to the number of variables, n, and the number of equations, m, are dealt with. A solution is sought not as a fuzzy vector, as usual, but as a fuzzy set of vectors. Each vector in the solution set solves the given fuzzy linear system with a certain possibility. Assuming that the coefficient matrix is a full rank matrix, three cases are considered: For m = n (square system), the solution set is shown to be a parallelepiped in coordinate space and is expressed by an explicit formula. For m > n (overdetermined system), the solution set is proved to be a convex polyhedron and a novel geometric method is proposed to compute it. For m < n (underdetermined system), by determining the contribution of free variables, general solution is computed. From the results of three cases mentioned above, a method is proposed to handle the general case, in which the coefficient matrix is not necessarily a full rank matrix. Comprehensive examples are provided and investigated in depth to illustrate each case and suggested method.

N. Gasilov, Şahin Emrah Amrahov, A. Golayoglu Fatullayev et al.

In this paper, linear systems with a crisp real coefficient matrix and with a vector of fuzzy triangular numbers on the right-hand side are studied. A new method, which is based on the geometric representations of linear transformations, is proposed to find solutions. The method uses the fact that a vector of fuzzy triangular numbers forms a rectangular prism in n-dimensional space and that the image of a parallelepiped is also a parallelepiped under a linear transformation. The suggested method clarifies why in general case different approaches do not generate solutions as fuzzy numbers. It is geometrically proved that if the coefficient matrix is a generalized permutation matrix, then the solution of a fuzzy linear system (FLS) is a vector of fuzzy numbers irrespective of the vector on the right-hand side. The most important difference between this and previous papers on FLS is that the solution is sought as a fuzzy set of vectors (with real components) rather than a vector of fuzzy numbers. Each vector in the solution set solves the given FLS with a certain possibility. The suggested method can also be applied in the case when the right-hand side is a vector of fuzzy numbers in parametric form. However, in this case, -cuts of the solution can not be determined by geometric similarity and additional computations are needed.