Piecewise Constant Controlled Linear Fuzzy Differential Inclusions

In this article we consider the some properties of the fuzzy R-solution of the controlled linear fuzzy differential inclusions. Also, many engineering systems use piecewise constant controls. However the majority of results of the theory of optimum control are received for measurable controls. In the given paper we introduce the algorithm of replacement of measurable control on piecewise constant control so that the corresponding fuzzy R-solutions of linear fuzzy systems would be close (with necessary accuracy)


Introduction
In 80th years of XX century the control theory in the conditions of uncertainty was generated. In the given theory, controlled differential inclusions [9,14,15,22,23,24,26], the usual controlled differential equations with a set-valued initial condition [16,38,39] or the set-valued differential equations containing control [3,20,35] are used at exposition of mathematical model. In recent years, the fuzzy set theory introduced by Zadeh [37] has emerged as an interesting and fascinating branch of pure and applied sciences. The applications of fuzzy set theory can be found in many branches of regional, physical, mathematical, differential equations, and engineering sciences. Recently there have been new advances in the theory of fuzzy differential equations [8,10,11,33] and inclusions [1,4,5,7,34] as well as in the theory of controlled fuzzy differential equations [6,19,21,28,27,30] and inclusions [12,13,29,28,32].
In this article we consider the some properties of the fuzzy R-solution of the controlled linear fuzzy differential inclusions. Also, in many engineering control systems piecewise constant controls, instead of measurable controls are used. In this paper we prove that for any measurable admissible control w(·) and for any ε > 0 there exists piecewise constant admissible controlw(·) such that the fuzzy R-solutions of the control fuzzy linear system are ε-close.

Preliminaries
We denote the set of real numbers by R and usual n−dimensional space of vectors x = (x 1 , . . . , x n ), where x i ∈ R, i = 1, n, by R n . The inner product of two vectors x and y in R n is expressed by < x, y >= x 1 y 1 + · · · + x n y n . The norm of a vector x ∈ R n is defined by ∥x∥ =< x, x > 1/2 . Let C(R n ) (CC(R n )) be the family of all nonempty compact (convex) subsets of R n with the Hausdorff metric Let A ⊂ R n be a compact set. The function Let E n be the family of mappings ξ : R n → [0, 1] satisfying the following conditions: 1) ξ is normal, i.e. there exists an x 0 ∈ R n such that ξ(x 0 ) = 1; 2) ξ is fuzzy convex, i.e. ξ(λx + (1 − λ)y) ≥ min{ξ(x), ξ(y)} whenever x, y ∈ R n and λ ∈ [0, 1]; 3) ξ is upper semicontinuous, i.e. for any x 0 ∈ R n and ε > 0 exists δ(x 0 , ε) > 0 such that ξ(x) < ξ(x 0 ) + ε whenever ||x − x 0 || < δ, x ∈ R n ; 4) the closure of the set {x ∈ R n : ξ(x) > 0 } is compact.
Define the metric D : E n × E n → R + by the equation Using the results of [36], we know that

Theorem 2.[17]
If a mapping f : I → E n is measurable and integrably bounded then f is integrable over I. Now, we consider the following controlled fuzzy systemẋ By an admissible control we mean any measurable function u : R + → R k satisfying the inclusion u(t) ∈ U everywhere. We denote the set of all admissible control of controlled fuzzy system (1) by U(R + ) (or U).
Obviously, the controlled fuzzy differential inclusion (1) turns into the ordinary fuzzy differential inclusioṅ if the controlũ(·) ∈ U is fixed and G(t, x) ≡ F (t, x,ũ(t)). We interpret [1,4,5] the fuzzy differential inclusion (2) as a family of differential inclusionṡ where the subscript α indicates that the α-level set of a fuzzy set is involved (the system (3) can only have any significance as a replacement for (1) if the solutions generate fuzzy sets (fuzzy R-solution) [7]).
Let X(t) denotes the fuzzy R-solution of the differential inclusion (2), then X(t,ũ) denotes the fuzzy Rsolution of the controlled differential inclusion (1) for the fixedũ(·) ∈ U.
∈ U } is called the attainable set of the fuzzy system (1).

The controlled fuzzy differential inclusion
In this article we consider linear controlled differential equations with fuzzy parameteṙ In correspondence to the controlled problem (4) we will put the following controlled differential inclusion with fuzzy right-hand sidė where f : for all t ∈ [0, T ] and u(·) ∈ U ; 2) X(t, u) ∈ E n for all t ∈ [0, T ] and u(·) ∈ U; 3) the fuzzy R-solution X(·, u) is continuous on the interval [0, T ] for all u(·) ∈ U; 4) the attainable set Y (T ) is nonempty convex compact subset of the space E n , i.e. Y (T ) ∈ CC(E n ), where Φ(·) is a fundamental matrix of the differential equatioṅ x = A(t)x such that Φ(0) = I, I is a unit matrix. Remark 1. Properties of space C(E n ) have been considered in work [34].
We obtained the basic properties of the fuzzy Rsolution of systems (5).
In many engineering controlled systems piecewise constant controls, instead of measurable controls are used.
. Now, we need to establish that for any measurable admissible control u(·) and for any ε > 0 there exists piecewise constant admissible control u(·) such that for fuzzy R-solutions of system (5) holds D(X(t, u), X(t,ū)) < ε for all t ∈ [0, T ].
Theorem 5. Let the conditions of the theorem 4 fulfill, and 7) There exists constant γ > 0 such that Then for every u(·) ∈ U there existsū(·) ∈ U such that 1)

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Piecewise Constant Controlled Linear Fuzzy Differential Inclusions By (9), we have This concludes the proof.

Remark 2.
Obviously, if we take Here we consider the numerical solution of problem by proposed approach in above.

Conclusions
In the last years, a number of works devoted to problems of optimal control of set-valued trajectories (fuzzy trajectories, trajectory bundles or an ensemble of trajectories) appeared. These works fall into a subdivision of the optimal control theory, namely, the theory of process control under uncertainty and fuzzy conditions. This is conditioned by the fact that, in actual problems arising in economy and engineering in the course of construction of a mathematical model, it is practically impossible to exactly describe the behavior of an object. It is explained by following facts. First, for some parameters of the object, it impossible to specify exact values and laws of their change, but it is possible to determine the domain of these changes. Second, for the sake of simplicity of the mathematical model being constructed, the equations that describe the behavior of the object are simplified and one should estimate the consequences of such a simplification. Therefore, if is possible to divide the articles devoted to this direction into two types characterized by the following distinctive features: 1) there exists an incomplete or fuzzy information on the initial data; 2) the equations describing the behavior of the object to be controlled are assumed to be inexact, for example, they can contain some parameters whose exact values and laws of variation are unknown but the domain of their values is fuzzy.
In the second case, fuzzy differential inclusions are frequently used to describe behavior of objects. The reason is that, first this approach is most obvious and, second, theory of fuzzy and ordinary differential inclusions is well found and is rapidly developed at the present time.
In this article, properties of R-solutions of controlled fuzzy linear differential inclusions are received, also we introduce the algorithm of replacement of measurable control on piecewise constant control so that the corresponding fuzzy R-solutions of linear fuzzy systems would be close (with necessary accuracy).
In the first, as to receive an exact solution for the fuzzy differential equations it is impossible, this result helps to construct the approached solutions for these equations.
In the second, the obtained result is useful for constructing the ε-optimal piecewise constant controls for optimal control fuzzy system (fuzzy Mayer problem [25], fuzzy time-optimal problem [29,30,31,32] and other).