Controllability of second order neutral impulsive fuzzy functional differential equations with Non-Local conditions

In this paper, the controllability of fuzzy solutions for a second-order nonlocal impulsive neutral functional differential equation with both nonlocal and impulsive conditions in terms of fuzzy are considered. The sufficient condition of controllability is developed using the Banach fixed point theorem and a fuzzy number whose values are normal, convex, upper semi-continuous, and compactly supported fuzzy sets with the Hausdorff distance between α -cuts at its maximum. The α -cut approaches allow to translate a system of fuzzy differential equations into a system of ordinary differential equations to the endpoints of the states. An example of the application is given at the end to demonstrate the results. These kinds of systems come in use for designing landing systems for planes and spacecraft, as well as car suspension systems.


Introduction
Control theory is a fascinating part of application-oriented mathematics that deals with the fundamental ideas that support control framework analysis and design.The main objective of the control theory is to perform specific tasks by the system applying appropriate control.Controllability is well recognized in the context of control systems and comprises a central location.In controllability, one analyses the possibility of changing a system from a given state (initial state) to a certain required final state by using a set of permissible controls.One main presumption in the control system is that all its components are involved with complete precision.Moreover, control systems related to reasonable circumstances are characterized by fuzziness.Fuzzy set theory, introduced by [21] is competent to take care of such kind of fuzziness.Since a fuzzy differential equation describes a fuzzy control system with some initial conditions (fuzzy or non-fuzzy).So, first, we study some results pertaining to fuzzy differential equations, see ( [10], [16], [11]), and references therein.
Neutral functional differential equations emerge in various disciplines of applied mathematics and so these equations have become more important in a few decades.For more detail on neutral functional differential equations, refer [13], and the references therein.Different kinds of mathematical models in the study of population dynamics, biology, ecology, and epidemics can be represented as impulsive neutral differential equations.The theory of these equations has been examined by many authors ( [9], [14], [20]).The vehicle industry has closely examined and is still curious about, the vehicle suspension system since it is the component that physically isolates the vehicle body from the wheels of the car to move forward the ride stability, comfort, and street dealing with of vehicles.
The issues which can be modeled in form of impulsive control systems experience sudden changes at certain focuses of time.Generally, impulses are not defined in a precise manner.So fuzzy impulsive condition may be better than to simple impulsive condition.For more details on fuzzy and non-fuzzy impulsive differential equations, we refer to see ( [5], [18]) and references therein.[19] studied the periodic boundary value problems for second-order impulsive integrodifferential equations.[7] studied the controllability for the following impulsive fuzzy neutral functional integrodifferential equations using Banach fixed point theorem.

d dt
x t g t x Ax t f t x q t s x ds u t t J The controllability of impulsive second-order semilinear fuzzy integrodifferential control systems with nonlocal initial conditions has been studied by [17].Controllability of second-order impulsive neutral integrodifferential systems with an infinite delay has been studied by [3].
Recently, [1] studied the controllability results of fuzzy solutions for the following first-order nonlocal impulsive neutral functional differential equation using the Banach fixed point theorem 3) [8] studied the existence of fuzzy solutions for nonlocal impulsive neutral functional differential equations.In this paper, we attempt to establish controllability results for a class of fuzzy control systems governed by a fuzzy differential equation of second order coupled with nonlocal and impulsive conditions using the α-cut technique.In fact, nonlocal conditions are more viable for depicting the physical measurement rather than classical conditions (see for instance ( [12], [4]), and references therein).We have discussed the controllability of fuzzy solutions for the following second-order non-local functional differential equations with an impulse which is an extension to the work done in [15].Here both nonlocal as well as impulsive conditions are considered fuzzy.
where A B J E n , : → is the fuzzy coefficient, E n is the set of all upper semi-continuous, convex, normal fuzzy numbers with bounded α levels.
The functions We define c c c o c : : M f o such that for any r > 0, (0) ϕ is bounded and measurable function on [ ,0] −r and , where B h is endowed with the norm The paper is organized as follows: Section 2 summarizes the fundamental heuristics.The controllability results of the fuzzy solutions to non-local second-order neutral functional differential equations with impulse are proved using the Banach fixed point theorem in Section 3.An example has been provided to support the theory in Section 4. Section 5 contains an application with a graphical representation of the solution and finally, the conclusion is given in Section 6.

Definition: Fuzzy Set
A fuzzy set A X zM is characterized by its membership function A X : [0,1] → and A x ( ) is interpreted as the degree of membership of element x in fuzzy set A for each x X ∈ .

Definition
Let CC n ( ) R denote the family of all nonempty, compact, and convex subsets of R n .Define addition and scalar multiplication in CC n ( ) R by such that satisfies a s below : 1. w is normal, that is, there exists an R is a complete and separable metric space [7].

Definition
The complete metric d ∞ on E n is defined by ∞ is a complete metric space [22].

Definition
The supremum metric H 1 on C J E n ( , ) is defined by Hence, ( ( , ), ) 1 C J E H n is a complete metric space [24].

Definition
The derivative ′ x t ( ) of a fuzzy process x E n ∈ is defined by provided that the equation defines a fuzzy set c x t E n ( ) [24].

Definition
The fuzzy integral provided that the Lebesgue integrals on the right-hand side exist [24].

Definition
A

Definition
A strongly measurable and integrably bounded map is strongly measurable and integrably bounded, then f is integrable [23].

Definition
A system is said to be controllable in E n , if there exists an admissible control function u(t) using which it is possible to steer a system from any arbitrary state to desired final state.

Definition
The α-cut or α-level set of A is denoted by A α or [A] α and is defined as

Assumptions
Assume the following hypothesis.
H1. [16] H2. [1] The H3. [1] If g is continuous and there exists constants G k p k , =1,2, , , such that ) H4. [16] There exists a non-negative d k and d k′ such that where, where, for all x y E n , ∈ and t J ∈ .

H5. [1] The nonlinear function f J E E n n
: u o is continuous and there exists a constant d 2 > 0 , satisfying the global Lipschitz condition, such that satisfies the following conditions: • For each positive number l ∈ », there exists a positive function w(l) dependent on l such that and lim inf w here sup s • f is completely continuous.

Definition
[7] The nonlocal problem (1.4) is said to be controllable on the interval J if there exists a control u(t), such that the fuzzy solution x(t) of (3.1) is controllable and satisfies Before proving the controllability of system (1.4), we define the fuzzy mapping where P( ) R is the set of all closed compact control functions in R and Γ u is the closure of support u.In [2], the support Γ u of a fuzzy number u is defined as a special case of the level set by Γ u = { : ( ) > 0} x x u µ .
Then, there exists W j l r , are bijective mappings.Now, the α-level set of u(s) is ( Substituting this in equation (3.1), we get an α -level set of x(T) as Hence, the fuzzy solution x(t) for equation (3.1) satisfies [ ( )] = [ ] . 1 x T x α α Now for each x(t) and t J ∈ , define ( )[ (0, )] ( , ) ( where ( ) 1 W − satisfies the previous statements.Observe Φ( ( )) x t = [ ] 1 x , which represents that the control u(t) steers the system (3.1) from the arbitrary stage to x 1 in time T, given that there must exist a fixed point of the nonlinear operator Φ. Similarly,

³ g y y y S T y h h T y AT s
The controllability of fuzzy solutions for the neutral impulsive functional differential equation with nonlocal conditions is discussed in the following theorem.
Proof: For x y , c :

³ ³ S T y h h T y A C T s h s y ds S T s
T T s T

C t g y y y S t y h H p d
] ,

S t t I y t S t t I y t
]

S T s v s ds CT t I y t ST t I y t t s ds S t s W x C T g y y
], [( ( )) (  )) ( ( ), ( )) (( ), ( ))

T y A T s h s y ds S T s v s ds CT t I
.
By condition (H6), Φ is a contraction mapping.Using the Banach fixed point theorem, equation (3.3) has a unique fixed point x c : .Hence the System (1.4) is controllable on J. Similarly, we proceed for ′ x , 1 Substituting this in equation (3.1), we get an α -level set of x′(T) as Hence, the fuzzy solution x(t) for equation ( where ( ) 1 W − satisfies the Equation (3.5).Observe )( ( )) [ ] x , which represents that the control u(t) steers the system (3.1) from the arbitrary stage to x 1 in time T, given that there must exist a fixed point of the nonlinear operator Φ. Similarly, For c c c x y , , : )   Proceeding in the similar fashion as in x t ( ), we get, )] ( , ).

Example
In this section, we apply the results proved in the previous section to study the controllability of the following partial differential equation: The α-level set of number 0 is given by, And the α-level set of number 2 is given by, satisfies the inequality which is given in condition (H5).Let the target state be 2. Now, from the definition of fuzzy solution Then the α-level set of x T ( ) is given by, x T x α α .Thus all the conditions stated in Theorem 3.1 are satisfied.So the system (4.1) is controllable on J.

Application
Consider a coil spring suspended from the ceiling with 8-lb weight placed upon the lower end of the spring.Stretching   From the graph, we conclude that as x 1 increases x 2 is decreasing function.Hence, the solution is a strong solution.

Conclusions
In this paper, we have proved the controllability of the fuzzy solutions for the second-order impulsive neutral functional differential equation by applying the contraction mapping principle.Further, we can extend the controllability results for the fuzzy inclusions.The numerical solution of the system is also useful for the study of a real-life phenomenon.For instance, we can consider a real-life phenomenon of a friction pendulum bearing specially designed for base isolators used in many heavy structures like bridges, buildings, towers, etc. to reduce the impact of earthquakes.Using the above system, we can also find critical points where the structure becomes unstable or gets damaged.

Figure 1 :
Figure 1: Represents the coil spring-suspended with 8-lb mass