Free Essay on Existence and Uniqueness of Control Theory

The existence and uniqueness of the control theory have become established through satisfaction of conditions on terms exhibiting non-linear qualities. In the justification of the various theorems that surround the existence and uniqueness in the theorem, differential equations that are derived from fractional order have become essential. In the specific cases such as the study of boundary value problem with respect to systems of fractional differential equations having orders of a (0 a 1), the conditions must be defined so that functions involving integral boundary conditions and two points are solvable.

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The integral role of the existence and uniqueness of the control theory has pointed out to its use to the study of impulsive systems that are ideally differential and in that case, its wide use has continued to find applicability in modeling processes. Most of the modeling processes that make use of mathematical concepts have continually become subject to slight changes in value and cannot become described through any classical differential problems (Luca, 2007, 1). In the study, the paper would use an initial value theorem (IVP) for its system of differential equations existing as fractions as shown below:

[Da g(t)=g(t, g(t))..(i) Caputo fractional derivative

[g(t0)=g0 (ii)The condition.

f= g (x, y), f(x0)= f0 ..(iii) function

The above value theorem aids in formulation of the basis for defining the derivative boundaries especially in their fraction form. The essence comes from the fact that they become vital in the modeling of various phenomena with mathematical backgrounds. Other disciplines whose practicality become of significance have: signal processing, viscoelasticity and dynamical processes. In the application of fractional analysis with regard to the use of fractional equations there exists variety of problems that have boundary value problems found in integral boundaries conditions (Mardanov, 20114, 1). All these have values ranging from two-point to multi-point value problems captured in the paper. The discussion would look into the control theory solving problems using differential equations that exhibit non-linearity. The existence and uniqueness theorem with regard to the control theory also play a critical role in the derivation of a single answer to a first order differential equation that satisfies the initial condition we have above.

In this case, having a unique initial value problem such as the one above also presents a unique solution y(x) definable through an integral, I. When the constant function is identified as (iii) it then becomes defined so as to aid in the generation of a sequence of functions following through an assumption:

g (x y) converges at the solution of y(x) from the initial problem: (iii) denoted by f= g(x.y) f(x0)= f0

Starting from the Caputo fractional derivative, the function below has:

t J'=J (t1.tp) K:=( 0, I)

k (ii+-k(ii-)= a1, I = (0,1, 2,p)

x (0) = x0, ak= R (Mardanov, 2014, 1)

Where cDa (Caputo derivative) the condition, a (0, 1) and the function:

k(t, k(t) 0tk t, xyxdx(iv)

Since it denotes a continuous function the function (iv) can then be compressed into f: K*B B bringing continuity of the function. Another aspect of discussion in the paper would point out to existence of answers for non-linear fractional equations having value conditions for their boundaries. For instance, the boundary problems below are a show of fractional derivatives with defined conditions giving rise to:

fig. 2 (Mardanov, 2014, 1)

t J'=J (t1.tp)

K:=( 0, I)

From figure 1 above, denoting the Caputo fractional derivation in the order of a (0, 1), the function is assigned a precise lower limit of zero f: K*B B showing joint continuity.

As tk satisfies 0= t<t1<..tp< tp+1 = T. x (tk +) = lim 0+ x(tk+ (v)

And x (tk-) = lim 0+ x(tk- .(vi)

The two set of equations represent the left and right limits that exist in functions of x(t) at t= tk, Ik C (R, R), and a, b, c are real constants with a+b 0 (Luca, 2007, 2439). From the equations and functions above, the paper limits itself to differential equations with conditions that narrow down to the form:

cD0+a x (t) = g (t, y (t) , i J

x (ij+) x (ij) = IJ (x(ij)), where j denoted values from 1,2,p

Ax (0) + Bx (T) = 0Tg s,xsds, [k(t0)=k0

Where C, D Rn*n have the matrices and therefore the det (C+ D) 0. Here f, g: (0,I) * Rn Rn and I j : Rn Rn are the given functions. From the above notations, the derivatives here termed as fractional is are defined in a. (0,1) and having a lower limit defines t0 by:

Da y(t)= 1G(1-a)t0tt-s)-y (s) ds C=[g Rn y-y0 b](vii)

J= [t0- a, t0+ a].. (viii)

With regard to the above function, x C1 (J, Rn), can be the solution to

[Da g(t)=g(t, h(t)) . (ix)

Only in the fulfillment of equation (ix), the conditionality of k (i0) =i0, solution of the IVP, brings the equation into perspective using different methods and techniques. The techniques involved in obtaining correct criteria on uniqueness and existence of answers to problems surrounding the control theory therefore fundamentally make use of (ix).

Preliminary considerations

The preliminaries set off through the introduction of definitions, notations and preliminary calculations in the rest of the paper.

Yc= max {y(t): t J}(x)

When A Rn*n then A becomes the norm for A. Recalling the following definitions and their respective conditions:

h D [a, b] with a > 0

g D1 (J, Rn

Defining Function 1

If h D [a, b] and condition a greater than 0, then the fractional integral becomes defined through:

Ia+ ah(t)= 1G(a)atg kg-h1-adg .(xi)

G in this case references to the a function becomes defined as

G(d)= 0td-1d-tdt

As a function of x C1 (J, Rn) brings a solution to the definition with the condition that x fulfills the equation of (ix) and that the condition x (t0) = x0 (Luca, 2007, 2448)..

Defining Function 2

For the second definition, the Caputo fractional derivative in the order of a > 0 of a continuous function g [a,b] R that is defined by:

cDa + ag(t)=1 G(t- a)atgnst-sa-t+1ds..(xii)

In this case, the notation t= a+1 and the a shows the largest integer can never go beyond a. Control theorys justification through the differential equations is showed from the instantaneous change when a function for instance has its definition through Caputo fractional derivative showing (ix) and its defined boundaries and conditions existing as a > 0.

As the provision of the control theory present natural conditions of g(t), the Caputo fractional derivative still becomes the right integer order derivative of g(t) as a. n that is the function. In another instance, when we let a, > 0 and n= [a] + 1: then

D0 + a,ct= G()G(-a,)t-1 and >n, D0 + actk=0 and k=0,1,2..n-1. (xiii)

The two notations therefore depict that the relations hold true for a control theory which seeks to show slight changes or variations in dynamical systems.

For a >0, ht F [0, T]L1[0,I] we then find a homogenous differential equation as:

CDa0 + g(t)=0 therefore its solution is got as

g(t) = f0+ f1t+ f2t+.fn-1tn-1(xiv)

In this case, we have fi K i=0.1.t-1and t= (a) +1

When we assume that g(t) C [0, T] ]L1[0,T], then the derivative of the order n belonging to C[0,T] L1[0,T] brings about:

Ia0+ fDa0 + g(t) = f0+ f1t+ f2t+.ft-1tn-1. (xv)

Where the function has ci K i=0.1.T-1and T= (a) +1

When letting q, r 0 g K [0, T] the results come to:

IP0+IQ0+ f(t) = IP+q0 f(t)= IP0+IQ0+ f(t)..(xvi)

A look at the above equation shows satisfaction everywhere based on [0,T]. In this case, if it happens then f C 0,T then (xvi) becomes true for all values of t [0, T]Main Results

Assuming that the function f: E Rn brings satisfaction with the conditions below such as

k (t, x) shows ease in measurement with respect to g and h

k (t, x) shows ease in measurement with respect x and B:

There exists a constant (0, a) with a real value in function k (t) such that k(g, x)m(t) for every g J and all the k B. Therefore a solution for the IVP relies on the condition when the function a (0, 1)

To prove this, the IVP (1) for the case exists when t [k0, k0+g] is used in verification for a case where t (k0-g, g0). According to the above conditions i) and ii), the calculation gives (t-s) a-1 L11-b (t0-t) for cases where t (k0+k0+ g). Using Holders inequality, the function comes to (t-s)a-1f (s,x (s)). With the Lebesgue inequality, it becomes easy to integrate with consideration of the expression s [ k0, k], for every k [t0, t+g

The Outcomes

The results would first of all appreciate the fact that there is a constant K>0 where f (t, g) f (t, h) Lx-y has almost both Jand x , y B. With regard to the case for every (0,1) a unique IVP solution is found, when considering the expression at interval [t0-h1,t0 + h1] as the h1 is regarded as the minimum. The proof of the IVP solution below presents:

t0t (f-g)-1 f(g, x))..........(xvii)

ds t0t((f-g)-1)11-bdf 1- [t0tms1df)a.(xviii)

Assuming that the y (t) and x(t) are the solution of IVP (1) [t0+t0 + h] then the

f (t) = x0+ G1 t0t((f-g)-1)11-bds t [t0+t0 + g]...(xix)

g (t) x0 + G1 t0t((f-g)-1)11-bds t [t0+t0 + g].(xx)

Hence we finally have, f(t)-g(t) = G1 t0t((t-s)-1)11-bds t [t0+t0 + h]

It narrows down to max s [t0+t0 + h] f(t)-g(t) c

Max s [t0+t0 + g] f(t)g(s) t [t0+t0 + g] (Mardanov, 2014, 1). The notation therefore implies that: x (t) y(t) t [t0+t0 + g] and this shows that the proof is complete with regard to the uniqueness and existence of the particular theorem.

References

Mardanov, M., Mahmudov, N., & Sharifov, Y. (2014). Existence and Uniqueness Theorems for Impulsive Fractional Differential Equations with the Two-Point and Integral Boundary Conditions. The Scientific World Journal, 2014, 1-8. http://dx.doi.org/10.1155/2014/918730

Luca, R. (2007). Existence and uniqueness for a nonlinear discrete hyperbolic system. Nonlinear Analysis: Theory, Methods & Applications, 67(8), 2433-2446. http://dx.doi.org/10.1016/j.na.2006.09.003