SUFFICIENT CONDITIONS FOR EXTREMUM OF FRACTIONAL VARIATIONAL PROBLEMS

. Sufficient conditions for extremum of fractional variational problems are formulated with the help of Caputo fractional derivatives. The Euler–Lagrange equation is defined in the Caputo sense and Jacobi conditions are derived using this. Again, Wierstrass integral for the considered functional is obtained from the Jacobi conditions and the transversality conditions. Further, using the Taylor’s series expansion with Caputo fractional derivatives in the Wierstrass integral, the Legendre’s sufficient condition for extremum of the fractional variational problem is established. Finally, a suitable counterexample is presented to justify the efficacy of the fresh findings.


Introduction and preliminaries
The fractional calculus deals with the problem of extremizing functionals, which are of non integer order and are differentiable.It's origin is more than three centuries old, and dates back to L'Hôpital's query to Leibniz about the significance of the fractional order derivative of a function.In his reply to the above question, Leibniz indicated it to be a paradox which can lead to useful consequences in the near future [36].The subject caught the attention of many eminent mathematicians including Euler, Fourier, Laplace, Lacroix, Abel, Riemann, Liouville and Caputo et al. for its advancement.Several fractional derivatives in the sense of Grünwald Letnikov, Riemann-Liouville and Caputo et al. are discussed in [32].However, it gained popularity in the last three decades for its wide applications in the field of science, engineering, statistics and mathematical biology [1-6,12,14,17,19-21, 24, 28, 34, 35, 37] etc.
The fractional derivatives are used in calculus of variations to generalize it to non integer orders.Agrawal [1] formulated the Euler-Lagrange equations for fractional variational problem and proved the necessary conditions for extremum of fractional variational problems using Riemann-Liouville fractional derivatives.Legendre's second order necessary optimality conditions for weak extremizers of the variational problems was subsequently established by Lazo and Torres [29], where the involved functionals are fractional differentiable in the sense of Riemann-Liouville.The isoperimetric problems have also been proved using the Caputo fractional derivatives [10,30].
Variational fuctionals with a Lagrangian are considered by Odzijewicz et al. [31] using classical as well as Caputo fractional derivatives.Wang and Xiao [39] defined the fractional variational integrators in terms of Caputo derivatives.The notion of Euler-Lagrange fractional extremals is used to prove the Noether type theorem by Frederico and Torres [16].Agrawal and Baleanu [7] studied fractional optimal control problems as an application of the Riemann-Liouville fractional derivatives.
Recently, Zhang [40] proved the necessary and sufficient optimality conditions for the fractional variational problems using the Caputo-Fabrizio fractional derivatives, and Baleanu et al. [11] presented suitable counterexamples to support it.Almeida [9] considered several problems of calculus of variations depending upon Lagrange function on a Caputo type fractional derivative.Again, Almeida [8], obtained the necessary and sufficient conditions, when the involved functionals are dependent upon the fractional integrals and fractional derivatives having indefinite integral and in the presence of time delay.Rahman et al. [33] established the generalized Riemann-Liouville fractional integrals in the sense of increasing, positive, monotone, and measurable function Ψ and also discussed the Chebyshev functionals.Goufo et al. [18] explored the differential fractional operators, which include Atangana-Baleanu derivative and the Caputo-Fabrizio derivative.
Several interesting applications of fractional derivatives in modeling the ecological systems are found in [23,26,28].Between the Haar wavelet and Adams-Bashforth-Moulton method, a comparative investigation for the fractional Lotka-Volterra (LV) system in the Caputo sense was done by Kumar et al. [22].The fractional derivatives such as the Caputo, Caputo-Fabrizio and Atangana-Baleanu derivatives are used to investigate a mathematical system numerically to figure out the possible dynamics for the spread of the diseases like HIV/AIDS and COVID-19 by Kumar et al. [25,27].
In this paper, sufficient optimality conditions for fractional variational problems are investigated with the help of Legendre's condition by using the Caputo fractional derivative.Again, the Jacobi condition and the transversality conditions are discussed to validate the aforesaid condition.With the help of fractional Taylor's series expansion, the Legendre's sufficient conditions for extremum of the fractional variational problems are obtained.A number of remarks are mentioned at appropriate places for verifying the proper generalizations of several classical equations and conditions.To support our new findings, a suitable counterexample is also provided.

Definition 1.1 ([15]
).Let  be a real valued function in the interval [, ],  be a positive real number,  be the integer satisfying ( − 1) ≤  < , and Γ be the Euler gamma function.Then, (i) the left Caputo fractional derivatives (CFD) of order  is defined by

and
(ii) the right CFD of order  is defined by Let  = (, ), be the solution curves of the Euler-Lagrange's equation having center at the point ( 0 ,  0 ).The -discriminant curve is defined by the equations; and 2) The curves of this family that are close to the extremals  = () moving through the points ( 0 ,  0 ) & ( 1 ,  1 ) and the -discriminant curves will intersect at points close to the tangent points of the extremal under consideration with the -discriminant curve.The function () will have continuous left and right Caputo fractional derivatives of order  and , respectively with the boundary points ( 0 ,  0 ) and ( 1 ,  1 ).Now let's consider the functional on the pencil of extremals i.e., 3) The Jacobi condition states that, to establish a central field of extremals containing the arc  of extremals it is sufficient that, there does not exist any common point (apart from ) between the -discriminant curve and the arc  of extremals.Now, let us define the Jacobi condition analytically in fractional variational sense.Denote the function  := (,)  , where  is given.
, since  is the function of  alone.The functions  = (, ); are solutions of the Euler-Lagrange's equation.∴ Differentiating the above equation with respect to  in the Caputo sense, we get, which is the Jacobi equation in the classical sense [13].
Note that for fractional calculus of variational problems the resulting Jacobi equation contains both the left and right CFD.This is to be expected, since the optimum function must satisfy both terminal conditions.If there exists a solution of the Jacobi equation that vanishes only at the point  =  0 and does not vanish at any other point in [ 0 ,  1 ], then no points are conjugate to the arc AB; and the Jacobi condition is fulfilled.)︁ , satisfies the Euler-Lagrange's equation (1.1) as well as the Jacobi equation (2.5) in CFD sense.Therefore, the above functional is taken into consideration.

The transversality condition
Consider the functional of the form (2.3) having fixed boundary points.This functional has a solution of the form (2.1).Now consider the case where one or both of the boundary points can move.Then we can get a larger class of acceptable curves comprising of the comparison curves having common boundary points with the curve under consideration and the curves that are generated by the extension of the boundary points.We establish the following result: Theorem 2.3.Let [()] be a functional of the form (2.3) defined on the set of function (), which have continuous left and right CFD of order  and  respectively with moving boundary points.Then the extremal of the given functional satisfies the transversality condition given by = 0 (in left CFD sense) and where  is a function of .
Proof.The variation of the functional [(, )] on the extremals  = (, ), when the boundary point is displaced from the position ( 1 ,  2 ) to the position ( 1 +  1 ,  2 +  2 ) may be calculated in the fractional sense and is given by, where ℎ is the difference between the original and the perturbed curve and is given by, ℎ =  −   D    (in left CFD sense) and ℎ =  +   D    (in right CFD sense).Then (2.9) Thus, (2.10) The second part of the integration of the above equation is and the third part of the integration is (2.12) Hence, )︁ ℎ = 0. (by EL equation) Now replacing the value of ℎ in the above equation (2.15) in left CFD sense, we have, Now the basic necessary condition for an extremum;  = 0 has the form (2.16).
If the variations are independent, then,  By substituting the value of  from the above equation in equation (2.16); Similarly, replacing the value of ℎ in equation (2.15) and proceeding in the same way, we will get, This establishes a relationship between the slopes of  and  at the boundary points.This condition is called the transversality condition.

Sufficient condition for extremum
In the preceding sections we have discussed the Euler-Lagrange equation, Jacobi condition and the transversality condition in the CFD sense.The Euler-Lagrange equation is the principal necessary condition which must be satisfied by the solution of the functional to achieve the extremum.The Jacobi condition determines the limitations to the extent of the region over which the integral may be extended.The transversality condition deals with the moving boundary conditions of the functional under consideration.
Next, we are going to prove the sufficient conditions for the solution of the functional to achieve the extremum.We now prove the following: Theorem 3.1.A problem involving an extremal of the functional of the form (2.3) having boundary points ( 0 ) =  0 and ( 1 ) =  1 satisfies the Jacobi condition in Caputo sense.If  0 () gives a relative minimum(or maximum) for , and for all close permissible curve κ.Here (, ) is the slope of the central field through which the extremal  has passed.
Proof.Consider the problem involving an extremal of the functional of the form (2.3) having boundary points ( 0 ) =  0 and ( 1 ) =  1 .Let the Jacobi conditions are satisfied.Suppose the extremal , passes through the points ( 0 ,  0 ) and ( 1 ,  1 ) is included in the central field with slope (, ).The increment ∆, the difference of the functional J passing from the extremal  to some close acceptable curve κ, is given by, The )︁ d taken along the arcs of the curves κ and , respectively.Now the auxiliary functional The functional ∆ is minimum on the curve  if the function  ≥ 0, and maximum if  ≤ 0. This condition is sufficient to determine the extremum of the given functional.The Taylor's series expansion [38]   The following example is given to understand some of our theoretical investigations considered in this paper.
The Euler-Lagrange equation for left CFD is given by equation (1.1); It's general solution is given by Applying the initial conditions, we get,  1 = 0 and  2 = 1.188396.The Jacobi condition for left CFD is given by,  where  1 is a constant.
The function vanishes at the points  = , where  is an integer.On the interval 0 ≤  ≤ 1, the function  vanishes only at the point  = 0 and the Jacobi equation is satisfied.
By the Legendre's condition, for any value of  0 D   .It follows that on the straight line  = 0 a strong minimum is achieved for  < .

Conclusion
The sufficient conditions for fractional variational problems of the type )︁ d, are formulated with the help of Cauputo fractional derivatives.The sufficient conditions for the proposed problems are established using the Weierstrass integral and the Legendre condition.It is observed that the present work generalizes many classical investigations [13].Again, with the help of discussed counterexample, our fresh findings have been justified.Many concepts of classical calculus of variations were found to be the extension of fractional calculus of variations with minor modifications.Thus, ample of opportunities are open for further investigation in this area.

. 5 )
which is called the Jacobi's equation.The solution of the above equation  = (,)  vanishes at the center of the extremals for  =  0 .If it does not vanishes further at any point in the interval  0 <  <  1 , then the Jacobi condition is fulfilled.If the solution of the equation (2.2), vanishes at any other point of the interval  0 <  <  1 , then the point  * conjugate to A is defined by the equations;  = (,  0 ) and (, )  = 0 or  = 0, which lies on the arc AB of the extremal.Remark 2.1.When  =  = 1,   D   = d d and   D   = −d d and the above equation reduces to
taken along the extremal  (when  =  = 1,   D   = d d and   D   = −d d ); since d d =  on extremals of the field.The auxiliary functional (3.3);