The variational problem of fractional-order control systems

This article discusses the control system of fractional endpoint variable variational problems. For this problem, we prove the Euler-Lagrange type necessary conditions which must be satisfied for the given functional to be extremum. Finally, one example is provided to show the application of our results.


Introduction
Fractional calculus is an old mathematical topic since the seventeenth century, yet it has only received much attention and interest in the past  years. For more details on the basic theory of fractional calculus, one can see the monographs [-] and the references therein.
As one of the important topics in control theory, the variational method plays an important role in the analysis control systems and is an important branch of mathematics study functional extremum. In recent years, the variational method is widely used in physics, economics, electrical engineering, image processing fields, etc.
Moreover, the numerical methods for solving fractional differential equations, optimal control and variational problems have a good development. In [], Bhrawy et al. investigated a new spectral collocation scheme, which obtained a numerical solution of this equation with variable coefficients on a semi-infinite domain. In [], Doha et al. introduced a numerical technique for solving a general form of the fractional optimal control problem. And in [], Bhrawy et al. used the Rayleigh-Ritz method for the necessary conditions of optimization and the operational matrix of fractional derivatives together with the help of the properties of the shifted Legendre orthonormal polynomials to reduce the fractional optimal control problem to solving a system of algebraic equations that greatly simplifies the problem. In [], Bhrawy and Zaky proposed and analyzed an efficient operational formulation of spectral tau method for a multi-term time-space fractional differential equation with Dirichlet boundary conditions. In [], based on the shifted Legendre orthonormal polynomials, Ezz-Eldien et al. employed the operational matrix of fractional derivatives, the Legendre-Gauss quadrature formula and the Lagrange multiplier method for reducing such a problem into a problem consisting of solving a system of algebraic equations.
However, at present, very little work has been done in the area of fractional calculus of variations [-]. In [], Jiao and Zhou used the critical point theory to solve the existence of solutions for a class of fractional boundary value problems. In [, ], the author presented a new approach to mechanics that allows one to obtain the equations for a nonconservative system using certain functionals. In [], the author proposed the fractional-order variational problem and gave solution for a class of variational problems with fixed boundary value did not discuss the problem of variable boundary conditions and transversal conditions. In other words, the boundary condition x(t  ) = x  is fixed, but x(t f ) = C(t f ) is variable, or vice versa. Inspired by the above-mentioned works, in this paper, we follow the ideas to investigate the optimality of control systems. This paper is organized as follows. In Section , we briefly review the definitions of Riemann-Liouville fractional integrals and derivatives and some lemmas. In Section , we give the necessary conditions for the fractional-order functional variational problem with fixed and variable boundary. In Section , we give an example to show the application of our results, and Section  briefly summarizes the results of this paper and future work.

Preliminaries and lemmas
In this section, we give some basic definitions and results that are used throughout this paper. For more details, please see [-].
] be a finite interval on the real axis R. The Riemann-Liouville fractional integrals I α a,t f and I α t,b f of order α >  are defined by and respectively. Here, (·) denotes the gamma function. These integrals are called the leftsided and the right-sided fractional integrals.
and their until m derivatives are zero at t = a, b, and m is less than the largest integer of α.

The fractional variational problems-variable endpoints
is a function with continuous first and second (partial) derivatives with respect to all its arguments. Then, among all functions x(t) which have continuous LRLFD of order α and RRLFD of order β for t  ≤ t ≤ t f and satisfy the boundary conditions Looking for a function x(t) such that the functional has extreme value, where  < α, β ≤  and the endpoint C(t f ) is variable.

defined on the set of functions x(t) which have continuous LRLFD of order α and RRLFD of order β in [t  , t f ] and satisfy the boundary conditions x(t  ) = x  and x(t f ) = C(t f ). Then a necessary condition for J[x] to have an extremum for a given function x(t) is that x(t) satisfies the following Euler-Lagrange equation and terminal transversality condition:
Proof To prove the necessary conditions for the extremum, assume that x * (t) is the desired function. Let ε ∈ R, and define a family of curves where η(t) is a continuous differentiable function for all given, which satisfy the boundary conditions, i.e., Due to the changing terminal time t f , each has its own trajectory terminal point t f . Therefore we must define a terminal times set corresponding to x(t), Since D α a,t and D β t,t f are linear operations, it follows that we find that for each η(t) there is only just a function of ε. Note that J(ε) is extremum at ε = , and the differential of Eq. (.) with respect to ε, we obtain

Equation (.) is also called the variations of J(x) at x(t) along η(t).
A necessary condition for J(ε) to have an extremum is that ∂J ∂ε | ε= = , and it is true for any admissible η(t). This leads to the condition that for J(x) to have an extremum for x = x * (t), for all admissible η(t). Using the definition of fractional derivatives and the formula for fractional integration by parts, the second and third integral in Eq. (.) can be written as In virtue of η(t  ) = , Eq. (.) can be written in the form Note that η(t * f ) and ξ (t f ) are not independent of each other, they are affected by the terminal constraint x(t)| t=t f = C(t f ), namely Differentiating Eq. (.) with respect to ε and letting ε → , we have That is, Since η(t) and ξ (t f ) are arbitrary, it follows from a well-established result in the calculus of variations that If C(t) is perpendicular to the t axis, sinceĊ(t) = ∞, we get By numerical simulation Eq. (.), we know that the equation has some solutions (see Figure ). This example with α = , for which the optimal trajectory is x(t) =   t, and then we have obtained the same result.

Conclusions
Necessary conditions for the optimality control of those systems are established. The case of piecewise continuous conditions and researching the minimum value of fractional differential equations principles will be considered in a future work.