Direct Method for Solution Variational Problems by using Hermite Polynomials

abstract: In this paper a new numerical method is presented for numerical approximation of variational problems. This method with variable coefficients is based on Hermite polynomials. The properties of Hermite polynomials with the operational matrices of derivative and integration are used to reduce optimal control problems to the solution of linear algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

One of the widely used methods to solve optimal control problems is the direct method. There is a large number of research papers that employ this method to solve optimal control problems (see for example [2-4, 8, 9, 14-17, 19, 27-34, 37-40]). Razzaghi, et. al. used direct method for variational problems by using hybrid of block-pulse and Bernoulli polynomials [32]. Optimal control of switched systems based on Bezier control points presented in [15]. A new approach using linear combination property of intervals and discretization is proposed to solve a class 2 A. Yari and M. Mirnia of nonlinear optimal control problems, containing a nonlinear system and linear functional [37,38]. Time varying quadratic optimal control problem was solved by using Bezier control points [14]. Hybrid functions approach for nonlinear constrained optimal control problems presented by Mashayekhi et. al. [27]. The optimal control problem of a linear distributed parameter system is studied via shifted Legendre polynomials (SLPs) in [20]. An accurate method is proposed to solve problems such as identification, analysis and optimal control using the Bernstein orthonormal polynomials operational matrix of integration [36]. In [19] Jaddu and Shimemura proposed a method to solve the linear-quadratic and the nonlinear optimal control problems by using Chebyshev polynomials to parameterize some of the state variables, then the remaining state variables and the control variables are determined from the state equations. Also Razzaghi and Elnagar [33] proposed a method to solve the unconstrained linear-quadratic optimal control problem with equal number of state and control variables. Their approach is based on using the shifted Legendre polynomials to parameterize the derivative of each of the state variables. The approach proposed in [27] is based on approximating the state variables and control variables with hybrid functions. In [39] operational matrices with respect to Hermite polynomials and their applications is presented for solving linear dfferential equations with variable coeffcients. Investigation of optimal control problems and solving them using Bezier polynomials has been presented [1]. In [2] Solution of optimal control problems with payoff term and fixet state endpoint by using Bezier polynomials has been presented. In this paper, we present a computational method for solving variational problems by using Hermite polynomials. The method is based on approximating the state variables with Hermite polynomials. Our method consists of reducing the variational problems into a set of linear algebraic equations by first expanding the state rate x(t) as a Hermite polynomial with unknown coefficients.
The paper is organized as follows: In Section 2 we describe the basic formulation of the Hermite functions required for our subsequent development. Section 3 is devoted to the formulation of optimal control problems. Section 4 summarizes the application of this method to the optimal control problems, and in Section 5, we report our numerical finding and demonstrate the accuracy of the proposed method.

Hermite Polynomials and Their Properties
Hermite polynomials are a classical orthogonal polynomial sequence that arise in probability. The explicit expression of Hermite polynomials of degree n is defined by [39] H n (t) = n!
[ n 2 ] = largest integer number≤ n 2 , weher t is a real number (t ∈ R), and Rodrigues formula is the following H n (t) = (−1) n e t 2 d n dt n (e −t 2 ). 2) are solutions for following equation Namely x(t) = H n (t). The first few Hermite polynomials are H 0 (t) = 1 ,

Some Properties of Hermite Polynomials
Hermite polynomials obey the recurrence relation An important property of the Hermite polynomials is the following derivative relation [39] H with respect to the weight function w(t) = e −t 2 and satisfy in the following relation Weher δ i,j is kronecker delta function. some property for Hermite polynomials are

The operational matrices for the Hermite Polynomials
A function x(t) ∈ L 2 w (Λ), can be expressed in terms of Hermite polynomials as where the coeffcients a i is given by In practice, only the frst n + 1 term of the Hermite polynomials are considered. Then we have: where T denotes transposition.
The operational matrix of derivative: The differentiation of vector Φ n (t) can be expressed as where D φ is the (n + 1)(n + 1) operational matrix of derivative for the Hermite polynomials and it is given as following: The operational matrix of integration: The integration of vector Φ n (t) can be expressed as where P φ is the (n + 1)(n + 1) operational matrix of integration for the Hermite polynomials. The integration of H i (x) of order i can be obtianed as following formula: where for a = 0 we get: finaly we can written P φ matrix as:

Approximations by Hermite polynomials
Now in this section, we present some useful theorems which show the approximations of functions by Hermite polynomials. For this purpose, let us defne S n = span{H 0 (t), H 1 (t), . . . , H n (t)}. Any polynomial h(t) of degree m can be expanded in terms of H i (t), i = 0 . . . n as follows (2.17) Direct Method for Solution Variational Problems

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Also the L 2 (Λ)-orthogonal projection p n : L 2 (Λ) → S n is a mapping in a way that for any y(t) ∈ L 2 (Λ), we have: p n (y) − y, φ = 0, ∀φ ∈ S n . Due to the orthogonality, we can write where c i are constants in the following form In the literature of spectral methods, p n (y) is named as Hermite expansion of y(t) and approximates y(t) on (−∞, +∞). Also estimating the distance between y(t) and it's Hermite expansion as measured in the weighted norm . w is an important problem in numerical analysis. The following theorem provide the basic approximation results for Hermite expansion. [41].

Performance Index Approximation for the Variational Problem
By expanding x (n) (t) using the Bezier polynomials we have whereX T is vector of order 1 × (n + 1), By integrating Eq.(4.9) from 0 to t we get where P φ is operational matrix of integration given in Eq. (2.5). By using Eqs.

Illustrative Examples
This section is devoted to numerical examples. We implemented the proposed method in last section with MALAB (2017) in personal computer. To illustrate our technique, we present four numerical examples, and make a comparison with some of the results in the literatures.
The boundary conditions The exact solution is obtained by using the Euler equation ( ∂F ∂x − d dt ( ∂F ∂ẋ ) = 0) as following: where F (t, x(t),ẋ(t)) =ẋ 2 (t) + tẋ(t). Here we solve this problem with Hermite polynomials by choosing n = 2. Let where X = [X 0 , X 1 , X 2 ] is unknown and d = [0, 1 2 , 0]. Using Eqs. (2.11) and (5.4) we getẋ where D φ is the operational matrix of derivative given in Eq. (2.12). By substituting Eqs. (5.5)-(5.6) in Eqs. (5.1) we obtain where V = We obtain the approximate solution as following which is the exact solution.   For n = 4 we obtain which is the exact solution, and also not required to Euler equation and natural boundary conditions. Example 4. It has been studied by using bezier parameterization [34] and also bezier polynomials [1] for optimal control by differential evolution min Z =

Conclusion
In this paper we presented a numerical scheme for solving variational problems. The Hemite polynomials was employed. Also several test problems were used to see the applicability and efficiency of the method. The obtained results show that the new approach can solve the problem effectively.