Kronecker operational matrices for fractional calculus and some applications

Dedicated to Professor H.M. Srivastava on the occasion of his 65th birthday
https://doi.org/10.1016/j.amc.2006.08.122Get rights and content

Abstract

The problems of systems identification, analysis and optimal control have been recently studied using orthogonal functions. The specific orthogonal functions used up to now are the Walsh, the block-pulse, the Laguerre, the Legendre, Haar and many other functions. In the present paper, several operational matrices for integration and differentiation are studied. we introduce the Kronecker convolution product and expanded to the Riemann–Liouville fractional integral of matrices. For some applications, it is often not necessary to compute exact solutions, approximate solutions are sufficient because sometimes computational efforts rapidly increase with the size of matrix functions. Our method is extended to find the exact and approximate solutions of the general system matrix convolution differential equations, the way exists which transform the coupled matrix differential equations into forms for which solutions may be readily computed. Finally, several systems are solved by the new and other approaches and illustrative examples are also considered.

Introduction

One of the principle reasons is that matrices arise naturally in communication systems, economic planning, statistics, control theory and other fields of pure and applied mathematics [2], [4], [7]. Another is because an m × n matrix can be valued as representing a linear map from an n-dimensional vector space to an m-dimensional vector space, and conversely, all such maps can be represented as m × n matrices. The operational matrices for integration and differentiation are studied by many authors [1], [3], [6], [10]. For example, Mouroutsos and Sparis [6] solved the control problem using Taylor series operational matrices for integration, Chen et al. [3] solved the distributed systems by using Walsh operational matrices and Wang [10] introduced the inversions of rational and irrational transfer functions by using the generalized block pulse operational matrices for differentiation and integration. The Kronecker convolution product is an interesting area of current research and in fact plays an important role in applications, for example, Sumita [9], established the matrix Laguerre transform to calculate matrix convolutions and evaluated a matrix renewal function.

In the present paper, several operational matrices for integration and differentiation are studied. we introduce the Kronecker convolution product and expanded to the Riemann–Liouville fractional integral of matrices. For some applications, it is often not necessary to compute exact solutions, approximate solutions are sufficient because sometimes computational efforts rapidly increase with the size of matrix functions. Our method is extended to find the exact and numerical solutions of the general system matrix convolution differential equations, the way exists which transform the coupled matrix differential equations into forms for which solutions may be readily computed. Finally, several systems are solved by the new and other approaches and illustrative examples are also considered.

As usual, the notations A{−1}(t) and det A(t) are the inverse and determinant of matrix function A(t), respectively, with respect to convolution. The notations AT(t) and Vec A(t) are transpose and vector-operator of matrix function A(t), respectively. The term “Vec A(t)” will be used to transform a matrix A(t) into a vector by stacking its column one underneath.The notations δ(t) and Qn(t) = Inδ(t) are the Dirac delta function and Dirac identity matrix, respectively, where In is the identity scalar matrix of order n × n. Finally, the notations A(t)  B(t) and A(t)  B(t) are the convolution and Kronecker convolution products, respectively. Finally, A  B stands for the Kronecker product ; A  B = [aijB]ij.

Section snippets

Taylor series operational matrix for integration

A function f(t) that is analytic function at the neighborhood of the point t0 can be expanded in the following formula:f(t)=n=1anφn(t),where an=1n!dnf(0)dtn and φn(t) = tn.

To obtain an approximate expression of the analytic function f(t) we may truncate the series (1) up to the (r + 1)th term:f(t)n=1ranφn(t).By defining the coefficient vector γT as:γT=(a0,,ar)and the power series basis vector φT(t) as:φT(t)=(φ0(t),,φr(t)).Eq. (2) can written in the following compact form:f(t)γTφ(t).The basis

Convolution product and Riemann–Liouville fractional integral of matrices

Definition 1

Let A(t) = [fij(t)] and B(t) = [gij(t)] be m × m absolutely integrable matrices on [0, b). The convolution and Kronecker convolution product s are matrix functions defined by

  • (i)

    Convolution product:A(t)B(t)=[hij(t)],hij(t)=k=1m0tfik(t-t1)gkj(t1)dt1=k=1mfik(t)gkj(t).

  • (ii)

    Kronecker convolution productA(t)B(t)=[fij(t)B(t)]ij.Here, fij(t)  B(t) is the ijth submatrix of order m × m and A(t)  B(t) is of order m2 × m2.

Definition 2

Let A(t)=1Γ(αij)tαij-1 and B(t) = [gij(t)] be m × m absolutely integrable matrices on [0, b). The

Solution of the state–space equations and optimal control problem using Taylor series and Kronecker product

The state–space equation is given byx(t)=Ax(t)+Bu(t),x(0)=x0,0tb,where x(t)Rn, u(t)Rm analytic and A, B are known constant matrices. The input vector u(t) may be expanded in Taylor series as follows:u(t)=u1(t)u2(t)um(t)=h10h11h12h1,r-1h20h21h22h2,r-1hm0hm1hm2hm,r-1·φ0(t)φ1(t)φr-1(t)=H·φ(t),where H is a known constant matrix.

Similarly, the state vector x(t) may also be expanded in Taylor series as follows:x(t)=x1(t)x2(t)xn(t)=f10f11f12f1,r-1f20f21f22f2,r-1fn0fn1fn2fn,r-1.φ

Acknowledgements

The present research has been partially supported by University Putra Malaysia (UPM) under the grant IRPA09-02-04-0259-EA001.

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