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LEGENDRE-TAU APPROXIMATION FOR FUNCTIONAL DIFFERENTIAL EQUATIONS PART III: EIGENVALUE APPROXIMATIONS AND UNIFORM STABILITY

IP.com Disclosure Number: IPCOM000149128D
Original Publication Date: 1899-Dec-30
Included in the Prior Art Database: 2007-Apr-13

Publishing Venue

Software Patent Institute

Related People

Ito, Kazufumi: AUTHOR [+2]

Abstract

LEGENDKE-TAU APPROXIMATION FOK FUNCTLONAL DIFFEKENTIAL EQUATIONS PAKT 111: EIGENVALUE APPROXIMATIONS AND UNIFORM STABILITY Kazufumi Ito

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LEGENDKE-TAU APPROXIMATION FOK FUNCTLONAL DIFFEKENTIAL EQUATIONS PAKT 111: EIGENVALUE APPROXIMATIONS AND UNIFORM STABILITY

Kazufumi Ito


Institute for Computer Applications in Science and Engineering

Abstract

   The stability and convergence properties of the Legendre-tau approximation for hereditary differential systems are analyzed. We derive a characteristic equation for the eigenvalues of the resulting approximate system. As a result of this derivation we are able to establish that the uniform exponential stability of the solution semigroup is preserved under approximation. It is the key to obtaining the convergence of approximate solutions of the algebraic Riccati equation in trace norm.

    Research was supported by the National Aeronautics and Space Administration under NASA Contract No. NAS1-17070 while the author was in residence at ICASE, NASA Langley Research Center, Hampton, VA 23665.

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Introduction

   It has been demonstrated in [6], [7] that the Legendre-tau approximation is a quite powerful approximation method for hereditary differential systems i n many instances. However, there remained an important question which had not been resolved. This is the question concerned with the presentation of exponential stability under approximation. We establish it in Section 5.

   As observed in [7], the Legendre-tau approximation scheme provides a good approximation technique for computation of eigenvalues for hereditary differential system as well as its optimal closed-loop system. We give a justification of this observation deriving a characteristic equation for eigenvalues of the approximating system and relating it to the pad: approximations of the exponential function. Moreover, it leads to a characterization of detectability and stabilizability conditions for the approximating system and the preservation of those properties under approximation.

   The results discussed i n this paper are similar to those for the 'averaging' approximation scheme [I] that have been obtained in [I], [5] and
[9], and a great deal of our discussions are motivated and inspired by those investigations. We refer to [I], [8], [2], and [5] for the the summary of the earlier contributions on optimal control and numerical approximation problems for hereditary differential systems.

   The following is a brief summary of the contents of this paper. In Section 2 we state the type of problems to be considered and review the equivalence results between hereditary differential equations and abstract Cauchy problems on the product space 3 x L2, and then describe the Legendre- tau approximation scheme within the abstract framework. The convergence of

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