El Teorema Kharitonov mediante polinomios ortogonales

Abstract

In the first part of this work I made a brief introduction of the advances that have been made throughout the history of mankind in the methods to calculate the number π. Studying methods from ancient cultures to modern computers of the digits of π. In the second part of this work I realized the geometric construction of some approximations of π I built in dynamic geometry software GeoGebra, where I built the approximations of π made by Hipias, Arquímedes, Gelder, Hobson, Ramanujan and Mascheron The study of the stability and the asymptotic stability of the system (1) are vital Importance for the understanding of the physical phenomenon studied. From the theory of ODEs it is known that systems of type (1) have asymptotically solutions Stable if and only if the characteristic polynomial associated with matrix A has all its Roots in 1 C -. In practice, the parameters p 0, p n-1 of matrix A are not exact, then a type of uncertainty arises which is called uncertainty Parametric analysis. The present thesis is devoted to the study of the asymptotic stability of Systems of the type (1) that is translated in the study of the behavior of the roots Of the characteristic polynomial representing the uncertainty of the system (1). The polynomial of type (2) is called Polynomial interval (PI).

El estudio de la estabilidad y de la estabilidad asintótica del sistema (1) son de vital importancia para el entendimiento del fenómeno físico estudiado. De la teoría de las EDO se conoce que los sistemas del tipo (1) tienen soluciones asintóticamente estables si y sólo si el polinomio característico asociado a la matriz A tiene todas sus raíces en 1 C −. En la práctica, los parámetros p 0,..., p n−1 de la matriz A no son exactos, surge entonces un tipo de incertidumbre a la que se le denomina incertidumbre paramétrica. La presente tesis se dedica al estudio de la estabilidad asintótica de los sistemas del tipo (1) que se traduce en el estudio del comportamiento de las raíces del polinomio característico que representa la incertidumbre del sistema (1). El polinomio de tipo (2) es llamado polinomio intervalo (PI).