Abstract not found

Consider the ane matrix family A(x) = A 0 + k=1 x k A k , mapping a design vector x 2 R into the space of n n real matrices.

This paper addresses the problems of detecting Hopf bifurcations in systems of ordinary differential equations and following curves of Hopf points in two parameter families of vector fields. The established approach to this problem relies upon augmenting the equilibrium condition so that a Hopf bifurcation occurs at an isolated, regular point of the extended system. We propose two new methods of this type, based on classical algebraic results regarding the roots of polynomial equations and properties of Kronecker products for matrices. In addition to their utility as augmented systems for use with standard Newton-type continuation methods, they are also particularly well-adapted for solution by computer algebra techniques for vector fields of small or moderate dimension.

This article provides a new proof of Barnett's Theorem giving the degree of the greatest common divisor of several univariate polynomials with coefficients in a field by means of the rank of a well precised matrix. The new proof is elementary and self-contained (no use of Jordan Form or invariant factors) and it is based in some easy to state properties of Subresultants. Moreover this proof allows to generalize Barnett's results to the case when the considered polynomials have their coefficients in an integral domain. 1 Introduction Let IF be a field and fA(x); B 1 (x); : : : ; B t (x)g a family of polynomials in IF [x] with A(x) monic and n = deg(A(x)) ? deg(B j (x)) for every j 2 f1; : : : ; tg. Barnett's Theorem (see [1] or [2]) assures that the degree of the greatest common divisor of A(x), B 1 (x), : : :, B t (x) verifies: deg(gcd(A(x); B 1 (x); : : : ; B t (x))) = n \Gamma rank i B 1 (\Delta A ); B 2 (\Delta A ); : : : ; B t (\Delta A ) j where \Delta A is the companion matr...

Robust stability conditions obtained through generalization of the notion of energy dissipation in physical systems are discussed in this report. Linear time-invariant (LTI) systems which dissipate energy corresponding to quadratic power functions are characterized in the time-domain and the frequency-domain, in terms of linear matrix inequalities (LMIs) and algebraic Riccati equations (AREs). A novel characterization of strictly dissipative LTI systems is introduced in this report. Sufficient conditions in terms of dissipativity and strict dissipativity are presented for (1) stability of the feedback interconnection of dissipative LTI systems, (2) stability of dissipative LTI systems with memoryless feedback nonlinearities, and (3) quadratic stability of uncertain linear systems. It is demonstrated that the framework of dissipative LTI systems investigated in this report unifies and extends small gain, passivity and sector conditions for stability. Techniques for selecting power funct...