Izchak Lewkowicz

Convex invertible cones and the Lyapunov equation

Abstract Convex cones of matrices which are closed under matrix inversion are defined, and their structure is studied. Various connections with the algebraic Lyapunov equation of general inertia are explored.

A necessary and sufficient criterion for the stability of a convex set of matrices

A Hurwitz stability criterion for the convex hull of two matrices, due to C.R. Johnson (1976), is generalized to the case of any compact convex set of matrices. It is shown that the same criterion is not available in the case of Schur stability. >

Convex invertible cones of state space systems

In the paper [CL1] the notion of a convex invertible cone,cic, of matrices was introduced and its geometry was studied. In that paper close connections were drawn between thiscic structure and the algebraic Lyapunov equation. In the present paper the same geometry is extended to triples of matrices andcics of minimal state space models are defined and explored. This structure is then used to study balancing, Hankel singular values, and simultaneous model order reduction for a set of systems. State spacecics are also examined in the context of the so-called matrix sign function algorithm commonly used to solve the algebraic Lyapunov and Riccati equations.

Maximal stability robustness for state equations

A measure for stability robustness of a linear time-invariant finite-dimensional system state equations is introduced. An upper bound for the measure is derived using the characteristic values of the system. It is shown that the set of optimal systems, namely, systems for which the stability robustness measure attains the bound, contains the normal set, which has been considered as the set of optimal robustness. >

A pair of matrices sharing common Lyapunov solutions—A closer look☆

Abstract Let A , B be a pair of matrices with regular inertia. If HA + A * H and HB + B * H are both positive definite for some Hermitian matrix H then all matrices in conv( A , A −1 , B , B −1 ) have identical regular inertia. This, in turn, implies that both conv( A , B ) and conv( A , B −1 ) consist of non-singular matrices. In general, neither of the converse implications holds. In this paper we seek situations where they do hold, in particular, when A and B are real 2×2 matrices. Several aspects of the above statements for n × n matrices are discussed. A connection to the characterization of the convex hull of matrices with regular inertia is introduced. Differences between the real and the complex case are indicated.

Convex invertible cones of matrices — a unified framework for the equations of Sylvester, Lyapunov and Riccati

Abstract Convex cones of matrices which are closed under matrix inversion were defined and various connections with the algebraic Lyapunov equation of general inertia were studied in (N. Cohen, I. Lewkowicz, Linear Algebra and its Appl. 250 (1997) 105–131). Here, in the price of doubling the size of the matrices involved, we introduce a unified framework for the equations of Sylvester. Lyapunov and Riccati. This enables one to extend the convex invertible cone structure to all these three equations and explore related properties. In particular, the use of the Matrix Sign Function for solving these equations is examined.

When are the complex and the real stability radii equal

Characterization of those real stable matrices for which the unstructured complex and real stability radii are equal is presented. Special cases of this equality are discussed. As application of this result, it is shown that, for 2*2 matrices with nonreal eigenvalues, neither the case of equality nor the case of strict inequality is generic. >

Representation Formulas for Hardy Space Functions Through the Cuntz Relations and New Interpolation Problems

We introduce connections between the Cuntz relations and the Hardy space H 2 of the open unit disk \(\mathbb{D}\). We then use them to solve a new kind of multipoint interpolation problem in H 2, where, for instance, only a linear combination of the values of a function at given points is preassigned, rather than the values at the points themselves.

An easy-to-compute factorization of rational generalized positive functions

We give a simple constructive proof of a factorization theorem for scalar rational functions with a non-negative real part on the imaginary axis. A Mathematica program, performing this factorization, is provided.

Exponential stability of triangular differential inclusion systems

Abstract We consider a differential inclusion system of the form dot x ϵ A x , where A is a collection of upper triangular matrices. Conditions for exponential stability of all the possible solutions are given.