Simultaneously Dissipative Operators And The Infinitesimal Moore Effect In Interval Spaces

Abstract

One of shortcomings of stepwise interval methods is the following. The intervals determining the solution of a system are often expanded in the course of time irrespective of the method and step used (the {\em Moore effect}). We introduce the notion of general {\em interval spaces} and study the infinitesimal Moore effect (IME) in these spaces. We obtain the local conditions of absence of the IME in terms of Jacobi matrices field. The relation between the absence of IME and simultaneous dissipativity of the Jacobi matrices is established. We study simultaneously dissipative operators in $\Bbb{R}^n$. A linear operator $A$ is {\em dissipative} with respect to a norm $\|...\|$ if $\| \exp (At) \| \leq 1$ at all $t \geq 0$. For each norm, the dissipative operator form a closed convex cone. An operator $A$ is {\em stable dissipative} if it belongs to the interior of this cone. The family of linear operators $\{A_α\}$ is called {\em simultaneously dissipative}, if there exists a norm with respect to which all the operators are dissipative. We studied general properties of such families. For example, let the family $\{A_α\}$ be finite and generate a nilpotent Lee algebra and let for each $A_α$ there exist a norm with respect to which it is dissipative. Then $\{A_α\}$ is simultaneously dissipative. Let the family $\{A_α\}$ be compact and generate solvable Lee algebra, and let the spectrum of each operator $A_α$ lie in the open left half-plane. Then $\{A_α\}$ is simultaneously stable dissipative, i.e. there exists a norm with respect to which all $A_α$ are stable dissipative. We study the conditions of simultaneous dissipativity of the matrices of rank one and discussed their application to equations of {\em mass action law} kinetics.