The behavioral approach to systems and modeling

An introduction to behavioral system theory, and a brief review of the content of the Special Issue are given

Pick matrix conditions for sign-definite solutions of the algebraic Riccati equation

We study the existence of positive and negative semidefinite solutions of algebraic Riccati equations (ARE) corresponding to linear quadratic problems with an indefinite cost functional. The problem to formulate reasonable necessary and sufficient conditions for the existence of such solutions is a long-standing open problem. A central role is played by certain two-variable polynomial matrices associated with the ARE. Our main result characterizes all unmixed solutions of the ARE in terms of the Pick matrices associated with these two-variable polynomial matrices. As a corollary of this result we obtain that the signatures of the extremal solutions of the ARE are determined by the signatures of particular Pick matrices

Hamiltonian and Variational Linear Distributed Systems

We use the formalism of bilinear- and quadratic differential forms in order to study Hamiltonian and variational linear distributed systems. It was shown in [1] that a system described by ordinary linear constant-coefficient differential equations is Hamiltonian if and only if it is variational. In this paper we extend this result to systems described by linear, constant-coefficient partial differential equations. It is shown that any variational system is Hamiltonian, and that any scalar Hamiltonian system is contained (in general, properly) in a particular variational system

Linear Hamiltonian behaviors and bilinear differential forms

We study linear Hamiltonian systems using bilinear and quadratic differential forms. Such a representation-free approach allows us to use the same concepts and techniques to deal with systems isolated from their environment and with systems subject to external influences and allows us to study systems described by higher-order differential equations, thus dispensing with the usual point of view in classical mechanics of considering first- and second-order differential equations only

Disorder regimes and equivalence of disorder types in artificial spin ice

The field-induced dynamics of artificial spin ice are determined in part by interactions between magnetic islands, and the switching characteristics of each island. Disorder in either of these affects the response to applied fields. Numerical simulations are used to show that disorder effects are determined primarily by the strength of disorder relative to inter-island interactions, rather than by the type of disorder. Weak and strong disorder regimes exist and can be defined in a quantitative way.Comment: The following article has been submitted to J. Appl. Phys. After it is published, it will be found at http://link.aip.org/link/?ja

Diversity enabling equilibration: disorder and the ground state in artificial spin ice

We report a novel approach to the question of whether and how the ground state can be achieved in square artificial spin ices where frustration is incomplete. We identify two types of disorder: quenched disorder in the island response to fields and disorder in the sequence of driving fields. Numerical simulations show that quenched disorder can lead to final states with lower energy, and disorder in the driving fields always lowers the final energy attained by the system. We use a network picture to understand these two effects: disorder in island responses creates new dynamical pathways, and disorder in driving fields allows more pathways to be followed.Comment: 5 pages, 5 figure

Vertex dynamics in finite two dimensional square spin ices

Local magnetic ordering in artificial spin ices is discussed from the point of view of how geometrical frustration controls dynamics and the approach to steady state. We discuss the possibility of using a particle picture based on vertex configurations to interpret time evolution of magnetic configurations. Analysis of possible vertex processes allows us to anticipate different behaviors for open and closed edges and the existence of different field regimes. Numerical simulations confirm these results and also demonstrate the importance of correlations and long range interactions in understanding particle population evolution. We also show that a mean field model of vertex dynamics gives important insights into finite size effects.Comment: 4 pages, 4 figures; v2: minor changes to text and figures. Accepted to Phys. Rev. Let

A Behavioral Approach to Passivity and Bounded Realness Preserving Balanced Truncation with Error Bounds

In this paper we revisit the problems of passivity and bounded realness preserving model reduction by balanced truncation. In the behavioral framework, these problems can be considered as special cases of balanced truncation of strictly half line dissipative system behaviors, where the number of input variables of the behavior is equal to the positive signature of the supply rate. Instead of input-state-output representations, the balancing algorithm uses normalized driving variable representations of the behavior. We show that the diagonal elements of the minimal solution of the balanced algebraic Riccati equation are the singular values of the map that assigns to each past trajectory the optimal storage extracting future continuation. Since the future behavior is only an indefinite inner product space, the term singular values should be interpreted here in a generalized sense. We establish some new error bounds for this model reduction method