Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userNetworks of open systems
67
0
0.0
(
0
)
Authors
Eugene Lerman
Ask ChatGPT about the research
No Arabic abstract
Many systems of interest in science and engineering are made up of interacting subsystems. These subsystems, in turn, could be made up of collections of smaller interacting subsystems and so on. In a series of papers David Spivak with collaborators formalized these kinds of structures (systems of systems) as algebras over presentable colored operads. It is also very useful to consider maps between dynamical systems, which in effect amounts to viewing dynamical systems as objects in an appropriate category. This is the point of view taken by DeVille and Lerman in the study of dynamics on networks. The goal of this paper is to describe an algebraic structure that encompasses both approaches to systems of systems. This allows us, on one hand, build new large open systems out of collections of smaller open subsystems and on the other keep track of maps between open systems. Consequently we obtain synchrony results for open systems which generalize the synchrony results of Golubitsky, Stewart and their collaborators for groupoid invariant vector fields on coupled cell networks.
rate research
Read More
We generalize the results of Networks of open systems by the first author to the setting of hybrid systems. In particular we introduce the notions of hybrid open systems, their networks and maps between networks. A network of systems is a blueprint for building a larger system out of smaller subsystems by specifying a pattern of interactions between subsystems --- an interconnection map. Maps between networks allow us to produce maps between complex hybrid dynamical systems by specifying maps between their subsystems
Many systems of interest in science and engineering are made up of interacting subsystems. These subsystems, in turn, could be made up of collections of smaller interacting subsystems and so on. In a series of papers David Spivak with collaborators formalized these kinds of structures (systems of systems) as algebras over presentable colored operads. It is also very useful to consider maps between dynamical systems. This amounts to viewing dynamical systems as objects in an appropriate category. This is the point taken by DeVille and Lerman in the study of dynamics on networks. The goal of this paper is to describe an algebraic structure that encompasses both approaches to systems of systems. To this end we replace the monoidal category of wiring diagrams by a monoidal double category whose objects are surjective submersions. This allows us, on one hand, build new large open systems out of collections of smaller open subsystems and on the other keep track of maps between open systems. As a special case we recover the results of DeVille and Lerman on fibrations of networks of manifolds.
In this paper, we investigate geometric properties of monotone systems by studying their isostables and basins of attraction. Isostables are boundaries of specific forward-invariant sets defined by the so-called Koopman operator, which provides a linear infinite-dimensional description of a nonlinear system. First, we study the spectral properties of the Koopman operator and the associated semigroup in the context of monotone systems. Our results generalize the celebrated Perron-Frobenius theorem to the nonlinear case and allow us to derive geometric properties of isostables and basins of attraction. Additionally, we show that under certain conditions we can characterize the bounds on the basins of attraction under parametric uncertainty in the vector field. We discuss computational approaches to estimate isostables and basins of attraction and illustrate the results on two and four state monotone systems.
At the quantum level, feedback-loops have to take into account measurement back-action. We present here the structure of the Markovian models including such back-action and sketch two stabilization methods: measurement-based feedback where an open quantum system is stabilized by a classical controller; coherent or autonomous feedback where a quantum system is stabilized by a quantum controller with decoherence (reservoir engineering). We begin to explain these models and methods for the photon box experiments realized in the group of Serge Haroche (Nobel Prize 2012). We present then these models and methods for general open quantum systems.
We study stability of multivariable control-affine nonlinear systems under sparsification of feedback controllers. Sparsification in our context refers to the scheduling of the individual control inputs one at a time in rapid periodic sweeps over the set of control inputs, which corresponds to round-robin scheduling. We prove that if a locally asymptotically stabilizing feedback controller is sparsified via the round-robin scheme and each control action is scaled appropriately, then the corresponding equilibrium of the resulting system is stabilized when the scheduling is sufficiently fast; under mild additional conditions, local asymptotic stabilization of the corresponding equilibrium can also be guaranteed. Moreover, the basin of attraction for the equilibrium of scheduled system also remains same as the original system under sufficiently fast switching. Our technical tools are derived from optimal control theory, and our results also contribute to the literature on the stability of switched systems in the fast switching regime. Illustrative numerical examples depicting several subtle features of our results are included.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
