197 research outputs found
A new measure of instability and topological entropy of area-preserving twist diffeomorphisms
We introduce a new measure of instability of area-preserving twist
diffeomorphisms, which generalizes the notions of angle of splitting of
separatrices, and flux through a gap of a Cantori. As an example of
application, we establish a sharp >0 lower bound on the topological entropy in
a neighbourhood of a hyperbolic, unique action-minimizing fixed point, assuming
only no topological obstruction to diffusion, i.e. no homotopically non-trivial
invariant circle consisting of orbits with the rotation number 0. The proof is
based on a new method of precise construction of positive entropy invariant
measures, applicable to more general Lagrangian systems, also in higher degrees
of freedom
Heisenberg Picture Approach to the Stability of Quantum Markov Systems
Quantum Markovian systems, modeled as unitary dilations in the quantum
stochastic calculus of Hudson and Parthasarathy, have become standard in
current quantum technological applications. This paper investigates the
stability theory of such systems. Lyapunov-type conditions in the Heisenberg
picture are derived in order to stabilize the evolution of system operators as
well as the underlying dynamics of the quantum states. In particular, using the
quantum Markov semigroup associated with this quantum stochastic differential
equation, we derive sufficient conditions for the existence and stability of a
unique and faithful invariant quantum state. Furthermore, this paper proves the
quantum invariance principle, which extends the LaSalle invariance principle to
quantum systems in the Heisenberg picture. These results are formulated in
terms of algebraic constraints suitable for engineering quantum systems that
are used in coherent feedback networks
Heisenberg Picture Approach to the Stability of Quantum Markov Systems
Quantum Markovian systems, modeled as unitary dilations in the quantum
stochastic calculus of Hudson and Parthasarathy, have become standard in
current quantum technological applications. This paper investigates the
stability theory of such systems. Lyapunov-type conditions in the Heisenberg
picture are derived in order to stabilize the evolution of system operators as
well as the underlying dynamics of the quantum states. In particular, using the
quantum Markov semigroup associated with this quantum stochastic differential
equation, we derive sufficient conditions for the existence and stability of a
unique and faithful invariant quantum state. Furthermore, this paper proves the
quantum invariance principle, which extends the LaSalle invariance principle to
quantum systems in the Heisenberg picture. These results are formulated in
terms of algebraic constraints suitable for engineering quantum systems that
are used in coherent feedback networks
Invariance-like theorems and “lim inf” convergence properties
International audienceSeveral theorems, inspired by the Krasovskii-LaSalle invariance principle, to establish “lim inf” convergence results are presented in a unified framework. These properties are useful to “describe” the oscillatory behavior of the solutions of dynamical systems. The theorems resemble “lim inf” Matrosov and Small-gain theorems and are based on a “lim inf” Barbalat's Lemma. Additional technical assumptions to have “lim” convergence are given: the “lim inf”/“lim” relation is discussed in-depth and the role of some of the assumptions is illustrated by means of examples
Robust stability theory for stochastic dynamical systems
In this work, we focus on developing analysis tools related to stability theory forcertain classes of stochastic dynamical systems that permit non-unique solutions. Thenon-unique nature of solutions arise primarily due to the system dynamics that aremodeled by set-valued mappings. There are two main motivations for studying suchclasses of systems. Firstly, understanding such systems is crucial to developing a robuststability theory. Secondly, such system models allow flexibility in control design problems.We begin by developing analysis tools for a simple class of discrete-time stochasticsystem modeled by set-valued maps and then extend the results to a larger class ofstochastic hybrid systems. Stochastic hybrid systems are a class of dynamical systemsthat combine continuous-time dynamics, discrete-time dynamics and randomness. Theanalysis tools are established for properties like global asymptotic stability in probabilityand global recurrence. We focus on establishing results related to sufficient conditions for stability, weak sufficient conditions for stability, robust stability conditions and converse Lyapunov theorems. In this work a primary assumption is that the stochastic system satisfies some mild regularity properties with respect to the state variable and random input. The regularity properties are needed to establish the existence of random solutions and results on sequential compactness for the solution set of the stochastic system.We now explain briefly the four main types of analysis tools studied in this work.Sufficient conditions for stability establish conditions involving Lyapunov-like functionssatisfying strict decrease properties along solutions that are needed to verify stability properties. Weak sufficient conditions relax the strict decrease nature of the Lyapunov like function along solutions and rely on either knowledge about the behavior of thesolutions on certain level sets of the Lyapunov-like function or use multiple nested non-strict Lyapunov-like functions to conclude stability properties. The invariance principleand Matrosov function theory fall in to this category. Robust stability conditions determinewhen stability properties are robust to sufficiently small perturbations of thenominal system data. Robustness of stability is an important concept in the presenceof measurement errors, disturbances and parametric uncertainty for the nominal system.We study two approaches to verify robustness. The first approach to establish robustnessrelies on the regularity properties of the system data and the second approach isthrough the use of Lyapunov functions. Robustness analysis is an area where the notionof set-valued dynamical systems arise naturally and it emphasizes the reason for ourstudy of such systems. Finally, we focus on developing converse Lyapunov theorems forstochastic systems. Converse Lyapunov theorems are used to illustrate the equivalencebetween asymptotic properties of a system and the existence of a function that satisfiesa decrease condition along the solutions. Strong forms of the converse theorem implythe existence of smooth Lyapunov functions. A fundamental way in which our resultsdiffer from the results in the literature on converse theorems for stochastic systems isthat we exploit robustness of the stability property to establish the existence of a smoothLyapunov function
Robust open-loop stabilization of Fock states by time-varying quantum interactions
A quantum harmonic oscillator (spring subsystem) is stabilized towards a
target Fock state by reservoir engineering. This passive and open-loop
stabilization works by consecutive and identical Hamiltonian interactions with
auxiliary systems, here three-level atoms (the auxiliary ladder subsystem),
followed by a partial trace over these auxiliary atoms. A scalar control input
governs the interaction, defining which atomic transition in the ladder
subsystem is in resonance with the spring subsystem. We use it to build a
time-varying interaction with individual atoms, that combines three
non-commuting steps. We show that the resulting reservoir robustly stabilizes
any initial spring state distributed between 0 and 4n+3 quanta of vibrations
towards a pure target Fock state of vibration number n. The convergence proof
relies on the construction of a strict Lyapunov function for the Kraus map
induced by this reservoir setting on the spring subsystem. Simulations with
realistic parameters corresponding to the quantum electrodynamics setup at
Ecole Normale Superieure further illustrate the robustness of the method
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