5,325 research outputs found
Asymptotic Expansions for Stationary Distributions of Perturbed Semi-Markov Processes
New algorithms for computing of asymptotic expansions for stationary
distributions of nonlinearly perturbed semi-Markov processes are presented. The
algorithms are based on special techniques of sequential phase space reduction,
which can be applied to processes with asymptotically coupled and uncoupled
finite phase spaces.Comment: 83 page
Bounds and Invariant Sets for a Class of Switching Systems with Delayed-state-dependent Perturbations
We present a novel method to compute componentwise transient bounds, ultimate
bounds, and invariant regions for a class of switching continuous-time linear
systems with perturbation bounds that may depend nonlinearly on a delayed
state. The main advantage of the method is its componentwise nature, i.e. the
fact that it allows each component of the perturbation vector to have an
independent bound and that the bounds and sets obtained are also given
componentwise. This componentwise method does not employ a norm for bounding
either the perturbation or state vectors, avoids the need for scaling the
different state vector components in order to obtain useful results, and may
also reduce conservativeness in some cases. We give conditions for the derived
bounds to be of local or semi-global nature. In addition, we deal with the case
of perturbation bounds whose dependence on a delayed state is of affine form as
a particular case of nonlinear dependence for which the bounds derived are
shown to be globally valid. A sufficient condition for practical stability is
also provided. The present paper builds upon and extends to switching systems
with delayed-state-dependent perturbations previous results by the authors. In
this sense, the contribution is three-fold: the derivation of the
aforementioned extension; the elucidation of the precise relationship between
the class of switching linear systems to which the proposed method can be
applied and those that admit a common quadratic Lyapunov function (a question
that was left open in our previous work); and the derivation of a technique to
compute a common quadratic Lyapunov function for switching linear systems with
perturbations bounded componentwise by affine functions of the absolute value
of the state vector components.Comment: Submitted to Automatic
On feedback stabilization of linear switched systems via switching signal control
Motivated by recent applications in control theory, we study the feedback
stabilizability of switched systems, where one is allowed to chose the
switching signal as a function of in order to stabilize the system. We
propose new algorithms and analyze several mathematical features of the problem
which were unnoticed up to now, to our knowledge. We prove complexity results,
(in-)equivalence between various notions of stabilizability, existence of
Lyapunov functions, and provide a case study for a paradigmatic example
introduced by Stanford and Urbano.Comment: 19 pages, 3 figure
Robust Stability Analysis of Nonlinear Hybrid Systems
We present a methodology for robust stability analysis of nonlinear hybrid systems, through the algorithmic construction of polynomial and piecewise polynomial Lyapunov-like functions using convex optimization and in particular the sum of squares decomposition of multivariate polynomials. Several improvements compared to previous approaches are discussed, such as treating in a unified way polynomial switching surfaces and robust stability analysis for nonlinear hybrid systems
Network synchronization: Spectral versus statistical properties
We consider synchronization of weighted networks, possibly with asymmetrical
connections. We show that the synchronizability of the networks cannot be
directly inferred from their statistical properties. Small local changes in the
network structure can sensitively affect the eigenvalues relevant for
synchronization, while the gross statistical network properties remain
essentially unchanged. Consequently, commonly used statistical properties,
including the degree distribution, degree homogeneity, average degree, average
distance, degree correlation, and clustering coefficient, can fail to
characterize the synchronizability of networks
Non-equilibrium almost-stationary states and linear response for gapped quantum systems
We prove the validity of linear response theory at zero temperature for
perturbations of gapped Hamiltonians describing interacting fermions on a
lattice. As an essential innovation, our result requires the spectral gap
assumption only for the unperturbed Hamiltonian and applies to a large class of
perturbations that close the spectral gap. Moreover, we prove formulas also for
higher order response coefficients.
Our justification of linear response theory is based on a novel extension of
the adiabatic theorem to situations where a time-dependent perturbation closes
the gap. According to the standard version of the adiabatic theorem, when the
perturbation is switched on adiabatically and as long as the gap does not
close, the initial ground state evolves into the ground state of the perturbed
operator. The new adiabatic theorem states that for perturbations that are
either slowly varying potentials or small quasi-local operators, once the
perturbation closes the gap, the adiabatic evolution follows non-equilibrium
almost-stationary states (NEASS) that we construct explicitly.Comment: v1->v2 section 4 on linear response added, presentation partly
reworked. v2->v3 slightly stronger statements for "fast" switching. Final
version as to appear in CM
Model Reduction Near Periodic Orbits of Hybrid Dynamical Systems
We show that, near periodic orbits, a class of hybrid models can be reduced
to or approximated by smooth continuous-time dynamical systems. Specifically,
near an exponentially stable periodic orbit undergoing isolated transitions in
a hybrid dynamical system, nearby executions generically contract
superexponentially to a constant-dimensional subsystem. Under a non-degeneracy
condition on the rank deficiency of the associated Poincare map, the
contraction occurs in finite time regardless of the stability properties of the
orbit. Hybrid transitions may be removed from the resulting subsystem via a
topological quotient that admits a smooth structure to yield an equivalent
smooth dynamical system. We demonstrate reduction of a high-dimensional
underactuated mechanical model for terrestrial locomotion, assess structural
stability of deadbeat controllers for rhythmic locomotion and manipulation, and
derive a normal form for the stability basin of a hybrid oscillator. These
applications illustrate the utility of our theoretical results for synthesis
and analysis of feedback control laws for rhythmic hybrid behavior
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