3,098 research outputs found
Stability analysis of a general class of singularly perturbed linear hybrid systems
Motivated by a real problem in steel production, we introduce and analyze a
general class of singularly perturbed linear hybrid systems with both switches
and impulses, in which the slow or fast nature of the variables can be
mode-dependent. This means that, at switching instants, some of the slow
variables can become fast and vice-versa. Firstly, we show that using a
mode-dependent variable reordering we can rewrite this class of systems in a
form in which the variables preserve their nature over time. Secondly, we
establish, through singular perturbation techniques, an upper bound on the
minimum dwell-time ensuring the overall system's stability. Remarkably, this
bound is the sum of two terms. The first term corresponds to an upper bound on
the minimum dwell-time ensuring the stability of the reduced order linear
hybrid system describing the slow dynamics. The order of magnitude of the
second term is determined by that of the parameter defining the ratio between
the two time-scales of the singularly perturbed system. We show that the
proposed framework can also take into account the change of dimension of the
state vector at switching instants. Numerical illustrations complete our study
A parameter uniform fitted mesh method for a weakly coupled system of two singularly perturbed convection-diffusion equations
In this paper, a boundary value problem for a singularly perturbed linear
system of two second order ordinary differential equations of convection-
diffusion type is considered on the interval [0, 1]. The components of the
solution of this system exhibit boundary layers at 0. A numerical method
composed of an upwind finite difference scheme applied on a piecewise uniform
Shishkin mesh is suggested to solve the problem. The method is proved to be
first order convergent in the maximum norm uniformly in the perturbation
parameters. Numerical examples are provided in support of the theory
An integrated approach to global synchronization and state estimation for nonlinear singularly perturbed complex networks
This paper aims to establish a unified framework to handle both the exponential synchronization and state estimation problems for a class of nonlinear singularly perturbed complex networks (SPCNs). Each node in the SPCN comprises both 'slow' and 'fast' dynamics that reflects the singular perturbation behavior. General sector-like nonlinear function is employed to describe the nonlinearities existing in the network. All nodes in the SPCN have the same structures and properties. By utilizing a novel Lyapunov functional and the Kronecker product, it is shown that the addressed SPCN is synchronized if certain matrix inequalities are feasible. The state estimation problem is then studied for the same complex network, where the purpose is to design a state estimator to estimate the network states through available output measurements such that dynamics (both slow and fast) of the estimation error is guaranteed to be globally asymptotically stable. Again, a matrix inequality approach is developed for the state estimation problem. Two numerical examples are presented to verify the effectiveness and merits of the proposed synchronization scheme and state estimation formulation. It is worth mentioning that our main results are still valid even if the slow subsystems within the network are unstable
The linear quadratic regulator problem for a class of controlled systems modeled by singularly perturbed Ito differential equations
This paper discusses an infinite-horizon linear quadratic (LQ) optimal control problem involving state- and control-dependent noise in singularly perturbed stochastic systems. First, an asymptotic structure along with a stabilizing solution for the stochastic algebraic Riccati equation (ARE) are newly established. It is shown that the dominant part of this solution can be obtained by solving a parameter-independent system of coupled Riccati-type equations. Moreover, sufficient conditions for the existence of the stabilizing solution to the problem are given. A new sequential numerical algorithm for solving the reduced-order AREs is also described. Based on the asymptotic behavior of the ARE, a class of O(√ε) approximate controller that stabilizes the system is obtained. Unlike the existing results in singularly perturbed deterministic systems, it is noteworthy that the resulting controller achieves an O(ε) approximation to the optimal cost of the original LQ optimal control problem. As a result, the proposed control methodology can be applied to practical applications even if the value of the small parameter ε is not precisely known. © 2012 Society for Industrial and Applied Mathematics.Vasile Dragan, Hiroaki Mukaidani and Peng Sh
Delayed loss of stability in singularly perturbed finite-dimensional gradient flows
In this paper we study the singular vanishing-viscosity limit of a gradient
flow in a finite dimensional Hilbert space, focusing on the so-called delayed
loss of stability of stationary solutions. We find a class of time-dependent
energy functionals and initial conditions for which we can explicitly calculate
the first discontinuity time of the limit. For our class of functionals,
coincides with the blow-up time of the solutions of the linearized system
around the equilibrium, and is in particular strictly greater than the time
where strict local minimality with respect to the driving energy gets
lost. Moreover, we show that, in a right neighborhood of , rescaled
solutions of the singularly perturbed problem converge to heteroclinic
solutions of the gradient flow. Our results complement the previous ones by
Zanini, where the situation we consider was excluded by assuming the so-called
transversality conditions, and the limit evolution consisted of strict local
minimizers of the energy up to a negligible set of times
Landscapes of Non-gradient Dynamics Without Detailed Balance: Stable Limit Cycles and Multiple Attractors
Landscape is one of the key notions in literature on biological processes and
physics of complex systems with both deterministic and stochastic dynamics. The
large deviation theory (LDT) provides a possible mathematical basis for the
scientists' intuition. In terms of Freidlin-Wentzell's LDT, we discuss
explicitly two issues in singularly perturbed stationary diffusion processes
arisen from nonlinear differential equations: (1) For a process whose
corresponding ordinary differential equation has a stable limit cycle, the
stationary solution exhibits a clear separation of time scales: an exponential
terms and an algebraic prefactor. The large deviation rate function attains its
minimum zero on the entire stable limit cycle, while the leading term of the
prefactor is inversely proportional to the velocity of the non-uniform periodic
oscillation on the cycle. (2) For dynamics with multiple stable fixed points
and saddles, there is in general a breakdown of detailed balance among the
corresponding attractors. Two landscapes, a local and a global, arise in LDT,
and a Markov jumping process with cycle flux emerges in the low-noise limit. A
local landscape is pertinent to the transition rates between neighboring stable
fixed points; and the global landscape defines a nonequilibrium steady state.
There would be nondifferentiable points in the latter for a stationary dynamics
with cycle flux. LDT serving as the mathematical foundation for emergent
landscapes deserves further investigations.Comment: 4 figur
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