Showing papers on "Uniform boundedness published in 2004"

A constraint‐stabilized time‐stepping approach for rigid multibody dynamics with joints, contact and friction

Abstract: We present a method for achieving geometrical constraint stabilization for a linear-complementarity-based time-stepping scheme for rigid multibody dynamics with joints, contact, and friction. The method requires the solution of only one linear complementarity problem per step. We prove that the velocity stays bounded and that the constraint infeasibility is uniformly bounded in terms of the size of the time step and the current value of the velocity. Several examples, including one for joint-only systems, are used to demonstrate the constraint stabilization effect. Copyright © 2004 John Wiley & Sons, Ltd.

Obstructions to the Existence of Universal Stabilizers for Smooth Control Systems

Abstract: This paper states necessary conditions for the existence of “universal stabilizers” for smooth control systems. Roughly speaking, given a control system and a set {\cal U} of reference input functions, by “universal stabilizer” we mean a continuous feedback law that stabilizes the state of the system asymptotically to any of the reference trajectories produced by (arbitrary) inputs in {\cal U}. For an example, consider Brockett’s nonholonomic integrator, with {\cal U} representing a set of uniformly bounded, piecewise continuous functions of time. This system’s state can be asymptotically stabilized to any reference trajectory provided the latter is persistently exciting (PE). By contrast, for constant trajectories (i.e., equilibria), which are not PE, asymptotic stabilization is impossible by means of continuous pure-state feedback, in view of Brockett’s obstruction. However, since this obstruction can be circumvented by the use of time-varying state feedback, one might reasonably expect to be able to design a (time-varying) continuous control law capable of asymptotically stabilizing the state to arbitrary reference trajectories, be they PE or not. Surprisingly, a consequence of the results in this paper is that, for systems with nonholonomic constraints frequently found in control applications, if {\cal U} contains reference functions that are not PE, then the “universal stabilization” problem cannot be solved, even if time-varying feedback is used.

Uniqueness for the two-dimensional Navier-Stokes equation with a measure as initial vorticity

Abstract: We show that any solution of the two-dimensional Navier-Stokes equation whose vorticity distribution is uniformly bounded in $L^1(R^2)$ for positive times is entirely determined by the trace of the vorticity at $t = 0$, which is a finite measure. When combined with previous existence results by Cottet, by Giga, Miyakawa, and Osada, and by Kato, this uniqueness property implies that the Cauchy problem for the vorticity equation in $R^2$ is globally well-posed in the space of finite measures. In particular, this provides an example of a situation where the Navier-Stokes equation is well-posed for arbitrary data in a function space that is large enough to contain the initial data of some self-similar solutions.

Random attractor for a damped sine-Gordon equation with white noise

Abstract: We prove the existence of a compact random attractor for the random dynamical system generated by a damped sine-Gordon with white noise. And we obtain a precise estimate of the upper bound of the Hausdorff dimension of the random attractor, which decreases as the damping grows and shows that the dimension is uniformly bounded for the damping. In particular, under certain conditions, the dimension is zero.

Rotating Fluids with Self-Gravitation in Bounded Domains

Abstract: In this paper, we study the steady solutions of Euler-Poisson equations in bounded domains with prescribed angular velocity. This models a rotating Newtonian star consisting of a compressible perfect fluid with given equation of state P=eSργ. When the domain is a ball and the angular velocity is constant, we obtain both existence and non-existence theorems, depending on the adiabatic gas constant γ. In addition we obtain some interesting properties of the solutions; e.g., monotonicity of the radius of the star with both angular velocity and central density. We also prove that the radius of a rotating spherically symmetric star, with given constant angular velocity and constant entropy, is uniformly bounded independent of the central density. This is physically striking and in sharp contrast to the case of the non-rotating star. For general domains and variable angular velocities, both an existence result for the isentropic equations of state and non-existence result for the non-isentropic equation of state are also obtained.

Moment conditions for a sequence with negative drift to be uniformly bounded in L^r

Abstract: Suppose a sequence of random variables {X_n} has negative drift when above a certain threshold and has increments bounded in L^p. When p>2 this implies that EX_n is bounded above by a constant independent of n and the particular sequence {X_n}. When p= =p-1. These results are motivated by questions about stability of queueing networks.

A Critical Points Theorem and Nonlinear Differential Problems

Abstract: The existence of two intervals of positive real parameters λ for which the functional Φ + λΨ has three critical points, whose norms are uniformly bounded in respect to λ belonging to one of the two intervals, is established. As an example of an application to nonlinear differential problems, a two point boundary value problem is considered and multiplicity results are obtained.

Transfer Functions of Regular Linear Systems Part III: Inversions and Duality

Abstract: We study four transformations which lead from one well-posed lin- ear system to another: time-inversion, flow-inversion, time-flow-inversion and duality. Time-inversion means reversing the direction of time, flow-inversion means interchanging inputs with outputs, while time-flow-inversion means do- ing both of the inversions mentioned before. A well-posed linear system Σ is time-invertible if and only if its operator semigroup extends to a group. The system Σ is flow-invertible if and only if its input-output map has a bounded inverse on some (hence, on every) finite time interval (0 ,τ )( τ> 0). This is true if and only if the transfer function of Σ has a uniformly bounded inverse on some right half-plane. The system Σ is time-flow-invertible if and only if on some (hence, on every) finite time interval (0 ,τ ), the combined operator Στ from the initial state and the input function to the final state and the output function is invertible. This is the case, for example, if the system is conser- vative, since then Στ is unitary. Time-flow-inversion can sometimes, but not always, be reduced to a combination of time- and flow-inversion. We derive a surprising necessary and sufficient condition for Σ to be time-flow-invertible: its system operator must have a uniformly bounded inverse on some left half- plane. Finally, the duality transformation is always possible. We show by some examples that none of these transformations preserves regularity in general. However, the duality transformation does preserve weak regularity. For all the transformed systems mentioned above, we give formulas for their system operators, transfer functions and, in the regular case and under additional assumptions, for their generating operators.

Event diagnosis of discrete-event systems with uniformly and nonuniformly bounded diagnosis delays

Abstract: Various notions of diagnosability reported in literature deal with uniformly bounded finite detection or counting delays. The uniformity of delays can be relaxed while delays remain finite. We introduce various notions of diagnosability allowing nonuniformly bounded finite delays. A polynomial-time verification algorithm for diagnosability with nonuniformly bounded finite indefinite-counting delays is presented. A similar technique is applied to give a computationally better verification algorithm for diagnosability with uniformly bounded finite indefinite-counting delays than algorithms previously reported in literature. Finally, we develop a new on-line diagnosis algorithm that has a lower time and space complexity than on-line diagnosis algorithms reported in literature for counting the occurrence of repeated/intermittent faults.

Uniqueness and bubbling of the 2-dimensional Landau-Lifshitz flow

Abstract: We consider the Landau-Lifshitz flow on a bounded planar domain. An $\epsilon$ -regularity type a-priori estimate provides the analytic tool for the subsequent geometric description of the flow at isolated singularities. At forward isolated singularities where the energy is not left continuous the flow concentrates energy and develops bubbles. As in J.Qing’s bubbling-energy-equality for the harmonic map flow, the energy loss at such a singularity can be recovered as a finite sum of energies of tangent bubbles. We then clarify a known uniqueness result for the Landau-Lifshitz flow and show how non-uniqueness of extensions of the flow after point singularities is related to backward bubbling. Finally the $\epsilon$ -regularity estimate also yields a partial compactness result for sequences of smooth solutions to the Landau-Lifshitz flow with uniformly bounded energy, defined on a planar domain.

Optimal Growth and Impatience: A Phase Diagram Analysis

Abstract: In this paper we show that we can replace the assumption of constant discount rate in the one-sector optimal growth model with the assumption of decreasing marginal impatience without losing major properties of the model. In particular, we show that the steady state exists, is unique, and has a saddle-point property. All we need is to assume that the discount function is strictly decreasing, strictly convex and has a uniformly bounded first-derivative.

Boundedness properties of nonlinear quasi-dissipative systems

Abstract: Boundedness results for feedback interconnections of quasi-dissipative systems are presented. A classical result due to Hill and Moylan is generalized to the case of finite power gain stability of feedback interconnection of quasi-dissipative systems. It is also shown that under the assumption of strong finite-time detectability, such an interconnection has uniformly ultimately bounded trajectories for any uniformly bounded input.

Finite Element Methods for a Modified Reissner-Mindlin Free Plate Model

Abstract: The solution of the Reissner--Mindlin plate problem with free boundary conditions presents a strong layer effect near the free edges. As a consequence, the solution is not even uniformly bounded even in H3/2, which implies that at most an O(h1/2) uniform convergence rate can be reached by finite element methods in the H1 norm. Following instead the modified free boundary model presented by Beirao da Veiga and Brezzi, which gives more regular solutions, better error estimates can be obtained in principle. In this paper we present and analyze the extension of different families of well-known optimal plate methods to this new model. All the modified methods presented are proved to be optimal and free of locking.

On the existence of Pettis integrable functions which are not Birkhoff integrable

Abstract: Let X be a weakly Lindelof determined Banach space. We prove that if X is non-separable, then there exist a complete probability space (Ω,Σ,μ) and a bounded Pettis integrable function f: Ω → X that is not Birkhoff integrable; when the density character of X is greater than or equal to the continuum, then f is defined on [0,1] with the Lebesgue measure. Moreover, in the particular case X = c 0 (I) (the cardinality of I being greater than or equal to the continuum) the function f can be taken as the pointwise limit of a uniformly bounded sequence of Birkhoff integrable functions, showing that the analogue of Lebesgue's dominated convergence theorem for the Birkhoff integral does not hold in general.

Study of lower bound functions for MAX-2-SAT

Abstract: Recently. several lower bound functions are proposed for solving the MAX-2-SAT problem optimally in a branch-and-bound algorithm. These lower bounds improve significantly the performance of these algorithms. Based on the study of these lower bound functions, we propose a new, linear-time lower bound function. We show that the new lower bound function is admissible and it is consistently and substantially better than other known lower bound functions. The result of this study is a high-performance implementation of an exact algorithm for MAX-2-SAT which outperforms any implementation of the same class.

Multigrid algorithm for the cell‐centered finite difference method II: discontinuous coefficient case

Abstract: We consider a multigrid algorithm for the cell centered finite difference scheme with a prolongation operator depending on the diffusion coefficient. This prolongation operator is designed mainly for solving diffusion equations with strong varying or discontinuous coefficient and it reduces to the usual bilinear interpolation for Laplace equation. For simple interface problem, we show that the energy norm of this operator is uniformly bounded by 11/8, no matter how large the jump is, from which one can prove that W-cycle with one smoothing converges with reduction factor independent of the size of jump using the theory developed by Bramble et al. (Math Comp 56 (1991), 1–34). For general interface problem, we show that the energy norm is bounded by some constant C* (independent of the jumps of the coefficient). In this case, we can conclude W-cycle converges with sufficiently many smoothings. Numerical experiment shows that even V-cycle multigrid algorithm with our prolongation works well for various interface problems. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004.

Bounded-from-below solutions of the Hamilton-Jacobi equation for optimal control problems with exit times: vanishing lagrangians, eikonal equations, and shape-from-shading

Abstract: We study the Hamilton-Jacobi equation for undiscounted exit time control problems with general nonnegative Lagrangians using the dynamic programming approach. We prove theorems characterizing the value function as the unique bounded-from-below viscosity solution of the Hamilton-Jacobi equation which is null on the target. The result applies to problems with the property that all trajectories satisfying a certain integral condition must stay in a bounded set. We allow problems for which the Lagrangian is not uniformly bounded below by positive constants, in which the hypotheses of the known uniqueness results for Hamilton-Jacobi equations are not satised. We apply our theorems to eikonal equations from geometric optics, shapefrom-shading equations from image processing, and variants of the Fuller Problem.

Asymptotic behavior and time discretization analysis for the non-stationary Navier-Stokes problem

Abstract: The asymptotic behavior and the Euler time discretization analysis are presented for the two-dimensional non-stationary Navier-Stokes problem. If the data ν and f(t) satisfy a uniqueness condition corresponding to the stationary Navier-Stokes problem, we then obtain the convergence of the non-stationary Navier-Stokes problem to the stationary Navier-Stokes problem and the uniform boundedness, stability and error estimates of the Euler time discretization for the non-stationary Navier-Stokes problem.

A semilinear parabolic boundary-value problem in bioreactors theory

Abstract: We consider a Plug Flow bioreactor with axial dispersion in which occurs a simple growth reaction (one biomass/one substrate). The dynamics of this bioreactor are described by a system of partial differential equations. This work is devoted to the analysis of this system: we aim at proving uniform boundedness of solutions and describing their omega-limit sets. To ease this analysis, we perform a linear change of state variables which transforms the system into two equations. One of them is nonlinear but the other one is linear. Thanks to the properties of the linear equation, we proved that the new system is equivalent to a nonautonomous semilinear parabolic equation of type du dt = Au(t) + f(t, u),

Robust Adaptive Control of a Cantilevered Flexible Structure with Spatiotemporally Varying Coefficients and Bounded Disturbance

Abstract: In this paper, a robust model reference adaptive control of a cantilevered flexible structure with unknown spatiotemporally varying coefficients and disturbance is investigated. Any mechanically flexible manipulators/structures are inherently distributed parameter systems whose dynamics are described by partial, rather than ordinary, differential equations. Robust adaptive control laws are derived by the Lyapunov redesign method on an infinite dimensional Hilbert space. Under the assumption that disturbances are uniformly bounded, the proposed robust adaptive scheme guarantees the boundedness of all signals in the closed loop system and the convergence of the state error near to zero. With an additional persistence of excitation condition, the parameter estimation errors are shown to converge near to zero as well.

Minimum distance between bent and 1-resilient Boolean functions

Abstract: In this paper we study the minimum distance between the set of bent functions and the set of 1-resilient Boolean functions and present a lower bound on that. The bound is proved to be tight for functions up to 10 input variables. As a consequence, we present a strategy to modify the bent functions, by toggling some of its outputs, in getting a large class of 1-resilient functions with very good nonlinearity and autocorrelation. In particular, the technique is applied upto 12-variable functions and we show that the construction provides a large class of 1-resilient functions reaching currently best known nonlinearity and achieving very low autocorrelation values which were not known earlier. The technique is sound enough to theoretically solve some of the mysteries of 8-variable, 1-resilient functions with maximum possible nonlinearity. However, the situation becomes complicated from 10 variables and above, where we need to go for complicated combinatorial analysis with trial and error using computational facility.

Absence of super-exponentially decaying eigenfunctions on Riemannian manifolds with pinched negative curvature

Abstract: Let (X,g) be a metrically complete, simply connected Riemannian manifold with bounded geometry and pinched negative curvature, i.e. there are constants a>b>0 such that -a^2

Convergence of the Ricci flow toward a soliton

Abstract: We will consider a {\it $\tau$-flow}, given by the equation $\frac{d}{dt}g_{ij} = -2R_{ij} + \frac{1}{\tau}g_{ij}$ on a closed manifold $M$, for all times $t\in [0,\infty)$. We will prove that if the curvature operator and the diameter of $(M,g(t))$ are uniformly bounded along the flow, then we have a sequential convergence of the flow toward the solitons.

Minimum Distance between Bent and 1-Resilient Boolean Functions

Abstract: In this paper we study the minimum distance between the set of bent functions and the set of 1-resilient Boolean functions and present a lower bound on that. The bound is proved to be tight for functions up to 10 input variables. As a consequence, we present a strategy to modify the bent functions, by toggling some of its outputs, in getting a large class of 1-resilient functions with very good nonlinearity and autocorrelation. In particular, the technique is applied upto 12-variable functions and we show that the construction provides a large class of 1-resilient functions reaching currently best known nonlinearity and achieving very low autocorrelation values which were not known earlier. The technique is sound enough to theoretically solve some of the mysteries of 8-variable, 1-resilient functions with maximum possible nonlinearity. However, the situation becomes complicated from 10 variables and above, where we need to go for complicated combinatorial analysis with trial and error using computational facility.