Showing papers on "Uniform boundedness published in 2022"

Prescribed-Time Robust Differentiator Design Using Finite Varying Gains

Abstract: A novel hybrid differentiator is proposed for any time-varying signal, whose second derivative is uniformly bounded. The exact real-time differentiation is obtained in prescribed time, and it is based on the robust observer design for the perturbed double integrator. The proposed observer strategy is in successive applications of rescaled and standard supertwisting observers with finite (time-varying and respectively constant) gains. The former observer aims to nullify the observation error dynamics in prescribed time whereas the latter observer is to extend desired robustness features to the infinite horizon. The resulting real-time differentiator uses the current signal measurement only and inherits the observer features of robust convergence to the estimated signal derivative in prescribed time regardless of the initial differentiator state. Tuning conditions to achieve the exact signal differentiation in prescribed time are explicitly derived. Theoretical results are supported by an experimental study of the exact prescribed-time velocity estimation of an oscillating pendulum, operating under uniformly bounded disturbances. The developed approach is additionally discussed to admit an extension to the sequential arbitrary order differentiation.

Asymptotics Near Extinction for Nonlinear Fast Diffusion on a Bounded Domain

Abstract: Abstract On a smooth bounded Euclidean domain, Sobolev-subcritical fast diffusion with vanishing boundary trace is known to lead to finite-time extinction, with a vanishing profile selected by the initial datum. In rescaled variables, we quantify the rate of convergence to this profile uniformly in relative error, showing the rate is either exponentially fast (with a rate constant predicted by the spectral gap), or algebraically slow (which is only possible in the presence of non-integrable zero modes). In the first case, the nonlinear dynamics are well-approximated by exponentially decaying eigenmodes up to at least twice the gap; this refines and confirms a 1980 conjecture of Berryman and Holland. We also improve on a result of Bonforte and Figalli by providing a new and simpler approach which is able to accommodate the presence of zero modes, such as those that occur when the vanishing profile fails to be isolated (and possibly belongs to a continuum of such profiles).

A Uniform Sobolev Inequality for Ancient Ricci Flows with Bounded Nash Entropy

Abstract: Abstract This note is a continuation of [7]. We shall show that an ancient Ricci flow with uniformly bounded Nash entropy also has uniformly bounded $ u $-functional. Consequently, on such an ancient solution, there are uniform logarithmic Sobolev and Sobolev inequalities. We emphasize that the main theorem in this paper is true so long as the theory in [3] is valid, and in particular, when the underlying manifold is closed.

Fixed-Time Prescribed Performance Adaptive Fixed-Time Sliding Mode Control for Vehicular Platoons With Actuator Saturation

Abstract: This paper investigates a prescribed tracking performance vehicular platoon control problem with actuator saturation, dynamics uncertainties, and unknown disturbances. First, a novel Gaussian error function (GEF)-based saturation function is introduced to approximate nonsmooth actuator saturation of each vehicle in a smooth way. Then, a fixed-time performance function (FPF) is proposed. Subsequently, an adaptive fixed-time sliding mode control scheme is developed based on the GEF and the FPF, which guarantees all signals of the closed-loop system are bounded and the tracking error converges to a predetermined region in fixed time. Meanwhile, string stability and traffic stability are also guaranteed. Finally, the effectiveness of the proposed algorithm is verified through numerical simulations.

On a generalized Aviles-Giga functional: compactness, zero-energy states, regularity estimates and energy bounds

Abstract: Abstract Given any strictly convex norm on that is C 1 in we study the generalized Aviles-Giga functional for and satisfying Using, as in the euclidean case the concept of entropies for the limit equation we obtain the following. First, we prove compactness in Lp of sequences of bounded energy. Second, we prove rigidity of zero-energy states (limits of sequences of vanishing energy), generalizing and simplifying a result by Bochard and Pegon. Third, we obtain optimal regularity estimates for limits of sequences of bounded energy, in terms of their entropy productions. Fourth, in the case of a limit map in BV, we show that lower bound provided by entropy productions and upper bound provided by one-dimensional transition profiles are of the same order. The first two points are analogous to what is known in the euclidean case and the last two points are sensitive to the anisotropy of the norm

Fixed-Time Bounded H Infinity Tracking Control of a Single-Joint Manipulator System with Input Saturation

Abstract: Based on Lyapunov finite-time stability theory and backstepping strategy, we put forward a novel fixed-time bounded H infinity tracking control scheme for a single-joint manipulator system with input saturation. The main control objective is to maintain that the system output variable tracks the desired signal at fixed time. The advantages of this paper are the settling time of the tracking error converging to the origin is independent of the initial conditions, and its convergence speed is more faster. Meanwhile, bounded H infinity control is adopted to suppress the influence of external disturbances on the controlled system. At the same time, the problem of input saturation control is considered, which effectively reduces the input energy consumption. Theoretical analysis shows that the tracking error of the closed-loop system converges to a small neighbourhood of the origin within a fixed time. In the end, a simulation example is presented to demonstrate the effectiveness of the proposed scheme.

Critical Transitions in Piecewise Uniformly Continuous Concave Quadratic Ordinary Differential Equations

Abstract: Abstract A critical transition for a system modelled by a concave quadratic scalar ordinary differential equation occurs when a small variation of the coefficients changes dramatically the dynamics, from the existence of an attractor–repeller pair of hyperbolic solutions to the lack of bounded solutions. In this paper, a tool to analyze this phenomenon for asymptotically nonautonomous ODEs with bounded uniformly continuous or bounded piecewise uniformly continuous coefficients is described, and used to determine the occurrence of critical transitions for certain parametric equations. Some numerical experiments contribute to clarify the applicability of this tool.

Boundedness and Asymptotic Stabilization in a Two-Dimensional Keller-Segel-Navier-Stokes System with Sub-Logistic Source

Abstract: This paper mainly deals with a Keller–Segel–Navier–Stokes model with sub-logistic source in a two-dimensional bounded and smooth domain. For a large class of cell kinetics including sub-logistic sources, it is shown that under an explicit condition involving the chemotactic strength, asymptotic “damping” rate and initial mass of cells, the associated no-flux/no-flux/Dirichlet problem possesses a global and bounded classical solution. Moreover, a systematical treatment has been conducted on convergence of bounded solutions toward constant equilibrium in [Formula: see text] for sub- and standard logistic sources. In such chemotaxis-fluid setting, our boundedness improves known blow-up prevention by logistic source to blow-up prevention by sub-logistic source, indicating standard logistic source is not the weakest damping source to prevent blow-up, and our stability improves known algebraic convergence under quadratic degradation to exponential convergence under log-correction of quadratic degradation, implying log-correction of quadratic degradation quickens the decay of bounded solutions. These findings significantly improve and extend previously known ones.

Regulating Constraint-Following Bound for Fuzzy Mechanical Systems: Indirect Robust Control and Fuzzy Optimal Design

Abstract: This article proposes an optimal indirect approach of constraint-following control for fuzzy mechanical systems. The system contains (possibly fast) time-varying uncertainty that lies in a fuzzy set. It aims at an optimal controller for the system to render bounded constraint-following error such that it can stay within a predetermined bound at all time and be sufficiently small eventually. First, for deterministic performance, the original system is transformed into a constructed system. A deterministic (not the usual IF-THEN rules-based) robust control is then designed for the constructed system to render it to be uniformly bounded and uniformly ultimately bounded, regardless of the uncertainty. Second, for optimal performance, a performance index, including the average fuzzy system performance and control effort, is proposed based on the fuzzy information. An optimal design problem associated with the control gain is then formulated and solved by minimizing the performance index. Finally, it is proved when the constructed system renders uniform boundedness and uniform ultimate boundedness, the original system achieves the desired performance of bounded constraint following.

Temporal decays and asymptotic behaviors for a vlasov equation with a flocking term coupled to incompressible fluid flow

Abstract: We are concerned with large-time behaviors of solutions for Vlasov–Navier–Stokes equations in two dimensions and Vlasov–Stokes system in three dimensions including the effect of velocity alignment/misalignment. We first revisit the large-time behavior estimate for our main system and refine assumptions on the dimensions and a communication weight function. In particular, this allows us to take into account the effect of the misalignment interactions between particles. We then use a sharp heat kernel estimate to obtain the exponential time decay of fluid velocity to its average in L ∞ -norm. For the kinetic part, by employing a certain type of Sobolev norm weighted by modulations of averaged particle velocity, we prove the exponential time decay of the particle distribution, provided that local particle distribution function is uniformly bounded. Moreover, we show that the support of particle distribution function in velocity shrinks to a point, which is the mean of averaged initial particle and fluid velocities, exponentially fast as time goes to infinity. This also provides that for any p ∈ [ 1 , ∞ ] , the p -Wasserstein distance between the particle distribution function and the tensor product of the local particle distributions and Dirac measure at that point in velocity converges exponentially fast to zero as time goes to infinity.

Edgeworth expansions for integer valued additive functionals of uniformly elliptic Markov chains

Abstract: We obtain asymptotic expansions for probabilities $\mathbb{P}(S_N=k)$ of partial sums of uniformly bounded integer-valued functionals $S_N=\sum_{n=1}^N f_n(X_n)$ of uniformly elliptic inhomogeneous Markov chains. The expansions involve products of polynomials and trigonometric polynomials, and they hold without additional assumptions. As an application of the explicit formulas of the trigonometric polynomials, we show that for every $r\geq1\,$, $S_N$ obeys the standard Edgeworth expansions of order $r$ in a conditionally stable way if and only if for every $m$, and every $\ell$ the conditional distribution of $S_N$ given $X_{j_1},...,X_{j_\ell}$ mod $m$ is $o_\ell(\sigma_N^{1-r})$ close to uniform, uniformly in the choice of $j_1,...,j_\ell$, where $\sigma_N=\sqrt{\text{Var}(S_N)}.$

Sharp moment estimates for martingales with uniformly bounded square functions

Abstract: We provide sharp bounds for the exponential moments and $p$-moments, $1\leqslant p \leqslant 2$, of the terminate distribution of a martingale whose square function is uniformly bounded by one. We introduce a Bellman function for the corresponding extremal problem and reduce it to the already known Bellman function on $\mathrm{BMO}([0,1])$. In the case of tail estimates, a similar reduction does not work exactly, so we come up with a fine supersolution that leads to sharp tail estimates.

Variations of generalised pairs

Abstract: In this paper we investigate various properties of generalised pairs in families, especially boundedness of several kinds. We show that many statements for usual pairs do not hold for generalised pairs. In particular, we construct an unexpected counter-example to boundedness of generalised lc models with fixed appropriate invariants. We also show that the DCC of Iitaka volumes and existence of nef reduction maps fail in families of generalised pairs. In a positive direction we show boundedness of bases of log Calabi-Yau fibrations with their induced generalised pair structure under natural assumptions. Roughly speaking we prove this boundedness for fibrations whose general fibres belong to a bounded family and whose Iitaka volume is fixed.

Bubble-Tree Convergence and Local Diffeomorphism Finiteness for Gradient Ricci Shrinkers

Abstract: We prove bubble-tree convergence of sequences of gradient Ricci shrinkers with uniformly bounded entropy and uniform local energy bounds, refining the compactness theory of Haslhofer-Mueller. In particular, we show that no energy concentrates in neck regions, a result which implies a local energy identity for the sequence. Direct consequences of these results are an identity for the Euler characteristic and a local diffeomorphism finiteness theorem.

Strong transience for one-dimensional Markov chains with asymptotically zero drifts

Abstract: For near-critical, transient Markov chains on the non-negative integers in the Lamperti regime, where the mean drift at $x$ decays as $1/x$ as $x \to \infty$, we quantify degree of transience via existence of moments for conditional return times and for last exit times, assuming increments are uniformly bounded. Our proof uses a Doob $h$-transform, for the transient process conditioned to return, and we show that the conditioned process is also of Lamperti type with appropriately transformed parameters. To do so, we obtain an asymptotic expansion for the ratio of two return probabilities, evaluated at two nearby starting points; a consequence of this is that the return probability for the transient Lamperti process is a regularly-varying function of the starting point.

Proximinality and uniformly approximable sets in $L^p$

Abstract: For any $p\in[1,\infty]$, we prove that the set of simple functions taking at most $k$ different values is proximinal in $L^p$ for all $k\geq 1$. We introduce the class of uniformly approximable subsets of $L^p$, which is larger than the class of uniformly integrable sets. This new class is characterized in terms of the $p$-variation if $p\in[1,\infty)$ and in terms of covering numbers if $p=\infty$. We study properties of uniformly approximable sets. In particular, we prove that the convex hull of a uniformly approximable bounded set is also uniformly approximable and that this class is stable under H\"older transformations. We also prove that, for $p\in [1,\infty)$, the unit ball of $L^p$ is uniformly approximable if and only if $L^p$ is finite-dimensional, while for $p=\infty$ the unit ball is always uniformly approximable.

On Discretization Methods for Indirect Adaptive Sliding Mode Control

Abstract: In this paper, discrete-time versions of the first-order indirect adaptive sliding mode control (SMC) algorithm are developed and analyzed. In particular, a scalar linear system with parametric uncertainty linear in the state and with bounded input disturbance is considered. The discretization of continuous-time first-order indirect adaptive SMC algorithm may not preserve continuous-time properties like asymptotic stability of the origin. In order to ensure boundedness, knowledge about where the actual parameter may be located is used in this work. Then, for each proposed discretization scheme, conditions on the sampling time are derived that guarantee convergence into a bounded set. The theoretical results are illustrated with simulation examples and the proposed discretization schemes are compared with regard to the condition on sampling time and precision.

A Berry–Esseen theorem and Edgeworth expansions for uniformly elliptic inhomogeneous Markov chains

Abstract: We prove a Berry–Esseen theorem and Edgeworth expansions for partial sums of the form $$\displaystyle S_N=\sum olimits _{n=1}^{N}f_n(X_n,X_{n+1})$$ , where $$\{X_n\}$$ is a uniformly elliptic inhomogeneous Markov chain and $$\{f_n\}$$ is a sequence of uniformly bounded functions. The Berry–Esseen theorem holds without additional assumptions, while expansions of order 1 hold when $$\{f_n\}$$ is irreducible, which is an optimal condition. For higher order expansions, we then focus on two situations. The first is when the essential supremum of $$f_n$$ is of order $$O(n^{-{\beta }})$$ for some $${\beta }\in (0,1/2)$$ . In this case it turns out that expansions of any order $$r<\frac{1}{1-2{\beta }}$$ hold, and this condition is optimal. The second case is uniformly elliptic chains on a compact Riemannian manifold. When $$f_n$$ are uniformly Lipschitz continuous we show that $$S_N$$ admits expansions of all orders. When $$f_n$$ are uniformly Hölder continuous with some exponent $${\alpha }\in (0,1)$$ , we show that $$S_N$$ admits expansions of all orders $$r<\frac{1+{\alpha }}{1-{\alpha }}$$ . For Hölder continues functions with $${\alpha }<1$$ our results are new also for uniformly elliptic homogeneous Markov chains and a single functional $$f=f_n$$ . In fact, we show that the condition $$r<\frac{1+{\alpha }}{1-{\alpha }}$$ is optimal even in the homogeneous case.

A uniform bound on costs of controlling semilinear heat equations on a sequence of increasing domains and its application

Abstract: In this paper, we first prove a uniform upper bound on costs of null controls for semilinear heat equations with globally Lipschitz nonlinearity on a sequence of increasing domains, where the controls are acted on an equidistributed set that spreads out in the whole Euclidean space R N . As an application, we then show the exact null-controllability for this semilinear heat equation in R N . The main novelty here is that the upper bound on costs of null controls for such kind of equations in large but bounded domains can be made uniformly with respect to the sizes of domains under consideration. The latter is crucial when one uses a suitable approximation argument to derive the global null-controllability for the semilinear heat equation in R N . This allows us to overcome the well-known problem of the lack of compactness embedding arising in the study of null-controllability for nonlinear PDEs in generally unbounded domains.

Asymptotics near extinction for nonlinear fast diffusion on a bounded domain

Abstract: On a smooth bounded Euclidean domain, Sobolev-subcritical fast diffusion with vanishing boundary trace is known to lead to finite-time extinction, with a vanishing profile selected by the initial datum. In rescaled variables, we quantify the rate of convergence to this profile uniformly in relative error, showing the rate is either exponentially fast (with a rate constant predicted by the spectral gap), or algebraically slow (which is only possible in the presence of non-integrable zero modes). In the first case, the nonlinear dynamics are well-approximated by exponentially decaying eigenmodes up to at least twice the gap; this refines and confirms a 1980 conjecture of Berryman and Holland. We also improve on a result of Bonforte and Figalli, by providing a new and simpler approach which is able to accommodate the presence of zero modes, such as those that occur when the vanishing profile fails to be isolated (and possibly belongs to a continuum of such profiles).