Showing papers on "Uniform boundedness published in 2019"
TL;DR: In this paper, the least squares regression function estimator over the class of real-valued functions on $[0, 1]^{d}$ that are increasing in each coordinate was studied and it was shown that the estimator achieves the minimax rate of order $n^{-\min\{2/(d+2),1/d\}}$ in the empirical loss, up to polylogarithmic factors.
Abstract: We study the least squares regression function estimator over the class of real-valued functions on $[0,1]^{d}$ that are increasing in each coordinate. For uniformly bounded signals and with a fixed, cubic lattice design, we establish that the estimator achieves the minimax rate of order $n^{-\min\{2/(d+2),1/d\}}$ in the empirical $L_{2}$ loss, up to polylogarithmic factors. Further, we prove a sharp oracle inequality, which reveals in particular that when the true regression function is piecewise constant on $k$ hyperrectangles, the least squares estimator enjoys a faster, adaptive rate of convergence of $(k/n)^{\min(1,2/d)}$, again up to polylogarithmic factors. Previous results are confined to the case $d\leq2$. Finally, we establish corresponding bounds (which are new even in the case $d=2$) in the more challenging random design setting. There are two surprising features of these results: first, they demonstrate that it is possible for a global empirical risk minimisation procedure to be rate optimal up to polylogarithmic factors even when the corresponding entropy integral for the function class diverges rapidly; second, they indicate that the adaptation rate for shape-constrained estimators can be strictly worse than the parametric rate.
87 citations
TL;DR: For networks with unknown external disturbances and unmodeled dynamics, neuro-adaptive-based coupling laws are designed to ensure that the synchronization error of the networks with undirected switching communication topologies under these laws is UUB.
Abstract: The evolution of the target system (leader) in pinning-controlled complex networks may need to be regulated by some control inputs for performing various practical tasks, e.g., obstacle avoidance, tracking highly maneuverable target, and so on. Motivated by this observation, we shall investigate the global pinning synchronization problems for complex switching networks for which the target system is subject to nonzero control inputs. First, using the idea of unit vector function method, a discontinuous coupling law is designed. With the aid of stability theory for switched systems, it is theoretically shown that synchronization in the network under this discontinuous coupling law can be achieved by choosing sufficiently large coupling strengths if the average dwell time (ADT) is bounded below by a positive constant. Second, we use the boundary layer method to design a continuous-coupling law. It has been theoretically shown that the synchronization error is ultimately uniformly bounded (UUB) under this continuous-coupling law. The chattering effect is also avoided in real implementation by using this continuous-coupling law. Furthermore, for networks with unknown external disturbances and unmodeled dynamics, neuro-adaptive-based coupling laws are designed to ensure that the synchronization error of the networks with undirected switching communication topologies under these laws is UUB. The obtained theoretical results are finally validated by performing numerical simulation on coupling Chua’s circuit systems.
59 citations
TL;DR: An event-triggered robust fuzzy control scheme, which also releases the constraint of input saturation is developed, and the existence of the minimal inter-event time and the uniform boundedness of all tracking errors in the closed-loop system is proved.
51 citations
TL;DR: A consensus Kalman filtering algorithm based on the leader–follower structure and weighted average strategy for sensor networks is proposed by introducing virtual estimation errors and confidence level functions that are fully distributively and adaptively designed in a proportion form of the sensors’ confidence levels.
44 citations
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TL;DR: In this article, the authors revisited the regularity theory of Escauriaza, Seregin, and Sverak for solutions to the Navier-Stokes equations which are uniformly bounded in the critical $L^3_x(mathbf{R}^3)$ norm.
Abstract: We revisit the regularity theory of Escauriaza, Seregin, and Sverak for solutions to the three-dimensional Navier-Stokes equations which are uniformly bounded in the critical $L^3_x(\mathbf{R}^3)$ norm. By replacing all invocations of compactness methods in these arguments with quantitative substitutes, and similarly replacing unique continuation and backwards uniqueness estimates by their corresponding Carleman inequalities, we obtain quantitative bounds for higher regularity norms of these solutions in terms of the critical $L^3_x$ bound (with a dependence that is triple exponential in nature). In particular, we show that as one approaches a finite blowup time $T_*$, the critical $L^3_x$ norm must blow up at a rate $(\log\log\log \frac{1}{T_*-t})^c$ or faster for an infinite sequence of times approaching $T_*$ and some absolute constant $c>0$.
37 citations
TL;DR: In this paper, a class of reaction diffusion systems of which weak solution exists global-in-time with relatively compact orbit in L 1 is introduced, and the reaction term in this class is quasi-positive, dissipative, and up to with quadratic growth rate.
Abstract: We introduce a class of reaction diffusion systems of which weak solution exists global-in-time with relatively compact orbit in L 1. Reaction term in this class is quasi-positive, dissipative, and up to with quadratic growth rate. If the space dimension is less than or equal to two, the solution is classical and uniformly bounded. Provided with the entropy structure, on the other hand, this weak solution is asymp-totically spatially homogeneous.
32 citations
TL;DR: Under the proposed robust control, the system performance is both deterministically guaranteed and fuzzily optimized from the Nash game perspective, as illustrated by the simulation control of a unicycle robot.
Abstract: The robust control design problem for uncertain dynamical systems is considered in this study. The uncertainty is time varying (possibly fast) and bounded, and the bound lies within a prescribed fuzzy set (hence the fuzzy dynamical system). We design the robust control in two steps. First, we propose a class of robust controls based on tunable parameters, which is in deterministic form and not conventionally IF–THEN fuzzy rule based. It is shown that these controls are able to guarantee deterministic system performance, namely uniform boundedness and ultimate uniform boundedness. Second, we seek the optima of tunable parameters in the control by formulating a two-player Nash game, which is based on two performance indexes (i.e., the cost functions). It is shown that the Nash equilibrium (i.e., the optima of tunable parameters) always exists. The procedure of obtaining the Nash equilibrium is provided. Under the proposed control, the system performance is both deterministically guaranteed and fuzzily optimized from the Nash game perspective. The effectiveness of the control design is illustrated by the simulation control of a unicycle robot.
31 citations
TL;DR: The optimal design problem of adaptive robust control for fuzzy mechanical systems with uncertainty is investigated and a novel concept: fuzzy confidence to measure the expectation value of a fuzzy number is proposed.
Abstract: The optimal design problem of adaptive robust control for fuzzy mechanical systems with uncertainty is investigated in this paper. The uncertainty that may be nonlinear and (possibly fast) time-varying is assumed to be bounded, and the knowledge of the bound only lies within a prescribed fuzzy set. Based on the Udwadia and Kalaba's approach, an adaptive robust controller, which is deterministic and is not the usual if-then rules-based is proposed to render the system to follow a class of prespecified constraints approximately. The adaptive law is of leakage type that can adjust the magnitude of the adaptive parameter based on the nonlinear performance-dependent gain. The resulting controlled system is uniformly bounded and uniformly ultimately bounded, which is proved via the Lyapunov minimax approach. Furthermore, we propose a novel concept: fuzzy confidence to measure the expectation value of a fuzzy number. Then, a fuzzy-based system performance index that includes the expectation value of the uniform ultimate boundedness (the average fuzzy performance) and the control cost is formulated. The optimal design problem associated with the control can then be solved by minimizing the performance index. As a result, the performance of the fuzzy mechanical system is both deterministically guaranteed and fuzzily optimized under this control.
31 citations
TL;DR: The importance of this new control scheme is that it can compensate the uncertainty in a very effective way and it also avoids over compensation and renders modest control effort.
30 citations
TL;DR: An intuitively plausible result is justified that the global iteration complexity of the Newton method is always better than that of the gradient method on the class of strongly convex functions with uniformly bounded second derivative.
Abstract: In this paper we study the iteration complexity of Cubic Regularization of Newton method for solving composite minimization problems with uniformly convex objective. We introduce the notion of second-order condition number of a certain degree and justify the linear rate of convergence in a nondegenerate case for the method with an adaptive estimate of the regularization parameter. The algorithm automatically achieves the best possible global complexity bound among different problem classes of uniformly convex objective functions with Holder continuous Hessian of the smooth part of the objective. As a byproduct of our developments, we justify an intuitively plausible result that the global iteration complexity of the Newton method is always better than that of the Gradient Method on the class of strongly convex functions with uniformly bounded second derivative.
30 citations
TL;DR: In this article, the inviscid limit of the Navier-Stokes equations to the Euler equations for compressible fluids in R 3 was studied and a Kolmogorov-type hypothesis for barotropic flows was introduced, in which the density and the sonic speed normally vary significantly.
TL;DR: The global existence and local boundedness of solution for the case β = 0 is investigated and it is shown that the solution exists globally and is uniformly bounded provided μ is large enough.
Abstract: We consider the prey-taxis system: u t = d 1 Δ u − χ ∇ ⋅ ( u ∇ v ) + u ( a − μ u ) + b u f ( v ) , x ∈ Ω , t > 0 , v t = d 2 Δ v + v ( c − β v ) − u f ( v ) , x ∈ Ω , t > 0 in a smoothly bounded domain Ω ⊂ R n , with zero-flux boundary condition, where a , d 1 , d 2 , χ , μ , b , c are positive constants and β is a non-negative constant. We first investigate the global existence and local boundedness of solution for the case β = 0 . Moreover, when β > 0 , we show that the solution exists globally and is uniformly bounded provided μ is large enough.
TL;DR: A linearized finite element method (FEM) for solving the cubic nonlinear Schrodinger equation with wave operator is proposed and it is proved that the proposed method keeps the energy conservation in the given discrete norm.
TL;DR: It is shown that the designed adaptive controller can guarantee that all the signals are ultimately uniformly bounded and the desired signal can be tracked with a small domain of the origin.
TL;DR: A novel adaptive neural tracking controller is developed by constructing the appropriately common Lyapunov function and applying backstepping technique and it is shown that all signals in the resulting closed-loop system are uniformly bounded and the tracking error converges to a small neighbourhood around zero.
TL;DR: In this paper, the authors investigated the behavior of the cost of controllability when the coefficient e goes to zero, according to the values of T. They showed that this cost is uniformly bounded with respect to e if T is greater than a minimal time TM.
Abstract: The advection-diffusion equation y e t − ey e xx + M y e x = 0, (x, t) ∈ (0, 1) × (0, T) is null controllable for any strictly positive values of the diffusion coefficient e and of the controllability time T. We discuss here the behavior of the cost of control when the coefficient e goes to zero, according to the values of T. It is actually known that this cost is uniformly bounded with respect to e if T is greater than a minimal time TM , with TM in the interval [1, 2 √ 3]/M for M > 0 and in the interval [2 √ 2, 2(1 + √ 3)]/|M | for M < 0. The exact value of TM is however unknown. We investigate in this work the determination of the minimal time TM employing two distincts but complementary approaches. In a first one, we numerically estimate the cost of controllability, reformulated as the solution of a generalized eigenvalue problem for the underlying control operator, with respect to the parameter T and e. This allows notably to exhibit the structure of initial data leading to large costs of control. At the practical level, this evaluation requires the non trivial and challenging approximation of null controls for the advection-diffusion equation. In the second approach, we perform an asymptotic analysis, with respect to the parameter e, of the optimality system associated to the control of minimal L 2-norm. The matched asymptotic expansion method is used to describe the multiple boundary layers.
TL;DR: It is proved that if a target probability density has bounded first-order weak derivatives and is uniformly bounded from below by a positive constant, then it can be reached in finite time using control inputs that are bounded in space and time.
Abstract: In this paper, we investigate the exact controllability properties of an advection–diffusion equation on a bounded domain, using time- and space-dependent velocity fields as the control parameters. This partial differential equation (PDE) is the Kolmogorov forward equation for a reflected diffusion process that models the spatiotemporal evolution of a swarm of agents. We prove that if a target probability density has bounded first-order weak derivatives and is uniformly bounded from below by a positive constant, then it can be reached in finite time using control inputs that are bounded in space and time. We then extend this controllability result to a class of advection–diffusion–reaction PDEs that corresponds to a hybrid switching diffusion process (HSDP), in which case the reaction parameters are additionally incorporated as the control inputs. For the HSDP, we first constructively prove controllability of the associated continuous-time Markov chain (CTMC) system in which the state space is finite. Then, we show that our controllability results for the advection–diffusion equation and the CTMC can be combined to establish controllability of the forward equation of the HSDP. Finally, we provide constructive solutions to the problem of asymptotically stabilizing an HSDP to a target nonnegative stationary distribution using time-independent state feedback laws, which correspond to spatially dependent coefficients of the associated system of PDEs.
TL;DR: This work proposes a novel adaptive robust control to control the UMR and compensate the uncertainties from the view of constraint-following and designs an adaptive law which is of leakage type.
Abstract: Underactuated mobile robot (UMR) is a typical nonlinear underactuated system with nonholonomic and holonomic constraints. Based on the model of UMR, we propose a novel adaptive robust control to control the UMR and compensate the uncertainties from the view of constraint-following. The uncertainties, which are (possibly fast) time-varying and bounded, include modeling error, initial condition deviation, friction force and other external disturbances. However, the bounds are unknown. To estimate the bounds of the uncertainties, we design an adaptive law which is of leakage type. The uniform boundedness and the uniform ultimate boundedness of the proposed control are verified by Lyapunov method. Furthermore, the effectiveness of the control is shown via numerical simulation of a case.
TL;DR: A general construction of strict ISS LKFs by using the positive definite and uniformly bounded solution to a scalar Lyapunov differential equation is established, which includes the proposed three classes of strictISS LKFS as special cases.
Proceedings Article•
11 Apr 2019TL;DR: In this paper, the authors proposed a projection-free algorithm for bandit convex optimization with a general convex constraint, which achieves a sublinear regret of O(nT^{4/5}) for any bounded convex function with uniformly bounded gradients.
Abstract: In this paper, we propose the first computationally efficient projection-free algorithm for bandit convex optimization (BCO) with a general convex constraint. We show that our algorithm achieves a sublinear regret of $O(nT^{4/5})$ (where $T$ is the horizon and $n$ is the dimension) for any bounded convex functions with uniformly bounded gradients. We also evaluate the performance of our algorithm against baselines on both synthetic and real data sets for quadratic programming, portfolio selection and matrix completion problems.
TL;DR: A novel adaptive dynamics surface control scheme is developed to remove the difficulties of pure-feedback characteristic, unknown nonlinearities, unknown control direction and “explosion of complexity”, which can guarantee that the proposed design is universal and low-complexity.
Abstract: In this paper, an adaptive state-feedback control technique is proposed for a class of unknown pure-feedback systems. A remarkable feature is that not only the problem of full-state constraints and prescribed performance tracking is solved together, but also the design is an approximation-free control scheme for pure-feedback systems with completely unknown nonlinearities. These properties will lead to a difficult task for designing a stable controller. To this end, a novel prescribed performance-barrier Lyapunov function is developed to guarantee that all the state constraints are not violated and the tracking error is preserved within a specified prescribed performance bound at all times, simultaneously. Then, by utilizing the mean value theorem, Nussbaum gain technique, a low-pass filter and a novel bounded estimation approach at each step of back-stepping procedure, a novel adaptive dynamics surface control scheme is developed to remove the difficulties of pure-feedback characteristic, unknown nonlinearities, unknown control direction and “explosion of complexity”, which can guarantee that the proposed design is universal and low-complexity. Moreover, it is proved that all the signals in the closed-loop system are global uniformly bounded. Two simulation studies are worked out to illustrate the performance of the proposed approach.
TL;DR: The fuzzy set theory is adopted to describe the uncertainty in the fuzzy dynamical system and a class of robust controls is proposed based on tunable parameters, able to guarantee deterministic system performance, namely uniform boundedness and uniform ultimate boundedness.
Abstract: This study investigates the robust control for uncertain dynamical systems. The uncertainty is (possibly fast) time-varying but bounded. We adopt the fuzzy set theory to describe the uncertainty in the system (hence, called the fuzzy dynamical system). A class of robust controls is proposed based on tunable parameters. The controls are deterministic and are not conventional IF–THEN fuzzy rules based (such as Mamdani type). The proposed controls are able to guarantee deterministic system performance, namely uniform boundedness and uniform ultimate boundedness. In the phase of searching for the optima from the pool of admissible control design parameters, we formulate this as a two-player cooperative game by developing two performance indexes (i.e., the cost functions), each of which is dominated by one tunable parameter (i.e., the player). By the cooperative game theory, we are able to obtain the Pareto-optimality (i.e., the optima of tunable parameters). Simulation results on an electric vehicle motion control problem are presented for demonstration.
TL;DR: This paper investigates the consensus asymptotic convergence problem for a class of nth-order strict-feedback multiagent systems, which include the input quantization, actuator faults, unknown nonlinear functions and directed communication topology.
TL;DR: The global solution to this optimal design problem is demonstrated to be always existent and unique and able to guarantee the uniform boundedness and uniform ultimate boundedness of the system in the presence of uncertainty, while minimizing a fuzzy-based performance index associated with both the fuzzy performance and the control cost.
Abstract: The position control and vibration suppression problems of the flexible solar array drive system containing uncertainty are considered in this paper. The uncertainty in system may be due to unknown parameters and external disturbance. The uncertainty bound can be described via a fuzzy set. In addition, there exists the flexible vibration in the system. A new optimal robust control approach with input shaping is proposed for the flexible solar array drive system. By designing the position command trajectory, the input shaper is proposed to suppress the flexible vibration. To enhance the position control performance, the optimal robust control is proposed by fuzzy description of the uncertainty bound. Neither the system nor the control is fuzzy if–then rule based. The global solution to this optimal design problem is demonstrated to be always existent and unique. The resulting control is able to guarantee the uniform boundedness and uniform ultimate boundedness of the system in the presence of uncertainty, while minimizing a fuzzy-based performance index associated with both the fuzzy performance and the control cost. In addition, the flexible vibration can be effectively suppressed. The novelty of this research is a systematic control approach by blending input shaping technology, control theory, fuzzy set theory, and optimization theory into an integrated framework, for solving the position control and vibration suppression problems of flexible solar array drive system with mismatched conditions.
Posted Content•
09 Dec 2019
TL;DR: In this paper, sufficient conditions for stability of nonlinear and time-varying impulsive systems are provided for input-to-state stability and zero-input global uniform asymptotic stability.
Abstract: We provide novel sufficient conditions for stability of nonlinear and time-varying impulsive systems. These conditions generalize, extend, and strengthen many existing results. Different types of input-to-state stability (ISS), as well as zero-input global uniform asymptotic stability (0-GUAS), are covered by employing a two-measure framework and considering stability of both weak (decay depends only on elapsed time) and strong (decay depends on elapsed time and the number of impulses) flavors. By contrast to many existing results, the stability state bounds imposed are uniform with respect to initial time and also with respect to classes of impulse-time sequences where the impulse frequency is eventually uniformly bounded. We show that the considered classes of impulse-time sequences are substantially broader than other previously considered classes, such as those having fixed or (reverse) average dwell times, or impulse frequency achieving uniform convergence to a limit (superior or inferior). Moreover, our sufficient conditions are not more restrictive than existing ones when particularized to some of the cases covered in the literature, and hence in these cases our results allow to strengthen the existing conclusions.
TL;DR: For a complete non-compact connected Riemannian manifold with bounded geometry, it was shown in this article that the isoperimetric profile function is a locally (1 − ε)-Holder continuous function and so in particular it is continuous.
Abstract: For a complete noncompact connected Riemannian manifold with bounded geometry $$M^n$$
, we prove that the isoperimetric profile function $$I_{M^n}$$
is a locally $$(1-\frac{1}{n})$$
-Holder continuous function and so in particular it is continuous. Here for bounded geometry we mean that M have Ricci curvature bounded below and volume of balls of radius 1, uniformly bounded below with respect to its centers. We prove also the equivalence of the weak and strong formulation of the isoperimetric profile function in complete Riemannian manifolds which is based on a lemma having its own interest about the approximation of finite perimeter sets with finite volume by open bounded with smooth boundary ones of the same volume. Finally the upper semicontinuity of the isoperimetric profile for every metric (not necessarily complete) is shown.
TL;DR: In this paper, the authors investigated the long time behavior of the error of numerical solutions to time-dependent partial differential equations and derived conditions guaranteeing that the errors are still bounded in time.
Abstract: For practical applications, the long time behaviour of the error of numerical solutions to time-dependent partial differential equations is very important. Here, we investigate this topic in the context of hyperbolic conservation laws and flux reconstruction schemes, focusing on the schemes in the discontinuous Galerkin spectral element framework. For linear problems with constant coefficients, it is well-known in the literature that the choice of the numerical flux (e.g. central or upwind) and the selection of the polynomial basis (e.g. Gaus–Legendre or Gaus–Lobatto–Legendre) affects both the growth rate and the asymptotic value of the error. Here, we extend these investigations of the long time error to variable coefficients using both Gaus–Lobatto–Legendre and Gaus–Legendre nodes as well as several numerical fluxes. We derive conditions guaranteeing that the errors are still bounded in time. Furthermore, we analyse the error behaviour under these conditions and demonstrate in several numerical tests similarities to the case of constant coefficients. However, if these conditions are violated, the error shows a completely different behaviour. Indeed, by applying central numerical fluxes, the error increases without upper bound while upwind numerical fluxes can still result in uniformly bounded numerical errors. An explanation for this phenomenon is given, confirming our analytical investigations.
TL;DR: In this paper, the weak-type boundedness of the Hardy-Littlewood maximal averaging operator on infinite graphs was studied and the connections between the doubling condition, having finite dilation and overlapping indices, uniformly bounded degree, the equidistant comparison property and the weak type boundedness were analyzed.
Abstract: We study different geometric properties on infinite graphs, related to the weak-type boundedness of the Hardy–Littlewood maximal averaging operator. In particular, we analyze the connections between the doubling condition, having finite dilation and overlapping indices, uniformly bounded degree, the equidistant comparison property and the weak-type boundedness of the centered Hardy–Littlewood maximal operator. Several non-trivial examples of infinite graphs are given to illustrate the differences among these properties.
TL;DR: In this article, it was shown that a connected and properly embedded translating soliton with uniformly bounded genus on compact sets which is $C^1$-asymptotic to two planes outside a cylinder either is flat or coincides with the grim reaper cylinder can be found.
Abstract: In this article we prove that a connected and properly embedded translating soliton in $\mathbb{R}^3$ with uniformly bounded genus on compact sets which is $C^1$-asymptotic to two planes outside a cylinder, either is flat or coincides with the grim reaper cylinder.
TL;DR: In this paper, the weak convergence to Gaussian processes of conditional empirical processes and conditional U-processes from stationary β-mixing sequences indexed by classes of functions satisfying some entropy conditions was studied.
Abstract: In this paper we are concerned with the weak convergence to Gaussian processes of conditional empirical processes and conditional U-processes from stationary β-mixing sequences indexed by classes of functions satisfying some entropy conditions. We obtain uniform central limit theorems for conditional empirical processes and conditional U-processes when the classes of functions are uniformly bounded or unbounded with envelope functions satisfying some moment conditions. We apply our results to introduce statistical tests for conditional independence that are multivariate conditional versions of the Kendall statistics.