Strict Inequality Research Articles

Robust control for a class of uncertain neutral‐type systems with time‐varying delays

AbstractThis paper investigates the problem of robust asymptotic stabilization for a class of uncertain neutral‐type systems with time‐varying delays. A sufficient condition is obtained to ensure the considered system is robust asymptotic stabilization with disturbance attenuation level , which is based on the sliding mode observer framework. Meanwhile, a novel integral sliding surface function with the compensator is proposed to make sure the considered system can be stabilized. And the gain matrix is obtained by solving a strict linear matrix inequality. Finally, two numerical examples are provided to demonstrate the validity and potential of the proposed method.

On asymptotic depth of integral closure filtration and an application

Let (A,m) be an analytically unramified formally equidimensional Noetherian local ring with depthA≥2. Let I be an m-primary ideal and set I⁎ to be the integral closure of I. Set G⁎(I)=⨁n≥0(In)⁎/(In+1)⁎ be the associated graded ring of the integral closure filtration of I. We prove that depthG⁎(In)≥2 for all n≫0. As an application we prove that if A is also an excellent normal domain containing an algebraically closed field isomorphic to A/m then there exists s0 such that for all s≥s0 and J is an integrally closed ideal strictly containing (ms)⁎ then we have a strict inequality μ(J)<μ((ms)⁎) (here μ(J) is the number of minimal generators of J).

A positive real lemma for singular hybrid systems

Abstract This paper introduces a positive real lemma for continuous-time singular hybrid systems. The necessary and sufficient condition of stochastic admissibility and strict passivity for the singular hybrid systems is obtained in terms of linear matrix inequalities. The sufficient condition is derived by using mode-dependent Lyapunov function. In this step, two slack variables are inserted to make the proposed condition be necessary and sufficient condition in terms of strict linear matrix inequalities. Then, to prove the necessary condition, the positive real lemma for the hybrid system is proposed. Since the admissible singular system can be reformulated into stable normal system, the positive real lemma for the hybrid system holds. Thus, we give a necessary condition by constructing the solution of the proposed lemma from that of the hybrid systems.

Finite-Time Stabilization for a Class of Nonlinear Singular Systems

The finite-time stabilization problem for a class of nonlinear singular systems is studied. Under the assumption that the considered system is impulse controllable, a sufficient condition is provided for the design of a state feedback control law guaranteeing the finite-time stability of the closed-loop system, and an explicit expression of the state feedback gain is also given. The proposed criterion is expressed in terms of strict matrix inequalities which is easy to be verified numerically. A numerical example is given to illustrate the effectiveness of the proposed method.

Delay-Dependent H∞ Control for Singular Markovian Jump Systems With Generally Uncertain Transition Rates

This article is devoted to the problem of $H_{\infty }$ control for a class of singular Markovian jump systems with time-varying delay and generally uncertain transition rates, which means each transition rate is completely unknown or only its estimated value is known. By using Lyapunov stability theory, a new delay-dependent $H_{\infty }$ admissible criterion in terms of strict linear matrix inequalities is obtained, which guarantees that the singular Markovian jump system with known transitions rates is regular, impulse-free and stochastically stable with a prescribed $H_{\infty }$ disturbance attenuation level $\gamma $ . Based on this obtained criterion, some suitable state feedback controllers are designed such that the closed-loop delayed singular Markovian jump system with generally uncertain transition rates is $H_{\infty }$ stochastically admissible. Finally, numerical examples are included to illustrate the effectiveness and the less conservativeness of the proposed method.

Edge Matching with Inequalities, Triangles, Unknown Shape, and Two Players

We analyze the computational complexity of several new variants of edge-matching puzzles. First we analyze inequality (instead of equality) constraints between adjacent tiles, proving the problem NP-complete for strict inequalities but polynomial-time solvable for nonstrict inequalities. Second we analyze three types of triangular edge matching, of which one is polynomial-time solvable and the other two are NP-complete; all three are #P-complete. Third we analyze the case where no target shape is specified and we merely want to place the (square) tiles so that edges match exactly; this problem is NP-complete. Fourth we consider four 2-player games based on 1 × n edge matching, all four of which are PSPACE-complete. Most of our NP-hardness reductions are parsimonious, newly proving #P and ASP-completeness for, e.g., 1 × n edge matching. Along the way, we prove #P- and ASP-completeness of planar 3-regular directed Hamiltonicity; we provide linear-time algorithms to find antidirected and forbidden-transition Eulerian paths; and we characterize the complexity of new partizan variants of the Geography game on graphs.

Indirect inference with(out) constraints

Indirect Inference (I‐I) estimation of structural parameters θ requires matching observed and simulated statistics, which are most often generated using an auxiliary model that depends on instrumental parameters β. The estimators of the instrumental parameters will encapsulate the statistical information used for inference about the structural parameters. As such, artificially constraining these parameters may restrict the ability of the auxiliary model to accurately replicate features in the structural data, which may lead to a range of issues, such as a loss of identification. However, in certain situations the parameters β naturally come with a set of q restrictions. Examples include settings where β must be estimated subject to q possibly strict inequality constraints g( β)&gt;0, such as, when I‐I is based on GARCH auxiliary models. In these settings, we propose a novel I‐I approach that uses appropriately modified unconstrained auxiliary statistics, which are simple to compute and always exists. We state the relevant asymptotic theory for this I‐I approach without constraints and show that it can be reinterpreted as a standard implementation of I‐I through a properly modified binding function. Several examples that have featured in the literature illustrate our approach.

A Relaxed Observer-Based Control for LPV Stochastic Systems Subject to H∞ Performance

A relaxed observer-based control problem of the disturbed Linear Parameter Varying (LPV) stochastic systems is addressed in this paper. For the control problem, some sufficient conditions are derived via a Lyapunov function whose positive definite matrix is not required as the diagonal case. Besides, a general H∞ performance index is proposed to constrain the effect of external disturbance on the state and estimation error. Furthermore, a lemma is proposed to eliminate the oscillation caused by determining performance level of state and estimation error. Another lemma is developed via extending projection lemma to convert the derived conditions into strict Linear Matrix Inequality (LMI) form. By using convex optimization algorithm, all feasible solutions can be obtained by a single-step calculation to establish an observer-based Gain-Scheduled (GS) controller. Based on the designed controller, the asymptotical stability and H∞ performance of the LPV stochastic systems can be guaranteed in the sense of mean square. Finally, two examples are provided to demonstrate the effectiveness and applicability of the proposed method.

Lower Assouad Dimension of Measures and Regularity

AbstractIn analogy with the lower Assouad dimensions of a set, we study the lower Assouad dimensions of a measure. As with the upper Assouad dimensions, the lower Assouad dimensions of a measure provide information about the extreme local behaviour of the measure. We study the connection with other dimensions and with regularity properties. In particular, the quasi-lower Assouad dimension is dominated by the infimum of the measure’s lower local dimensions. Although strict inequality is possible in general, equality holds for the class of self-similar measures of finite type. This class includes all self-similar, equicontractive measures satisfying the open set condition, as well as certain “overlapping” self-similar measures, such as Bernoulli convolutions with contraction factors that are inverses of Pisot numbers.We give lower bounds for the lower Assouad dimension for measures arising from a Moran construction, prove that self-affine measures are uniformly perfect and have positive lower Assouad dimension, prove that the Assouad spectrum of a measure converges to its quasi-Assouad dimension and show that coincidence of the upper and lower Assouad dimension of a measure does not imply that the measure is s-regular.

Ample Spectrum Contractions and Related Fixed Point Theorems

Simulation functions were introduced by Khojasteh et al. as a method to extend several classes of fixed point theorems by a simple condition. After that, many researchers have amplified the knowledge of such kind of contractions in several ways. R-functions, ( R , S ) -contractions and ( A , S ) -contractions can be considered as approaches in this direction. A common characteristic of the previous kind of contractive maps is the fact that they are defined by a strict inequality. In this manuscript, we show the advantages of replacing such inequality with a weaker one, involving a family of more general auxiliary functions. As a consequence of our study, we show that not only the above-commented contractions are particular cases, but also another classes of contractive maps correspond to this new point of view.

Dynamic output-feedback control for singular T–S fuzzy systems using fuzzy Lyapunov functions

In light of fuzzy weighting-dependent Lyapunov functions, a dynamic output-feedback control for singular T–S fuzzy systems is introduced. Based on a set-equivalence technique and fuzzy weighting-dependent Lyapunov functions, this paper derives a sufficient condition ensuring the singular T–S fuzzy systems to be admissible (regular, impulse-free and stable) in the form of strict parametric linear matrix inequalities (PLMIs). Then, for the closed-loop systems with a dynamic, rather than static, output-feedback controller, an admissibility criterion is given with strict PLMIs by specially representing the block entries of matrices in Lyapunov function. By appropriately choosing the structures of PLMI variables, the strict LMIs are obtained, where slack variables are included in the relaxation process fully exploiting the properties of the fuzzy weighting functions. Three numerical examples are provided to show the effectiveness of the introduced dynamic control.

The Hilali conjecture on product of spaces

The Hilali conjecture claims that a simply connected rationally elliptic space $X$ satisfies the inequality $\dim (\pi_*(X)\otimes \mathbb{Q} ) \leqq \dim H_*(X;\mathbb{Q} )$. In this paper we show that for any such space $X$ there exists a positive integer $n_0$ such that for any $n \geqq n_0$ the strict inequality $\dim (\pi_*(X^n)\otimes \mathbb{Q} ) \lt \dim H_*(X^n; \mathbb{Q} )$ holds, where $X^{n}$ is the product of $n$ copies of $X$.

Reliable LPS features of multiple airborne radars against PRS

Low probability of sorting (LPS) is an important step of the low probability of interception. This study proposes a way to keep reliable LPS features of signals based on the information–theoretical framework. First, the LPS process is mapped as features estimation and classification processes by the passive reconnaissance system (PRS). Second, with the Markov chain and the strict Fano's inequality, this study suggests that the misclassification performance of the PRS depends on the number of signals received by the PRS simultaneously. Third, this study establishes a single LPS feature model which is convenient to be extended to multiple features. For multiple LPS features under the misclassification constraint, this study provides a relatively reasonable model and suggests that the suitable model depends on the experience. Finally, with conditional entropy, this study illustrates that the classification performance of the PRS is proportional to the repeat number of the same signal, and suggests that the features of multiple airborne radars should be changed rapidly. With several clustering algorithms, simulations show that the misclassification performance of the PRS is guaranteed based on LPS features, the knowledge of the PRS and the multiple LPS features solution model.

Existence of Nonimpulsive Unique Solution and Stability for Discrete-Time Linear Rectangular Descriptor Markov Jump Systems

This paper investigates the existence of nonimpulsive unique solution and stochastic stability problems for the discrete-time linear rectangular descriptor Markov jump systems at two cases. For every case, the sufficient and necessary conditions, in terms of strict linear matrix inequalities, are eventually proposed to guarantee that the discrete-time linear rectangular descriptor Markov jump systems are column (row) regular, column (row) causal, stochastically stable, and have a unique solution. In addition, some assumptions are introduced to ensure that there is no impulse in the solution at the instant when the Markov process jumps. Moreover, the concrete procedures on how to apply the results obtained in this paper are presented as well. Finally, an example is provided to demonstrate the effectiveness of the results.

Observer-Based Sliding Mode Control for T–S Fuzzy Descriptor Systems With Time Delay

This paper examines the problem of observer-based sliding mode control designs for a class of descriptor Takagi–Sugeno fuzzy systems with time delay and uncertainties. Specifically, based on the detailed discussions on the existence conditions, a reduced-order robust observer is designed first where the influences of the uncertainties are totally removed. Second, by choosing appropriate coordinate transformations and matrix decompositions, an actual and a virtual sliding mode variables are constructed, and an observer-based sliding mode controller is developed to handle the uncertainties such that the virtual sliding mode surface can be reached and maintained in a finite time, whereas the actual sliding mode variable approaches to zeros asymptotically. And then, we prove that the system asymptotic stability can be guaranteed after the virtual sliding mode surface has been reached or the actual sliding mode variable approached to zero. In addition, the existence conditions for both the observer and the sliding mode controller are given in strict linear matrix inequality forms. Finally, a simulation example is given to demonstrate the effectiveness of the proposed method.

A proof of Saitoh's conjecture for conjugate Hardy $H^{2}$ kernels

In this article, we obtain a strict inequality between the conjugate Hardy $H^{2}$ kernels and the Bergman kernels on planar regular regions with $n > 1$ boundary components, which is a conjecture of Saitoh.

Observer-based [formula omitted] control design for singular switching semi-Markovian jump systems with random sensor delays

This paper studies the problem of H∞ control design for a class of singular switching semi-Markovian jump systems, in which the transition rates of jumping process are uncertain and the output measurements suffer from random sensor delays. Firstly, a Luenberger observer is designed to estimate the system state components, based on which an error dynamic is obtained. Then, stochastic stability analysis for the overall closed-loop system is given based on the upper and lower bounds of transition rates. Further, the state feedback controller gain matrices are given by solving a set of sufficient conditions in terms of strict linear matrix inequalities (LMIs), which also guarantee the closed-loop system to be stochastically stabilizable and has a prescribed H∞ performance index γ. Finally, a numerical example is provided to verify the effectiveness of the obtained result.

Robust generalized observer design for uncertain one-sided Lipschitz systems

This paper develops a novel generalized observer design approach for the uncertain descriptor systems with one-sided Lipschitz nonlinearities, parametric uncertainties, and external perturbations. The nonlinearities, uncertainties, and external perturbations are considered both in input and output equations to consider a matter-of-fact observer design. A generalized structure for the observer is employed to deal with a large number of systems, considering both non-singular and singular systems. The proposed observer scheme is based on non-strict and strict linear matrix inequalities (LMIs), which are derived using the concepts of generalized Lyapunov theory, uncertainty bounds, quadratic inner-boundedness, one-sided Lipschitz condition, matrix transformations, and L2 gain minimization criteria. The proposed state filtering approach is robust for disturbances with asymptotically stable estimation error dynamics under zero external perturbations and attenuation of disturbance effects to keep the estimation error within prescribed limits. In comparison to the conventional observer designs for the one-sided Lipschitz systems, the presented scheme is based on a generalized observer and can deal with parametric uncertainties. A numerical simulation example and an application example of spring-mass-damper system are provided to verify the effectiveness of the suggested filtering schemes.

Stabilization of singular T-S fuzzy Markovian jump system with mode-dependent derivative-term coefficient via sliding mode control

This paper introduces sliding mode control for singular Takagi–Sugeno (T-S) fuzzy Markovian jump system with mode-dependent derivative-term coefficient. Firstly, a new continuous switching function is designed. Then sufficient conditions about admissibility of sliding motion under the case of completely known transition rate and uncertain transition rate are presented, respectively. The obtained conditions are strict linear matrix inequalities. Furthermore, a sliding mode controller is presented. Finally, an example is given to illustrate the effectiveness of the obtained results in this paper.

Strict density inequalities for sampling and interpolation in weighted spaces of holomorphic functions

Answering a question of Lindholm, we prove strict density inequalities for sampling and interpolation in Fock spaces of entire functions in several complex variables defined by a plurisubharmonic weight. In particular, these spaces do not admit a set that is simultaneously sampling and interpolating. To prove optimality of the density conditions, we construct sampling sets with a density arbitrarily close to the critical density.The techniques combine methods from several complex variables (estimates for ∂¯) and the theory of localized frames in general reproducing kernel Hilbert spaces (with no analyticity assumed). The abstract results on Fekete points and deformation of frames may be of independent interest.