Quadratic Inequality Research Articles

Transportation cost inequalities for SDEs with irregular drifts

In this paper, the quadratic transportation cost inequality for SDEs with Dini continuous drift and the W1-transportation cost inequality for SDEs with singular coefficients are established via the stability of the Wasserstein distance and relative entropy of measures under the homeomorphism induced by Zvonkin’s transformation.

Talagrand’s quadratic transportation cost inequalities for SPDEs driven by fractional noises with two reflection walls

Using the method of Girsanov’s transformation, we investigate Talagrand’s quadratic transportation cost inequalities for the laws of the solutions of stochastic partial differential equations (SPDEs) with two reflection walls under the uniform norm on the continuous functions space. These equations are driven by fractional noises.

On critical dipoles in dimensions n ⩾ 3

We reconsider generalizations of Hardy's inequality corresponding to the case of (point) dipole potentials Vγ(x)=γ(u,x)|x|−3, x∈Rn﹨{0}, γ∈[0,∞), u∈Rn, |u|=1, n∈N, n⩾3. More precisely, for n⩾3, we provide an alternative proof of the existence of a critical dipole coupling constant γc,n>0, such thatfor all γ∈[0,γc,n], and all u∈Rn, |u|=1,∫Rndnx|(∇f)(x)|2⩾±γ∫Rndnx(u,x)|x|−3|f(x)|2,f∈D1(Rn) with D1(Rn) denoting the completion of C0∞(Rn) with respect to the norm induced by the gradient. Here γc,n is sharp, that is, the largest possible such constant. Moreover, we discuss upper and lower bounds for γc,n>0 and develop a numerical scheme for approximating γc,n.This quadratic form inequality will be a consequence of the fact[−Δ+γ(u,x)|x|−3]|C0∞(Rn﹨{0})‾⩾0 if and only if 0⩽γ⩽γc,n in L2(Rn) (with T‾ the operator closure of the linear operator T).We also consider the case of multicenter dipole interactions with dipoles centered on an infinite discrete set.

Exact robust D-stability analysis for linear dynamical systems with polynomial parameter perturbation

This paper provides the exact, complete, and possibly non connected robust -stability domain for linear uncertain dynamical systems, which polynomially depend upon a given number of real parameters. Derived in terms of a pencil of multiple degrees pertaining to the perturbed system’s dynamics and to the characteristic function of the considered quadratic matrix inequality (QMI) region, the bounds of these robust -stability domains, in the parameter space, turn out to be delimited by the finite, real and unduplicated generalised eigenvalues of this pencil, indicating when the system’s eigenvalues cross the boundaries of the considered QMI region. The paper also shows that these generalised eigenvalues coincide with those of an augmented pencil of degree just one, thus rendering the computation of these generalised eigenvalues, in general, a standard and relatively simple operation.

SVARs Identification Through Bounds on the Forecast Error Variance

This article identifies structural vector autoregressions (SVARs) through bound restrictions on the forecast error variance decomposition (FEVD). First, the article shows FEVD bounds correspond to quadratic inequality restrictions on the columns of the rotation matrix transforming reduced-form residuals into structural shocks. Second, the article establishes theoretical conditions such that bounds on the FEVD lead to a reduction in the width of the impulse response identified set relative to only imposing sign restrictions. Third, this article proposes a robust Bayesian approach to inference. Fourth, the article shows that elicitation of the bounds could be based on DSGE models with alternative parameterizations. Finally, an empirical application illustrates the potential usefulness of FEVD restrictions for obtaining informative inference in set-identified monetary SVARs and remove unreasonable implications of models identified through sign restrictions.

Transportation cost inequality for backward stochastic differential equations with mean reflection

Using the method of Girsanov’s transformation, we investigate Talagrand’s quadratic transportation cost inequalities for the law of the solution of backward stochastic differential equations with mean reflection, under the uniform norm and L2-norm. These equations are driven by a Brownian motion.

Exponential stabilization of sampled-data fuzzy systems via a parameterized fuzzy Lyapunov-Krasovskii functional approach

This paper devotes to stabilize nonlinear systems based on Takagi-Sugeno (T-S) fuzzy models, where only sampled state is available. Note that the membership functions (MFs) between T-S fuzzy models and fuzzy controllers are asynchronous under a sampling mechanism, a fuzzy state feedback controller with asynchronous MFs is introduced. A new parameterized fuzzy Lyapunov-Krasovskii functional (PFLKF) approach is presented to ensure that the closed-loop system is exponentially stable and there exists a large well-defined domain of attraction. In general, it is difficult to compute in advance the upper bounds of the time derivatives of MFs, where these time derivatives appear in the time derivative of the PFLKF or the errors of the asynchronous MFs. To this end, a novel quadratic inequality is established to characterize the time derivatives of MFs. Then an MF-dependent exponential stability criterion is given in terms of linear matrix inequalities, and a co-design method for the controller gains and the domain of attraction is presented. Finally, three illustrative examples show the effectiveness and advantages of the proposed approach.

Quadratic transportation inequalities for SDEs with measurable drift

Let X X be the solution of a stochastic differential equation in Euclidean space driven by standard Brownian motion, with measurable drift and Sobolev diffusion coefficient. In our main result we show that when the drift is measurable and the diffusion coefficient belongs to an appropriate Sobolev space, the law of X X satisfies Talagrand’s inequality with respect to the uniform distance.

Observer Design for One-sided Lipschitz Uncertain Descriptor Systems with Time-varying Delay and Nonlinear Uncertainties

This paper investigates observer design for a class of one-sided Lipschitz descriptor systems with time-varying delay and uncertain parameters. In order to provide a general framework for large-scale systems, the paper considers uncertainties, nonlinearities, disturbance and time-varying delay at both output and state. By constructing Lyapunov–Krasovskii functional, and using the one-sided Lipschitz condition and the quadratic inner-boundedness inequality, we establish the sufficient condition which guarantees that the observer error dynamics is asymptotically stable, and the proposed observer ensures the $$L_2$$ gain bounded by a scalar $$\gamma .$$ Then, we change the condition into a strict matrix inequality condition. Furthermore, based on the obtained results, we establish the linear matrix inequality-based condition to ensure the asymptotically convergence of state estimation error and to accomplish robustness against $$L_2$$ norm bounded disturbances by utilizing change of variables. We propose the computing method of observer gain. Finally, a simulation example is provided to demonstrate the effectiveness of the proposed method.

Quadratic transportation cost inequality for scalar stochastic conservation laws

In this paper, we established a quadratic transportation cost inequality for scalar stochastic conservation laws driven by multiplicative noise. The doubling variables method plays an important role.

Strict $2$-convexity of convex solutions to the quadratic Hessian equation

We prove that convex viscosity solutions to the quadratic Hessian inequality $\sigma_2(D^2u) \geq 1$ are strictly $2$-convex. As a consequence we obtain short proofs of smoothness and interior $C^2$ estimates for convex viscosity solutions to $\sigma_2(D^2u) = 1$, which were proven using different methods in recent works of Guan-Qiu \cite{GQ}, McGonagle-Song-Yuan \cite{MSY} and Shankar-Yuan \cite{SY2}.

BMI-Based Load Frequency Control in Microgrids Under False Data Injection Attacks

In this article, the power fluctuation problem caused by adding renewable energy sources to microgrids (MGs) is addressed. These power fluctuations which can cause several adverse effects, including system instability, load shedding, and eventually blackouts are attenuated by designing a load frequency controller (LFC) to provide a constant and uniform frequency in different operation cases of MGs. This article designs a dynamic output feedback controller (DOFC) for LFC in MGs while considering the contraction observer for attack mitigation. We suggest a new formulation of the sufficient conditions for the DOFC design problem in the form of bilinear matrix inequalities (BMIs) and try to minimize the frequency deviation subject to load, solar, and wind energy changes. The proposed formulation effectively changes the problem to a new BMI model which can easily be divided into two problems, one is a convex optimization problem and the remained part is a quadratic matrix inequality problem which is linearized around the feasible solution of the linear matrix inequality part by a Taylor-like expansion. This leads to a more exact and less conservative solution as compared to the previous methods suggested in the literature. To illustrate the merits of the developed approach, numerical simulations on a sample ac–MG are carried out. Comparing norms of the frequency variations in the closed-loop system with multiple prominent methods in previous works, including <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${{\boldsymbol{H}}_\infty }, \ {\boldsymbol{\mu }}$</tex-math></inline-formula> , fuzzy type-1, and intelligent-PI, shows that our proposed method reduces the two and infinity norms of frequency fluctuations by 84% and 80%, respectively compared to the traditional <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${{\boldsymbol{H}}_\infty }$</tex-math></inline-formula> method which is the best among the analyzed approaches. This further verifies the efficacy of the designed DOFC in our article.

Area minimizing surfaces of bounded genus in metric spaces

AbstractThe Plateau–Douglas problem asks to find an area minimizing surface of fixed or bounded genus spanning a given finite collection of Jordan curves in Euclidean space. In the present paper we solve this problem in the setting of proper metric spaces admitting a local quadratic isoperimetric inequality for curves. We moreover obtain continuity up to the boundary and interior Hölder regularity of solutions. Our results generalize corresponding results of Jost and Tomi-Tromba from the setting of Riemannian manifolds to that of proper metric spaces with a local quadratic isoperimetric inequality. The special case of a disc-type surface spanning a single Jordan curve corresponds to the classical problem of Plateau, in proper metric spaces recently solved by Lytchak and the second author.

Transportation inequalities for doubly perturbed stochastic differential equations with Markovian switching

&lt;abstract&gt;&lt;p&gt;In this paper, we focus on a class of doubly perturbed stochastic differential equations with Markovian switching. Using the Girsanov transformation argument we establish the quadratic transportation inequalities for the law of the solution of those equations with Markovian switching under the $ d_2 $ metric and the uniform metric $ d_{\infty} $.&lt;/p&gt;&lt;/abstract&gt;

Some general quadratic numerical radius inequalities for the off-diagonal parts of $2 ⨉ 2$ block operator matrices

Some general quadratic numerical radius inequalities for the off-diagonal parts of $2 ⨉ 2$ block operator matrices

Global optimality condition for quadratic optimization problems under data uncertainty

In this paper, we establish a robust version of the S-lemma that we use to characterize robust solutions for classes of homogeneous and non-homogeneous quadratic problems with a quadratic inequality constraint under interval uncertainty and a linear equality constraint. Necessary and sufficient conditions of global optimality of robust solution of these problems are given.

Convex Optimization for Finite-Horizon Robust Covariance Control of Linear Stochastic Systems

This work addresses the finite-horizon robust covariance control problem for discrete time, partially observable, linear systems affected by random zero mean noise and deterministic uncertain-but-bounded disturbances restricted to lie in what is called ellitopic uncertainty set (e.g., finite intersection of centered at the origin ellipsoids/elliptic cylinders). Performance specifications are imposed on the random state-control trajectory via averaged convex quadratic inequalities, linear inequalities on the mean, chance constrained linear inequalities as well as convex-monotone constraints on the covariance matrix. For this problem we develop a computationally tractable procedure for designing affine control policies, in the sense that the parameters of the policy that guarantees the aforementioned performance specifications are obtained as solutions to an explicit convex program. Our theoretical findings are illustrated by a numerical example.

Quadratically adjustable robust linear optimization with inexact data via generalized S-lemma: Exact second-order cone program reformulations

• Studies adjustable robust linear optimization problems with quadratic decision rules. • Incorporates inexactness of the data in the construction of uncertainty sets. • Establishes generalizations of S-Lemma for classes of nonconvex quadratic systems. • Presents exact conic programming reformulations via a generalized S-Lemma. • Illustrates results with numerical experiments on lot-sizing problems with demand uncertainty. Adjustable robust optimization allows for some variables to depend upon the uncertain data after its realization. However, the uncertainty is often not revealed exactly. Incorporating inexactness of the revealed data in the construction of ellipsoidal uncertainty sets, we present an exact second-order cone program reformulation for robust linear optimization problems with inexact data and quadratically adjustable variables. This is achieved by establishing a generalization of the celebrated S-lemma for a separable quadratic inequality system with at most one non-homogeneous function. It allows us to reformulate the resulting separable quadratic constraints over an intersection of two ellipsoids in terms of second-order cone constraints. We illustrate our results via numerical experiments on adjustable robust lot-sizing problems with demand uncertainty, showing improvements over corresponding problems with affinely adjustable variables as well as with exactly revealed data.

Finite-time anti-synchronization for delayed inertial neural networks via the fractional and polynomial controllers of time variable

<abstract> This paper focuses on the finite-time anti-synchronization for a class of delayed master-slave inertial neural networks. By means of using the property of quadratic inequality of one variable and designing the fractional and polynomial controllers of time variable, two sufficient conditions to assure the finite-time anti-synchronization for the master-slave delayed inertial neural networks are established. Our controllers designed related to time variable t and the study method on the finite-time anti-synchronization are different from these in the existing papers. </abstract>

Quadratic inequalities for functionals in l^{∞}

Quadratic inequalities for functionals in l^{∞}