Direct Adaptive Control Based on Gradient Estimation

A control method of direct adaptive control based on gradient estimation is proposed in this article. The dynamic system is embedded in a linear model set. Based on the embedding property of the dynamic system, an adaptive optimal control algorithm is proposed. The robust convergence of the proposed control algorithm has been proved and the static control error with the proposed method is also analyzed. The application results of the proposed method to the industrial polypropylene process have verified its feasibility and effectiveness.

GLOBAL BOUND ON THE GRADIENT OF SOLUTIONS TO p-LAPLACE TYPE EQUATIONS WITH MIXED DATA

In this paper,the study of gradient regularity for solutions of a class of elliptic problems of p-Laplace type is offered.In particular,we prove a global result concerning Lorentz-Morrey regularity of the non-homogeneous boundary data problem:-div((s^(2)+|▽u|^(2)p-2/2)▽u)=-div(|f|^(p-2)f)+g inΩ,u=h in■Ω,with the(sub-elliptic)degeneracy condition s∈[0,1]and with mixed data f∈L^(p)(Q;R^(n)),g∈Lp/(p-1)(Ω;R^(n))for p∈(1,n).This problem naturally arises in various applications such as dynamics of non-Newtonian fluid theory,electro-rheology,radiation of heat,plastic moulding and many others.Building on the idea of level-set inequality on fractional maximal distribution functions,it enables us to carry out a global regularity result of the solution via fractional maximal operators.Due to the significance of M_(α)and its relation with Riesz potential,estimates via fractional maximal functions allow us to bound oscillations not only for solution but also its fractional derivatives of orderα.Our approach therefore has its own interest.

THE GRADIENT ESTIMATE OF SUBELLIPTIC HARMONIC MAPS WITH A POTENTIAL

In this paper,we investigate subelliptic harmonic maps with a potential from noncompact complete sub-Riemannian manifolds corresponding to totally geodesic Riemannian foliations.Under some suitable conditions,we give the gradient estimates of these maps and establish a Liouville type result.

The UNIFORM C^(0)ESTIMATE AND WEIGHTED ESTIMATE OF GENERALIZED CHRISTOFFEL-MINKOWSKI PROBLEMS

In this paper,we consider generalized Christo®el-Minkowski problems as followsσ_(k)(u_(ij)+uδ_(ij))/σ_(l)(u_(ij)+uδ_(ij))=|u^(p-1)f(x),x∈S^(n),where 0≤l≤k≤n,p-1>0 and f is positive,and we establish the weighted gradient estimate and uniform C^(0)estimate for the positive convex even solutions,which is a generalization of Guan-Xia[1]and Guan[2].

GRADIENT ESTIMATES AND ENTROPY FORMULAE FOR WEIGHTED p-HEAT EQUATIONS ON SMOOTH METRIC MEASURE SPACES

Let （M,g, e^-fdv） be a smooth metric measure space. In this paper, we con- sider two nonlinear weighted p-heat equations. Firstly, we derive a Li-Yau type gradient estimates for the positive solutions to the following nonlinear weighted p-heat equationand f is a smooth function on M under the assumptionthat the m-dimensional nonnegative Bakry-Emery Ricci curvature. Secondly, we show an entropy monotonicity formula with nonnegative m-dimensional Bakry-Emery Ricci curva- ture which is a generalization to the results of Kotschwar and Ni [9], Li [7].

GRADIENT ESTIMATES FOR POSITIVE SMOOTH f-HARMONIC FUNCTIONS

For Riemannian manifolds with a measure, we study the gradient estimates for positive smooth f-harmonic functions when the ∞-Bakry-Emery Ricci tensor and Ricci tensor are bounded from below, generalizing the classical ones of Yau （i.e., when ： is constant）.

GRADIENT ESTIMATES FOR SOLUTIONS TO QUASILINEAR ELLIPTIC EQUATIONS WITH CRITICAL SOBOLEV GROWTH AND HARDY POTENTIAL

This note is a continuation of the work[17].We study the following quasilinear elliptic equations- △pu-μ/｜x｜p ｜u｜p-2 u=Q（x）｜u｜Np/N-p -2u,x∈R N,where 1 〈 p 〈 N,0 ≤ μ 〈（（N-p）/p）p and Q ∈ L∞（RN）.Optimal asymptotic estimates on the gradient of solutions are obtained both at the origin and at the infinity.

A sharp gradient estimate for the weighted p-Laplacian

Let M be an n-dimensional complete noncompact Riemannian manifold with sectional curvature bounded from below, dμ = e^h（x） dV（x） the weighted measure and △μ,p the weighted p-Laplacian. In this paper we consider the non-linear elliptic equation △μ,pu=-λμ,p｜u｜^p-2ufor p ∈ （1, 2）. We derive a sharp gradient estimate for positive smooth solutions of this equation. As applications, we get a Harnack inequality and a Liouville type theorem..

Hamilton-Souplet-Zhang's gradient estimates for two weighted nonlinear parabolic equations

In this paper, we consider gradient estimates for positive solutions to the following weighted nonlinear parabolic equations on a complete smooth metric measure space with only Bakry-Émery Ricci tensor bounded below: One is $${u_t} = {\Delta _f}u + au\log u + bu$$ with a, b two real constants, and another is $${u_t} = {\Delta _f}u + \lambda {u^\alpha }$$ with λ, α two real constants. We obtain local Hamilton-Souplet-Zhang type gradient estimates for the above two nonlinear parabolic equations. In particular, our estimates do not depend on any assumption on f.

GRADIENT ESTIMATES AND LIOUVILLE THEOREMS FOR LINEAR AND NONLINEAR PARABOLIC EQUATIONS ON RIEMANNIAN MANIFOLDS

In this article, we will derive local elliptic type gradient estimates for positive solutions of linear parabolic equations （△-e/et）u（x,t）＋q（x,t）u^p（x,t）=0 and nonlinear parabolic equations （△-e/et）u（x,t）＋h（x,t）u^p（x,t）=0（p 〉 1） on Riemannian manifolds.As applications, we obtain some theorems of Liouville type for positive ancient solutions of such equations. Our results generalize that of Souplet-Zhang （[1], Bull. London Math. Soc. 38（2006）, 1045-1053） and the author （[2], Nonlinear Anal. 74 （2011）, 5141-5146）.

ELLIPTIC GRADIENT ESTIMATES FOR DIFFUSION OPERATORS ON COMPLETE RIEMANNIAN MANIFOLDS

In this note, we obtain the elliptic estimate for diffusion operator L = △＋△Ф·△ on complete, noncompact Riemannian manifolds, under the curvature condition CD（K, m）, which generalizes B. L. Kotschwar＇s work [5]. As an application, we get estimate on the heat kernel. The Bernstein-type gradient estimate for SchrSdinger-type gradient is also derived.

A STUDY ON GRADIENT BLOW UP FOR VISCOSITY SOLUTIONS OF FULLY NONLINEAR,UNIFORMLY ELLIPTIC EQUATIONS

We investigate sharp conditions for boundary and interior gradient estimates of continuous viscosity solutions to fully nonlinear, uniformly elliptic equations under Dirichlet boundary conditions. When these conditions are violated, there can be blow up of the gradient in the interior or on the boundary of the domain. In particular we de- rive sharp results on local and global Lipschitz continuity of continuous viscosity solutions under more general growth conditions than before. Lipschitz regularity near the boundary allows us to predict when the Dirichlet condition is satisfied in a classical and not just in a viscosity sense, where detachment can occur. Another consequence is this： if interior gra- dient blow up occurs, Perron-type solutions can in general become discontinuous, so that the Dirichlet problem can become unsolvable in the class of continuous viscosity solutions.

Gradient estimates for porous medium equations under the Ricci flow

A local gradient estimate for positive solutions of porous medium equations on complete noncompact Riemannian manifolds under the Ricci flow is derived. Moreover, a global gradient estimate for such equations on compact Riemannian manifolds is also obtained.

An Elliptic Gradient Estimate for A Non-homogeneous Heat Equation on Complete Noncompact Manifolds

Let M be an n-dimensional complete noncompact Riemannian manifold. In this paper, we will give the elliptic gradient estimate for positive smooth solutions to the non-homogeneous heat equation(?_t-△)u(x, t) = A(x, t)when the metric evolves under the Ricci flow. As applications, we get Harnack inequalities to compare solutions at the same time.

Local Hamilton type Gradient Estimates and Harnack Inequalities for Nonlinear Parabolic Equations on Riemannian Manifolds

In this paper,we prove a local Hamilton type gradient estimate for positive solution of the nonlinear parabolic equation ut(x,t)=Δu(x,t)+au(x,t) ln u(x,t)+bu^(α)(x,t),on M×(-∞,∞) with α∈R,where a and b are constants.As application,the Harnack inequalities are derived.

BAS-ADAM:An ADAM Based Approach to Improve the Performance of Beetle Antennae Search Optimizer

In this paper,we propose enhancements to Beetle Antennae search(BAS)algorithm,called BAS-ADAIVL to smoothen the convergence behavior and avoid trapping in localminima for a highly noin-convex objective function.We achieve this by adaptively adjusting the step-size in each iteration using the adaptive moment estimation(ADAM)update rule.The proposed algorithm also increases the convergence rate in a narrow valley.A key feature of the ADAM update rule is the ability to adjust the step-size for each dimension separately instead of using the same step-size.Since ADAM is traditionally used with gradient-based optimization algorithms,therefore we first propose a gradient estimation model without the need to differentiate the objective function.Resultantly,it demonstrates excellent performance and fast convergence rate in searching for the optimum of noin-convex functions.The efficiency of the proposed algorithm was tested on three different benchmark problems,including the training of a high-dimensional neural network.The performance is compared with particle swarm optimizer(PSO)and the original BAS algorithm.

THE EIGENVALUE PROBLEM FOR THE LAPLACIAN EQUATIONS

This article studies the Dirichlet eigenvalue problem for the Laplacian equations △u = -λu, x ∈Ω , u = 0, x ∈δΩ, where Ω belong to R^n is a smooth bounded convex domain. By using the method of appropriate barrier function combined with the maximum principle, authors obtain a sharp lower bound of the difference of the first two eigenvalues for the Dirichlet eigenvalue problem. This study improves the result of S.T. Yau et al.

Enhanced multi-baseline unscented Kalman filtering phase unwrapping algorithm

This paper presents an enhanced multi-baseline phase unwrapping algorithm by combining an unscented Kalman filter with an enhanced joint phase gradient estimator based on the amended matrix pencil model, and an optimal path-following strategy based on phase quality estimate function. The enhanced joint phase gradient estimator can accurately and effectively extract the phase gradient information of wrapped pixels from noisy interferograms, which greatly increases the performances of the proposed method. The optimal path-following strategy ensures that the proposed algorithm simultaneously performs noise suppression and phase unwrapping along the pixels with high-reliance to the pixels with low-reliance. Accordingly, the proposed algorithm can be predicted to obtain better results, with respect to some other algorithms, as will be demonstrated by the results obtained from synthetic data.

On line parameter estimation based on gradient algorithm for fractional order systems

This article investigates the problem of on line parameter estimation for fractionalorder linear systems.Based on the theory of fractional-order calculus,the conventional gradient estimator is extended to the fractional-order areas.The stability of the estimator and the convergence are analysed using the continuous frequency distributed model and indirect Lyapunov method.Finally,numerical simulation examples are given to demonstrate the effectiveness of the proposed schemes.

Interior Gradient Estimates for General Prescribed Curvature Equations

In this paper,we derive the interior gradient estimate for solutions to general prescribed curvature equations.The proof is based on a fundamental observation of G°arding’s cone and some delicate inequalities under a suitably chosen coordinate chart.As an application,we obtain a Liouville type theorem.