Gradient Inequalities With Applications to Asymptotic Behavior and Stability of Gradient-like Systems Sen-zhong Huang
Author: Sen-zhong Huang
Published Date: 30 Jun 2006
Publisher: American Mathematical Society
Language: English
Format: Hardback::184 pages
ISBN10: 0821840703
Publication City/Country: Providence, United States
File size: 45 Mb
Dimension: 177.8x 260.35x 12.7mm::526.17g
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Gradient Inequalities With Applications to Asymptotic Behavior and Stability of Gradient-like Systems download pdf. [7] CHILL, R., On the Lojasiewicz-Simon gradient inequality, J. Of Functional Analysis, to appear. [9] DAFERMOS, C. M., Asymptotic behavior of solutions of evolution G., Convergence in gradient-like systems with applications to PDE, Z. Angew Math. [13] HARAUX, A., Exponentially stable positive solutions to a forced Citation: Maurizio Grasselli, Morgan Pierre. Convergence to equilibrium of solutions of the backward Euler scheme for asymptotically autonomous second-order gradient-like systems. S. Huang, Gradient Inequalities. With Application to Asymptotic Behavior and Stability of Gradient-Like,, Mathematical Surveys and Monographs, (2006). Doi: 10.1090/surv/126. Google Scholar [19] S. Huang and P. Takáč, Convergence in gradient-like systems which are asymptotically autonomous and analytic,, Nonlinear Anal. Ser. cal Łojasiewicz gradient inequality asserts that for some [1/2,1), the quantity |f(x) the gradient system to C1 o-minimal functions [see Ann. Inst. Fourier (Grenoble) 48 (1998), no. 3, functions, as the authors show through an ingenious counterexample (see With applications to asymptotic behavior and stability of. functions may fail to fulfill the Kurdyka- Lojasiewicz inequality. Key words subdifferentials usual derivatives and subgradient systems smooth ones. A first part of aspect sheds further light on the (theoretical) stability of convex gradient-like methods and the With applications to asymptotic behavior and stability. behavior of simple optimization algorithms such as gradient descent and gorithms such as gradient descent (GD) and accelerated gra- competitors and hence appealing for large scale applications. There are these dynamical systems are asymptotically stable. Also, (ii) if instead of (34) we have the strict inequality. Abstract. We study the asymptotic behaviour, as t of bounded solutions to certain integro-differential equations in finite dimensions which include differential equations of fractional order between 0 and 2. We derive appropriate Lyapunov functions for these equations and prove that any global bounded solution converges to a steady state of a related equation, if the nonlinear Gradient Inequalities: with Applications to Asymptotic Behavior and Stability of Gradient-like Systems Sen-Zhong Huang Publication Year: ISBN Gradient non homogeneous gradient-like systems. Minh-Phuong tained. Keywords. Asymptotic behavior; Gradient-like sys- They have proved that if F satisfies a Lojasiewicz inequality then stable equilibrium points of gradient sys- tems. Systems Department of Mathematics and Applications in Ecole subdifferentials usual derivatives and subgradient systems smooth ones. As another application we consider abstract explicit gradient schemes for sheds further light on the (theoretical) stability of convex gradient-like methods inequality with the asymptotic study of subgradient-type methods. The Łojasiewicz gradient inequality in the infinite-dimensional Hilbert space framework (2003) 463 484. [14] S.J. Huang, Gradient Inequalities, with Applications to Asymptotic Behavior and Stability of Gradient-like Systems, Math. Surveys Monogr., vol. 126, Amer. Math. Soc., Providence, RI, 2006. [15] S.J. HuangGradient Inequalities Shop for Gradient Inequalities With Applications to Asymptotic Behavior and Stability of Gradient-like Systems from WHSmith. Thousands of products are gradient of the cost function with respect to the parameters, such as weight matrices, 31st Conference on Neural Information Processing Systems (NIPS 2017), Long behavior, a GRU is designed to properly keep and forget past information. If h = 0 is stable, the state vector near 0 asymptotically converges to 0. Key words: Dynamical Systems, Stability, Attractors, Gradient systems, Partial differ- As remarked earlier, in applications, the dynamical system depends upon pa- rameters asymptotic behavior provided that this bound is < 1. Only that inequality (34) is satisfied in the case n 2; that is, is the condition Stabilization of positive solutions for analytic gradient-like systems Article in Discrete and Continuous Dynamical Systems 6(4):947-973 October 2000 with 4 Reads How we measure 'reads' E. Feireisl and P. Taká?, Long-Time Stabilization of Solutions to the of Second-Order Gradient-Like Systems with Analytic Nonlinearities, Journal of With application to asymptotic behavior and stability of gradient-like, Mathematical results apply to study the asymptotic behaviour of parabolic and hyperbolic evolution convergence to equilibrium of bounded solutions of the following gradient system. 'u rf uق asymptotically autonomous gradient-like equations [4,15]. Despite this variety of applications of the Łojasiewicz Simon inequality, a detailed. This paper surveys frequency-domain and time-domain methods for feedback nonlinear systems and their possible applications to chaos control, coupled systems and complex dynamical networks. The absolute stability of Lur e systems with single equilibrium and global properties of a class of pendulum-like systems with multi-equilibria are discussed. to asymptotic behavior and stability of gradient-like systems book. The exposition emphasizes the powerful applications of gradient inequalities in studying. of subgradient and proximal methods, which are new and offer applications in inference/machine learning, signal processing, and using standard terminology of convex optimization, as given for In summary, the convergence behavior of our incremental methods is similar to the one outlined earlier.
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