scholarly journals Radial Symmetry of Classical Solutions for Bellman Equations in Ergodic Control

1999 ◽  
pp. 255-264
Author(s):  
Hiroaki Morimoto ◽  
Yasuhiro Fujita
Keyword(s):  
Radial Symmetry ◽  
Classical Solutions ◽  
Ergodic Control ◽  
Bellman Equations

scholarly journals Stability and convergence of second order backward differentiation schemes for parabolic Hamilton–Jacobi–Bellman equations

2021 ◽  
Author(s):  
Olivier Bokanowski ◽  
Athena Picarelli ◽  
Christoph Reisinger
Keyword(s):  
Parabolic Equations ◽  
Linear Equations ◽  
Second Order ◽  
Stability And Convergence ◽  
Classical Solutions ◽  
Hjb Equations ◽  
Bellman Equations ◽  
Linear Parabolic Equations ◽  
Viscosity Sense ◽  
Hamilton Jacobi Bellman

AbstractWe study a second order Backward Differentiation Formula (BDF) scheme for the numerical approximation of linear parabolic equations and nonlinear Hamilton–Jacobi–Bellman (HJB) equations. The lack of monotonicity of the BDF scheme prevents the use of well-known convergence results for solutions in the viscosity sense. We first consider one-dimensional uniformly parabolic equations and prove stability with respect to perturbations, in the $$L^2$$ L 2 norm for linear and semi-linear equations, and in the $$H^1$$ H 1 norm for fully nonlinear equations of HJB and Isaacs type. These results are then extended to two-dimensional semi-linear equations and linear equations with possible degeneracy. From these stability results we deduce error estimates in $$L^2$$ L 2 norm for classical solutions to uniformly parabolic semi-linear HJB equations, with an order that depends on their Hölder regularity, while full second order is recovered in the smooth case. Numerical tests for the Eikonal equation and a controlled diffusion equation illustrate the practical accuracy of the scheme in different norms.

Ergodic Control of Diffusion Processes

2009 ◽  
Author(s):  
Ari Arapostathis ◽  
Vivek S. Borkar ◽  
Mrinal K. Ghosh
Keyword(s):  
Diffusion Processes ◽  
Ergodic Control

Remarks on nonlinear elastic waves with null conditions

2020 ◽  
Vol 26 ◽  
pp. 121
Author(s):  
Dongbing Zha ◽  
Weimin Peng
Keyword(s):  
Initial Data ◽  
Global Existence ◽  
Elastic Waves ◽  
Wave Equations ◽  
Alternative Proof ◽  
Classical Solutions ◽  
Small Initial Data ◽  
Hyperelastic Materials ◽  
Variational Structure ◽  
Nonlinear Elastic

For the Cauchy problem of nonlinear elastic wave equations for 3D isotropic, homogeneous and hyperelastic materials with null conditions, global existence of classical solutions with small initial data was proved in R. Agemi (Invent. Math. 142 (2000) 225–250) and T. C. Sideris (Ann. Math. 151 (2000) 849–874) independently. In this paper, we will give some remarks and an alternative proof for it. First, we give the explicit variational structure of nonlinear elastic waves. Thus we can identify whether materials satisfy the null condition by checking the stored energy function directly. Furthermore, by some careful analyses on the nonlinear structure, we show that the Helmholtz projection, which is usually considered to be ill-suited for nonlinear analysis, can be in fact used to show the global existence result. We also improve the amount of Sobolev regularity of initial data, which seems optimal in the framework of classical solutions.

On the Radial Symmetry Property for Harmonic Functions

2020 ◽  
Vol 64 (10) ◽  
pp. 9-19
Author(s):  
V. V. Volchkov ◽  
Vit. V. Volchkov
Keyword(s):  
Harmonic Functions ◽  
Symmetry Property ◽  
Radial Symmetry

Pollen characterization of ornamental species of Mammillaria Haw. (Cactaceae/Cactoideae)

2019 ◽  
Vol 6 (1) ◽  
pp. 13 ◽  
Author(s):  
Denise M. D. S. Mouga ◽  
Gabriel R. Schroeder ◽  
Nilton P. Vieira Junior ◽  
Enderlei Dec
Keyword(s):  
Pollen Morphology ◽  
Pollen Grains ◽  
Radial Symmetry ◽  
Morphological Data ◽  
Medium Size ◽  
Ornamental Species

The pollen morphology of thirteen species of Cactaceae was studied: M. backebergiana F.G. Buchenau, M. decipiens Scheidw, M. elongata DC, M. gracilis Pfeiff., M. hahniana Werderm., M. marksiana Krainz, M. matudae Bravo, M. nejapensis R.T. Craig & E.Y. Dawson, M. nivosa Link ex Pfeiff., M. plumosa F.A.C. Weber, M. prolifera (Mill.) Haw, M. spinosissima var. “A Peak” Lem. and M. voburnensis Scheer. All analysed pollen grains are monads, with radial symmetry, medium size (M. gracilis, M. marksiana, M. prolifera, large), tricolpates (dimorphs in M. plumosa [3-6 colpus] and M. prolifera [3-6 colpus]), with circular-subcircular amb (quadrangular in M. prolifera and M. plumosa with six colpus). The pollen grains presented differences in relation to the shape and exine thickness. The exine was microechinate and microperforated. The pollen morphological data are unpublished and will aid in studies that use pollen samples. These pollen grains indicate ornamental cacti.

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