Nonlocal reaction—diffusion equations and nucleation

Rubinstein, Jacob; Sternberg, Peter

doi:10.1093/imamat/48.3.249

Cited by 321 publications

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“…However, system (14) is iIOSS. To see this, consider the partial coordinates and feedback transformation (17) with ( 1 ) = o (0log(1+ 1 )).As the linear part of the -subsystem of system (17) (17) we obtain _ V 0kV () + v v 2 + y y 2 + pj1 + 1 jj 2 (z)jjj (18) for some positive real number p. Consider now the positive-definite and radially unbounded function…”

Section: Examplesmentioning

confidence: 99%

Norm Estimators and Global Output Feedback Stabilization of Nonlinear Systems With ISS Inverse Dynamics

Kaliora

Astolfi

Praly

2006

IEEE Trans. Automat. Contr.

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Abstract-A preliminary result on the construction of norm estimators for general nonlinear systems that do not necessarily admit a input output to state stable (IOSS)-Lyapunov characterization is given. Furthermore, an output feedback stabilization scheme is presented that makes use of norm estimators. This construction extends some previous results allowing for more general nonlinearities. Two examples complete the work.Index Terms-Input-output-to-state stability, nonlinear systems, norm estimators, output feedback. I. INTRODUCTIONIt has been clear for years now that, for nonlinear systems, global uniform observability alone does not imply the existence of a convergent observer, or even more so, the existence of a (globally) stabilizing (dynamic) output feedback control law. On the contrary, it has been shown in [5] that globally observable systems that do not however possess the unboundedness observability property cannot be stabilized by any dynamic output feedback scheme.It is now a growing trend to use high-gain observers as part of an output feedback stabilization architecture for a variety of nonlinear systems that exhibit a triangular structure [1]. In [6], a high-gain technique was introduced where the gain was time varying, i.e., tuned on line. Motivated by the above reference the authors of [4] considered an output feedback made of the combination of high, variable-gain observer and controller. Both the previous output feedbacks are inherently nonlinear, while in [8] a linear observer/controller (again with varying high gain) proves to be sufficient for the output feedback stabilization for a class of nonlinear systems.On the other hand, for nonlinear systems written in observability canonical form and that are input output to state stable (IOSS) globally convergent observers can be designed via the idea of norm estimators (see [3] for this and other related definitions), as shown in [7], where again a "high-gain" idea is used, but the gain is this time tuned via the norm estimator.Motivated by [7] in this note we provide an approach toward the design of norm estimators for systems that are not necessarily IOSS. As applications, we examine a nonlinear system that-in open loop-exhibits finite escape time and nonlinear systems that are linear in the unmeasured state.Finally, we extend the result of [8]. Namely, under the assumption of existence of a norm estimator, it is shown that the restrictions on the growth of the system nonlinearities can be relaxed, allowing, thus, for a larger class of nonlinear systems to be stabilized with this approach. where x 2 n is the state, u 2 is the input, and y 2 is the output, respectively. In [3] it is explained how the assumption that system (1) is input output to state stable (IOSS) can be used for the design of a first order dynamical system _ ! = (!; u; y) such that a function of !(t) serves asymptotically as an upper limit of the norm of x. In this design it is instrumental to assume the existence of a IOSS-Lyapunov function V (x) that satisfies the dynami...

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Section: Examplesmentioning

confidence: 99%

Norm Estimators and Global Output Feedback Stabilization of Nonlinear Systems With ISS Inverse Dynamics

Kaliora

Astolfi

Praly

2006

IEEE Trans. Automat. Contr.

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“…Given any collection of spheres which evolve under the averaged mean curvature flow, one finds-provided the spheres are far enough from each other so that they do not coalesce-that the smaller spheres shrink, and the larger ones grow [64]. This has been termed-somewhat humorously-the survival of the fattest.…”

Section: The Example Of Grayson In Three Dimensionsmentioning

confidence: 99%

A numerical scheme for moving boundary problems that are gradient flows for the area functional

Mayer

2000

Eur. J. Appl. Math

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Many moving boundary problems that are driven in some way by the curvature of the free boundary are gradient flows for the area of the moving interface. Examples are the Mullins-Sekerka flow, the Hele-Shaw flow, flow by mean curvature, and flow by averaged mean curvature. The gradient flow structure suggests an implicit finite differences approach to compute numerical solutions. The proposed numerical scheme will allow to treat such free boundary problems in both IR 2 and IR 3 . The advantage of such an approach is the re-usability of much of the setup for all of the different problems.As an example of the method we will compute solutions to the averaged mean curvature flow that exhibit the formation of a singularity. IntroductionIn this paper we will study geometric evolution problems for surfaces driven by curvature. These moving boundary problems arise from models in physics and the material sciences, and they describe phase changes, or more generally, phenomena in fluid flow.The Hele-Shaw model (named after H.S. ) describes the pressure of two immiscible viscous fluids trapped between two parallel glass plates and has attracted considerable attention in the literature, both on the analytical side [23,27,28,29,33,38,51,54,63] and on the numerical side [2,6,7,8,12,25,26,49,50,65]. This list of references also encompasses work related to the one-sided Hele-Shaw model, which arises as a limit when the viscosity of one of the fluids approaches zero. Furthermore, the Hele-Shaw model has classically been considered on a bounded domain, but many of the references above also address the problem on an unbounded domain. We shall only consider the problem on all of IR n ; for the precise formulation see (4.4).The Mullins-Sekerka model was first proposed by (and much later named after) Mullins and Sekerka to study solidification of materials of negligible specific heat [58]. For a while this model was also called the Hele-Shaw model, leading to a somewhat confused literature. Pego [61], and then Alikakos, Bates, and Chen [3], and also Stoth [68], established this model as a singular limit of the Cahn-Hilliard equation [17], a fourth order partial differential equation modelling nucleation and coarsening phenomena in a melted binary alloy. The Mullins-Sekerka model is considered to be a good model to describe the stage of Ostwald ripening in phase transitions, which is the stage after the initial nucleation has 2 U. F. Mayer essentially been completed and where some particles grow at the cost of others in an effort to decrease interfacial energy (surface area). The literature focuses mainly on the modelling and analytical aspects of the problem [11,16,18,19,21,32,34,46,56,66,71,72] though there are numerical simulations for the two-dimensional case [10,73]. We shall consider the two-phase problem on all of IR n , see (4.3).The mean curvature flow is probably the most studied geometric moving boundary problem, it is in some way also the simplest: the normal velocity is equal to the mean curvature of the interface. This sha...

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“…For the scalar Allen-Cahn variational inequality (and also for the vector-valued Allen-Cahn variational inequality with N = 2) we obtain an explicit solution for the following problem: Given two circles with radii r 1 and r 2 which do not intersect, then the sharp interface problem for volume conserved motion by curvature results in the following system of ODEs: r 1 = − 1 r 1 + λ, r 2 = − 1 r 2 + λ together with the condition of volume conservation 0 = 1 2 (r 2 1 + r 2 2 ) , where the initial radii r 1 (0) and r 2 (0) are known. This can be solved analytically, see [36]. We can use this problem in the case of N = 3 by considering two decoupled systems, that is four circles that do not intersect where phase 1 occupies two circles, phase 2 occupies the other two circles and phase 3 is present outside these four circles, see Figure 4.…”

Section: Vector-valued Allen-cahn Variational Inequality With Volume mentioning

confidence: 99%

Nonlocal Allen-Cahn systems: analysis and a primal-dual active set method

Blank

Garcke

Sarbu

et al. 2013

IMA Journal of Numerical Analysis

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We show existence and uniqueness of a solution for the non-local vector-valued Allen-Cahn variational inequality in a formulation involving Lagrange multipliers for local and non-local constraints. Furthermore, we propose and analyze a primal-dual active set method for local and non-local vector-valued Allen-Cahn variational inequalities. Convergence of the primal-dual active set algorithm is shown by interpreting the approach as a semi-smooth Newton method and numerical simulations are presented demonstrating its efficiency.

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Nonlocal reaction—diffusion equations and nucleation

Cited by 321 publications

References 0 publications

Norm Estimators and Global Output Feedback Stabilization of Nonlinear Systems With ISS Inverse Dynamics

Norm Estimators and Global Output Feedback Stabilization of Nonlinear Systems With ISS Inverse Dynamics

A numerical scheme for moving boundary problems that are gradient flows for the area functional

Nonlocal Allen-Cahn systems: analysis and a primal-dual active set method

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