Step-like contrast structure for class of nonlinear singularly perturbed optimal control problem

The step-like contrast structure for a class of nonlinear singularly perturbed optimal control problems is considered. The existence of the step-like contrast structure for the singularly perturbed optimal control problem is proved by equivalence, which is based on the necessary conditions. The authors not only give the conditions under which there exists a step-like contrast structure, but also determine where the internal transition time is. Meanwhile, the uniformly valid asymptotic expansion of the step-like contrast structure solution is constructed by the direct scheme method. Finally, an example is presented to show the result.     

Solving Nonlinear Delay-Differential-Algebraic Equations with Singular Perturbation via Block Boundary Value Methods

Block boundary value methods (BBVMs) are extended in this paper to obtain the numerical solutions of nonlinear delay-differential-algebraic equations with singular perturbation (DDAESP). It is proved that the extended BBVMs in some suitable conditions are globally stable and can obtain a unique exact solution of the DDAESP. Besides, whenever the classic Lipschitz conditions are satisfied, the extended BBVMs are preconsistent and $p$th order consistent. Moreover, through some numerical examples, the correctness of the theoretical results and computational validity of the extended BBVMs is further confirmed.     

A gap in the essential spectrum of a cylindrical waveguide with a periodic aperturbation of the surface

It is proved that small periodic singular perturbation of a cylindrical waveguide surface may open a gap in the essential spectrum of the Dirichlet problem for the Laplace operator. If the perturbation period is long and the caverns in the cylinder are small, the gap certainly opens. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Persistence of traveling wave solutions in a bio-reactor model with nonlocal delays

By considering the random migration of individuals and the period of consuming the captured nutrient, we first introduce the nonlocal delays into a bio-reactor model. The persistence of nontrivial traveling wave solutions then is proved by combining the geometric singular perturbation theory with the center manifold theorem. From the viewpoint of biology, our results indicate that the nonlocality induced by small average delays is harmless to the growth of the species.     

Asymptotic Behavior of Forced Nonlinear Delay Differential Equations with Impulses

The purpose of this paper is to investigate the asymptotic behavior of solutions of the forced nonlinear delay differential equations with impulses Our results, which hold for linear and nonlinear equations, forced and unforced equations, impulsive and nonimpulsive equations, improve and generalize the known results recently obtained in [8]. Received September 7, 1997, Revised May 26, 1998, Accepted July 15, 1998     

Singularly perturbed boundary value problems

1.IntroductionThesingularlypedurbednonlineartwO-pointboundaryvalueproblemsinthisarticleareofthefollowingtypewhereconstantparametere>0,constantsx?ER,asERe.Thestatesofthesystem(XI(t),Xz(t),YI(t),YZ(t)),whichtakevaluesinRxRexRadxRe,containonepart(XI(t),YI(t))calledslowvariablesandonepart(XZ(t),YZ(t))calledfastvariables.Welookforthesolutions(XI,X2,YI,Y2)EL'(0,T;R m m)whichsatisfysystem(1).Suchasituationiscommonintheapplications.Itappearsforexampleineconomicmodelstotakeintoaccoun…     

The Dirichlet problem for a conservation law with a multiplicative stochastic perturbation

In this paper, we are interested in the Dirichlet boundary value problem for a multi-dimensional nonlinear conservation law with a multiplicative stochastic perturbation. Using the concept of measure-valued solutions and Kruzhkov?s semi-entropy formulations, a result of existence and uniqueness of the entropy solution is proved.     

Generalizing the Paley-Wiener perturbation theory for Banach spaces

We extend the Paley-Wiener pertubation theory to linear operators mapping a subspace of one Banach space into another Banach space.

     

BOUNDARY VALUE PROBLEMS TO A CLASS OF SINGULARLY PERTURBED NONLINEAR SYSTEM WITH TURNING POINTS

ConsidertheboundaryvalueproblemtothesingularlyperturbednonlinearsystemintheformofwhereEisasmallpositiveparameter,y=(yi,'..)y.),f=(fi,...)fn),A=(Al,'',A.)andB=(BI,sn).MohasdiscussedtheseDroblemsundervarioussituationsusingthemethodofdifferentialinequalities[1--'].Inthispaperweconsiderthecaseoffi.:(0,y,y')=0anduseaclassofnewboundarylayerfunctions.TheoremAssumethat(a)fi(x,y,z)Hfi(x,yi,'',yn,if)''jz.)eC'{xE[~l,1],yie[di,ei],ifER}andtherearepositivenondecreasingcontinuousfunctionsif,i=1,2,'…     

Hybrid dynamical control based on consensus algorithms for current sharing in DC-bus microgrids

The main objective of this work is to propose a novel paradigm for the design of two layers of control laws for DC-bus microgrids in islanded mode. An intensive attention will be paid to the inner control level for the regulation of DC–DC electronic power converters, where the use of Hybrid Dynamical System theory will be crucial to formulate and exploit switching control signals in view of reducing the dissipated energy and improving system performance. Indeed, this recent theory is well suited for analysing power electronic converters, since they combine continuous (voltage and currents) and discrete (on–off state of switches) signals avoiding, in this way, the use of averaged models. Likewise, an outer control level for controlling DC-bus microgrids will be developed to provide a distributed strategy that makes the microgrid scalable and robust with respect to blackouts of sources and/or loads, as well as, to provide a balance in the system of charge of the batteries, following the principle of Multi-Agent System theory. In this distributed strategy, they are several crucial and innovative aspects to be regarded such as the different converter architectures, the hybrid and nonlinear nature of these converters. Stability properties are guaranteed by using singular perturbation analysis.