parameter uniform convergence of a numerical method for a coupled system of singularly perturbed semilinear reaction diffusion equations with boundary and interior layers abstract in this paper we consider a coupled system of m 2 singularly perturbed semilinear reaction diffusion equations with a discontinuous source term having a discontinuity at a point in the interior of the domain the diffusion term of each equation is multiplied by small singular perturbation parameter but these parameters are assumed to be different in magnitude a numerical method is constructed on a variant of shishkin mesh the approximations generated by this method are shown to be almost second order uniformly convergent with respect to all perturbation parameters numerical results are in support of the theoretical results

parameter uniformly convergent exponential spline difference scheme for singularly perturbed semilinear reaction diffusion problems abstract we consider a dirichlet boundary value problem for singularly perturbed semilinear reaction diffusion equation the problem is discretized using an exponential spline difference scheme derived on the basis of splines in tension on piecewise uniform shishkin type mesh the convergence analysis is given and the method is shown to be almost second order accurate in the discrete maximum norm uniformly in the perturbation parameter e numerical experiments are conducted to demonstrate the theoretical results

new interaction solutions for the 3 1 dimensional jimbo miwa equation abstract in this paper based on the hirota bilinear form and a rational cosh cos type test function we will study new interaction solutions among the lump waves kinky waves and periodic waves for the 3 1 dimensional jimbo miwa equation we analyze the dynamic behaviors of the constructed solutions in four special cases in detail and the dynamic characteristics of the obtained solutions are illustrated by some interesting figures with the help of maple

a new approach to the perfect regulation and the bounded peaking in linear multivariable control systems this paper presents a new approach to the perfect regulation p r and the bounded peaking b p in linear multivariable control systems based on the pole and the eigenvector assignment technique roughly speaking the p r represents an ideal control action which reduces the settling time to 0 while the b p simply means the bounded overshoot in this ideal situation simple frequency domain characterizations of the p r and the b p are derived they reveal some invariance properties of the p r and the b p that provide a powerful tool for achieving the p r and the b p the existence condition for the p r is derived which turns out to be identical to the result obtained in the optimal regulator theory this result is extended to a more general control objective of attaining the p r for one output while keeping the b p for another output a condition on which the p r is realized by an output feedback is also derived a simple design algorithm is proposed for achieving the p r which is essentially the eigenvector assignment procedure finally an application to a real system is discussed

new variable separation solutions of two dimensional burgers system abstract in this paper with the help of g g expansion method we obtain some exact solutions of the riccati equation based on the riccati equation and its exact solutions we find some variable separation solutions with two arbitrary functions of two dimensional burgers system as some special examples these exact solutions can be reduced to variable separation solutions in kink solution soliton solution periodic solution and rational function solution forms

perfect and subperfect regulation in linear multivariable control systems this paper is concerned with the synthesis of a high gain feedback control called perfect regulation p r for linear multivariable systems with external signals the existence condition for p r is derived in which the minimum phase property plays an essential role the use of an observer is discussed for achieving p r under restricted state observation some asymptotically ideal feedback properties are demonstrated such as the complete desensitization the complete servo performance with decoupling and the complete disturbance rejection which formulate the loop tightness for multivariable systems a design method for applying p r to non minimum phase systems called subperfect regulation s p r is proposed based on the factorization of the plant transfer function matrix into the minimum phase part and the totally non minimum phase part it is a multivariable extension of a well known design technique for scalar systems to overcome the difficulty of phase non minimality computational procedures for p r and s p r are discussed an illustrative example is shown

approximation and decomposition of singularly perturbed stochastic hybrid systems this paper considers a singularly perturbed stochastic hybrid system whose state equations are governed by a stochastic switching process which is modelled as a nearly completely decomposable continuous time finite state markov chain approximate models for the slow mode subsystem are derived over the intervals of rapid switching and their accuracy the mean squared errors between the approximate models and the original systems is quantified aggregate models for the subsystem are obtained over a longer period the almost surely exponential stability of the subsystem is studied the behaviour of the fast subsystem which depends on the relative size of the perturbation parameters is analysed the decomposition of the overall system and the switching process together into slow and fast subsystems is investigated the results derived in the paper are shown to hold when the switching process is stationary and irreducible and each group of strongly interacting states is irreducible and time reversible fina

reduced order feedback control of distributed parameter systems via singular perturbation methods abstract implementable feedback control of distributed parameter systems must often be based on reduced order models due to the infinite dimensional nature of the actual open loop system the behavior of controllers designed via reduced order models obtained with singular perturbation techniques is analyzed when such controllers are used in the actual distributed parameter system the closed loop stability is in question the results presented here provide bounds on the smallness of the singular perturbation parameter to ensure stable operation such a priori bounds may be used to evaluate potential reduced order controllers for distributed parameter systems

on the non existence of e uniform finite difference methods on uniform meshes for semilinear two point boundary value problems in this paper fitted finite difference methods on a uniform mesh with internodal spacing h are considered for a singularly perturbed semilinear two point boundary value problem it is proved that a scheme of this type with a frozen fitting factor cannot converge e uniformly in the maximum norm to the solution of the differential equation as the mesh spacing h goes to zero numerical experiments are presented which show that the same result is true for a number of schemes with variable fitting factors

novel solitary wave solutions for the 3 1 dimensional extended jimbo miwa equations abstract this paper retrieves new periodic solitary wave solutions for the 3 1 dimensional extended jimbo miwa equations based on the hirota bilinear method by utilizing maple software as a result the hirota bilinear method is successfully employed and acquired several classes of solitary wave solutions in terms of a new combination of exponential function trigonometric function and hyperbolic functions all solutions have been verified back into its corresponding equation by maple we depicted the physical explanation of the extracted solutions with the free choice of the different parameters by plotting some 3d and 2d illustrations finally it has been illustrated that the executed method is robust and more efficient than other methods and the obtained solutions are trustworthy in the applied sciences

exact three wave solutions for the kp equation a new type of three wave solution periodic two solitary wave solutions for 1 2 d kadomtsev petviashvili kp equation is obtained using the extended three soliton method and with the help of maple

order and the index of singular time invariant linar systems various concepts of order have been proposed in the literature it is shown that many of these are directly related to the index of a system

backlund transformation and solutions of a 3 1 dimensional nonlinear evolution equation in this paper bell polynomial method is used to study the integrability of a 3 1 dimensional nonlinear evolution equation we get the bilinear representation and backlund transformation of the equation the exact solutions are obtained by means of linear superposition principle and homoclinic test approach

a singular perturbation canonical form of invertible systems determination of multivariate root loci a new canonical form of multivariable invertible systems is given this form is suitable for the singular perturbation analysis of such singular problems as multi variable asymptotic root loci under high gain feedback cheap control state variable estimation with weak measurement noise etc appropriate selection and decomposition of state variables into slow and fast changing groups is a key feature that loads to well defined singular perturbation models the selection of state variables is accomplished without explicitly calculating any eigenvalues of the given system but by knowing the direct relationship of the output and its differential coefficients with the input the use of the canonical form to characterize the multivariable root loci under high gain feedback is demonstrated

on the robustness of output feedback control methods to modeling errors singular perturbation analysis is employed to study the robustness of output feedback control schemes for linear time invariant systems the effect of designing the control strategy using a model that neglects unknown parasitic elements is considered it is shown that such a strategy may in general be destabilizing even when the neglected modes are asymptotically stable special structures that will not permit this situation and will guarantee asymptotic stability are illustrated

comment on on the extended applications of homogeneous balance method in this letter we show that all results presented in senthilvelan s paper can be actually obtained by a direct method

subsystems time scales and multimodeling through a couple of naive examples the control theorists are invited to re examine the role of modeling in the study of large scale dynamic systems instead of assuming the existence of n diagonally dominant blocks it is possible to justify one strongly coupled slow core and n weakly coupled fast subsystems this structure is exhibited with a physically meaningful choice of state variables the controls are introduced following the recent concept of multimodeling

exact three wave solutions for the 3 1 dimensional jimbo miwa equation exact three wave solutions including periodic cross kink wave solutions doubly periodic solitary wave solutions and breather type of two solitary wave solutions for the 3 1 dimensional jimbo miwa equation are obtained using the generalized three wave method it is shown that the generalized three wave method with the help of symbolic computation provides an effective and powerful mathematical tool for solving high dimensional nonlinear evolution equations in mathematical physics

two layer fluid formation and propagation of periodic solitons induced by 3 1 dimensional kp equation abstract in this paper multi wave solutions of kadomtsev petviashvili kp equation with time variable coefficient are obtained it is found that the formation of two layer with bright and dark soliton which are symmetric these solitons are transversely periodic as periodic of two soliton and two antisoliton in time with cusps localized in space these two types of waves propagate in an opposite direction relative the higher layer the autonomous behavior may be argued to the fact that the kp is composed of the derivative and the anti derivative terms

breather wave solutions for the generalized shallow water wave equation with variable coefficients in the atmosphere rivers lakes and oceans abstract herein a 3 1 dimensional generalized shallow water wave equation with variable coefficients is studied which describes the flow below a pressure surface in a fluid abundant breather wave solutions are constructed via the hirota bilinear form and a novel test function velocity amplitude and shape changes of the breather wave solutions are shown in some 3d graphs and contour plots by selecting different values of the variable coefficients the novel test function is very effective for obtaining the breather wave solutions of the variable coefficient nonlinear partial differential equations

bifurcation in the presence of small noise the methods of deterministic bifurcation arc sensitive to the addition of small amounts of white noise thus for example in systems whose macroscopic deterministic description arises from an aggregation of microscopically fluctuating dynamics the predictions of deterministic bifurcation may be incorrect here we use laplace s method of steepest descent to study bifurcation in the presence of small noise motivation for this study arises from maxwell s equal area rule for phase transitions in van der waals gases the theory is then applied to the study of the dynamics of noisy constrained or implicitly defined dynamical systems

direct singular perturbation analysis of high gain and cheap control problems a new and direct approach to the analysis and design of high gain feedback and cheap control problems is developed transformation of such singular problems to singular perturbation models of conventional type early at the onset of analysis or design allows a direct application of the readily available singular perturbation literature the approach permits a simple characterization of a multivariable root loci under a high gain feedback and b asymptotic behavior of optimal closed loop poles state and control trajectories performance index and optimal transfer function as the control cost coefficient in the performance index goes to zero

on the solutions of linear differential equations with singular coefficients abstract equations of the form e ax u with e and a square matrices and e singular are considered the controversy that exists in the literature concerning the solutions of such equations is investigated solutions are arrived at through an application of singular perturbation theory

a uniformly convergent method for a singularly perturbed semilinear reaction diffusion problem with discontinuous data this paper deals with a uniform in a perturbation parameter convergent difference scheme for solving a nonlinear singularly perturbed two point boundary value problem with discontinuous data of a reaction diffusion type an error analysis is based on locally exact schemes uniform convergence of the proposed difference scheme on piecewise uniform and log meshes is proven a monotone iterative method which is based on the method of upper and lower solutions is applied to computing the nonlinear difference scheme numerical experiments are presented

convergence in the boundary layer for singularly perturbed equations a singularly perturbed linear finite dimensional ordinary differential equation is considered on the half line 0 the reduced system is assumed to have a unique solution in the sense of distributions it is proved that under reasonable conditions the solution of the full system converges to that of the reduced in the distributional sense and the order of the limiting distribution is computed an application to the optimal linear quadratic regulator with cheap control is described

variable structure control of synchronous generator singularly perturbed analysis synchronous generators have a natural different time scale dynamics that is why for modelling and control design in such systems the methods of singular perturbations are widely used in this paper the possibilities of sliding mode control design for synchronous generators are analysed with this aim the concept of singular perturbation is revised in order to use it for relay control systems the obtained results are used for sliding mode control of synchronous generator

on the existence of positive diagonal p such that pa a t p l 0 given a real square matrix a we address the question does there exist a positive diagonal matrix p such that pa a t p is negative definite we present an algorithm which when applied to a gives a definite yes or no answer to this question if the answer is yes the algorithm will provide one such p

on a class of optimal control problems with an almost cost free solution for a class of linear singular optimal control problems with a nonunique singular arc the solution of the corresponding nearly singular problem is analyzed and a limit solution based on formal singular perturbations is derived a rigorous proof of the correctness of the result is given by constructing a convergent power series satisfying the riccati equation of the nearly singular problem

perturbation analysis of a system of quasi variational inequalities for optimal stochastic scheduling optimal scheduling of continuously evolving dynamical systems involves the analysis of certain partial differential inequalities on solution dependent domains these are the quasi variatlonai inequalities qvi s introduced for such problems by a bensoussan and j l lions they arise naturally in such applications as inventory control and unit commitment scheduling in electric energy systems qvi s are in these contexts the analog of the dynamic programming partial differential equations of r bellman for continuous control problems analytical treatment of qvi s involves not only the solution of partial differential inequalities but also the simultaneous analysis of the free boundaries of the domain of the solution in effect the optimal return function is the solution of the inequalities and the optimal control switching policy is defined by the free boundaries this paper considers a system of two singularly perturbed second order quasi variational inequalities with a turning point in a bounded domain in r1 asymptotic approximations for the free boundaries are derived using standard perturbation analysis for boundary value problems also local regularity properties of the limit solution are obtained the results are illustrated by a parametric study of a simple model of commitment scheduling for a single unit in a power generation system

singular perturbational model reduction in the frequency domain singular perturbational approximations for linear continuous time and discrete time systems are developed in the frequency domain it is shown that the familiar singular perturbational result is an approximation at the origin in the complex plane however if the system has multiple time scale effects other approximations can be obtained at different locations on the negative real axis to emphasize such behavior the relationship between singular perturbational approximations and direct subsystem elimination is also investigated

a class of singularly perturbed semilinear differential equations with interior layers in this paper singularly perturbed semilinear differential equations with a discontinuous source term are examined a numerical method is constructed for these problems which involves an appropriate piecewise uniform mesh the method is shown to be uniformly convergent with respect to the singular perturbation parameter numerical results are presented that validate the theoretical results

brief paper singular perturbation analysis of a receding horizon controller a new receding horizon approach to the linear regulator problem is presented at each instant of time the control strategy is based on the minimization of a quadratic performance index extending over an interval of fixed length into the future connections with standard results are discussed and under general assumptions the closed loop system is shown to be asymptotically stable a potential advantage of our method lies in the fact that a singular perturbation analysis is well posed this indicates that for plants containing slow and fast modes the design procedure can be split up into two stages the resulting composite control exhibits attractive engineering properties use and effectiveness of the proposed techniques are demonstrated by investigating a practical example

singular perturbation analysis of linear systems with scalar quantized control abstract the effect of a scalar quantized control input on an otherwise linear singularly perturbed control system is analyzed the cases of open loop and closed loop control are studied analytical techniques are derived that allow the decoupling of such systems into fast and slow subsystems for analysis and design the results of a numerical example are given to demonstrate these techniques

homoclinic breather and rogue wave solutions to maccari equation abstract in this work a new extended homoclinic test combining with breather period limit method bplm for seeking homoclinic breather and rogue wave solution of nonlinear evolution 2 1 d maccari equation is proposed two new families of homoclinic breather wave solution and rational homoclinic solution homoclinic rogue wave for these equations are obtained respectively moreover dynamical features of homoclinic breather and rogue wave solution are exhibited this result shows that both homoclinic breather and rogue wave can occur in the same nonlinear system

asymptotic stability of nonlinear multiparameter singularly perturbed systems sufficient conditions are derived to guarantee the asymptotic stability of a class of nonlinear singularly perturbed systems with several perturbation parameters of the same order estimates of the region of attraction and bounds on the small parameters are obtained

a study of lump type and interaction solutions to a 3 1 dimensional jimbo miwa like equation abstract using generalized bilinear equations we construct a 3 1 dimensional jimbo miwa like equation which possesses the same bilinear type as the standard 3 1 dimensional jimbo miwa equation classes of lump type solutions and interaction solutions between lump type and kink solutions to the resulting jimbo miwa like equation are generated through maple symbolic computation we discuss the conditions guaranteeing analyticity and positiveness of the solutions by taking special choices of the involved parameters 3d plots are presented to illustrate the dynamical features of the solutions

a reduction process for perturbed markov chains given a perturbed markov chain with generator a varepsilon a 0 varepsilon a 1 varepsilon 2 a 2 cdots it is shown that to each time scale t t varepsilon t varepsilon 2 cdots is associated a reduced markov chain this process of reduction has only finitely many steps it is given in a recursive way and each reduced chain admits as state space the recurrent classes of the reduced chain defined at the previous stage it is shown that if lambda is an eigenvalue different from zero of the generator of the kth reduced chain with multiplicity p then a varepsilon admits p eigenvalues lambda varepsilon such that lambda varepsilon varepsilon k lambda o varepsilon k a basis of ker a 0 resp ker a 0 which is robust to the perturbation is constructed

state decomposition for singular perturbation order reduction a modal approach singular perturbation techniques for order reduction are based on decomposing the high order system into two parts approximation of the high order system dynamics by the low order model is mainly dependent on this partition this paper introduces a measure for the degree of dependence of the state variables on the high order dominant eigenvalues by means of this measure it is not only possible to find an appropriate decomposition yielding a good low order model but also to get an indication of the order of the reduced model two illustrative examples demonstrate the main results

uniform fourth order difference scheme for a singular perturbation problem the numerical solution of a nonlinear singularly perturbed two point boundary value problem is studied the developed method is based on hermitian approximation of the second derivative on special discretization mesh numerical examples which demonstrate the effectiveness of the method are presented

a singular perturbation approach to modeling and control of markov chains finite state continuous time markov processes with weak interactions are modeled as singularly perturbed systems aggregate states are obtained using a grouping algorithm two time scale expansions simplify cost equations and lead to decentralized optimization algorithms

sliding mode control for singularly perturbed systems this paper addresses the design of sliding mode controllers for singularly perturbed systems the full order system is separated into slow and fast subsystems and a sliding mode controller is designed for each subsystem a composite of these controls is then implemented on the full order system it is shown that if specific reaching conditions are satisfied for the reduced order models and an additional condition is met then the reaching conditions are also satisfied for the full order model furthermore sufficient conditions are derived which guarantee that the approximation error between the controlled reduced order response and the controlled full order model response is small a numerical example is given which successfully demonstrates the techniques developed in this paper

a multimodel approach to stochastic nash games a decentralized filtering and control scheme is presented for obtaining low order nash equilibrium strategies for decision makers using different models of the same large scale system multiparameter singular perturbations are employed to capture the multimodel nature of the fast dynamic subsystems interconnected through slow dynamic variables the small parameters are appropriately scaled so that the variables in both time scales are well defined the decision makers have decentralized information structures and are constrained to use only finite dimensional compensators of a specified dimension it is shown that the proposed scheme is in fact the asymptotic limit of the exact solution as the small parameters go to zero

sub optimal control of discrete regulator problems via time scale decomposition the paper investigates an infinite time discrete regulator problem with a two time scale property sufficient conditions are given for the existence of a suboptimal solution and a near optimum design method is proposed via the time scale decomposition approach the method reduces the system order and has a mode decoupling property

singular perturbations and time scale methods in control theory survey 1976 1983 recent progress in the use of singular perturbation and two time scale methods of modeling and design for control systems is reviewed over 350 references are organized into major problem areas representative issues and results are discussed with a view to outlining research directions and indicating potential areas of application the survey is aimed at engineers and applied mathematicians interested in model order reduction separation of time scales and allied simplified methods of control system analysis and design the exposition does not assume prior knowledge of singular perturbation methods