a note on the error analysis of time compression approximations the accuracy of the time compression analysis tca is analyzed by comparison with a numerical solution both the standard tca and a new modified tca are considered for a power law diffusivity and constant surface flux as expected the error of the approximations decreases with increasing power and the error of the modified tca is about half the error of the standard tca in a second part the errors of the two tcas are measured using a simple analytical solution instead of a numerical solution it is shown that the conclusions remain the same for the analytical and numerical solutions the advantage of using the analytical solution is to obtain simple analytical expressions showing the influence of parameters this is done to estimate the maximum error of both tcas a practical estimate of the errors can be obtained from equations which only require knowledge of the soil water diffusivity it appears that for real soils the errors of the tca are always 1 and thus are a very reliable tool for practical problems although not studied systematically it also appears that gravity effects reduce the errors of the tca so that the error obtained in the absence of gravity provides a conservative estimate when gravity is present

integrable forms of the one dimensional flow equation for unsaturated heterogeneous porous media the equation for the horizontal transport of a liquid in an unsaturated scale heterogeneous porous medium is t x x c x x e x x c e d a systematic search for lie backlund symmetries leads to the requirement that c a b 2 as in the homogeneous 1 case more generally e may be 1 mx 1 3 2 c or exp mx 3c 2 in these cases the transport equation may be linearized and solved exactly examples of more complicated heterogeneous extensions are presented for the integrable nonlinear diffusion equations and for burgers equation

scope for further analytical solutions for constant flux infiltration into a semi infinite soil profile or redistribution in a finite soil profile we attempt to generate new solutions for the moisture content form of the one dimensional richards 1931 equation using the lisle 1992 equivalence mapping this mapping is used as no more general set of transformations exists for mapping the one dimensional richards equation into itself starting from a given solution the mapping has the potential to generate an infinite number of new solutions for a series of nonlinear diffusivity and hydraulic conductivity functions we first seek new analytical solutions satisfying richards equation subject to a constant flux surface boundary condition for a semi infinite dry soil starting with the burgers model the first iteration produces an existing solution while subsequent iterations are shown to endlessly reproduce this same solution next we briefly consider the problem of redistribution in a finite length soil in this case lisle s equivalence mapping is generalized to account for arbitrary initial conditions as was the case for infiltration however it is found that new analytical solutions are not generated using the equivalence mapping although existing solutions are recovered

reciprocal transformations of two component hyperbolic systems and their invariants abstract criteria for the equivalence under reciprocal transformations of two component hyperbolic systems are established in terms of reciprocal invariants these results are applied to a 1 1 dimensional isentropic gasdynamics system to determine reciprocal equivalence to the canonical system with a polytropic gas law the four parameter model pressure density relations for which this reciprocal equivalence holds are parameterized in terms of the lagrangian signal speed

the problem of absolute stability a dynamic programming approach we consider the problem of absolute stability of a feedback system composed of a linear plant and a single sector bounded nonlinearity pyatnitskiy and rapoport used a variational approach and the maximum principle to derive an implicit characterization of the most destabilizing nonlinearity in this paper we address the same problem using a dynamic programming approach we show that the corresponding value function is composed of simple building blocks which are the generalized first integrals of appropriate linear systems we demonstrate how the results can be used to design stabilizing switched controllers

brief paper a full characterization of stabilizability of bimodal piecewise linear systems with scalar inputs this paper studies open loop stabilization problem for bimodal systems with continuous vector field it is based on the earlier work of the authors on the controllability problem for the same class of systems a full characterization of stabilizability is established by presenting algebraic necessary and sufficient conditions it is also shown that controllability implies stabilizability for these systems in a very similar fashion to the linear case

stability criteria for switched and hybrid systems the study of the stability properties of switched and hybrid systems gives rise to a number of interesting and challenging mathematical problems the objective of this paper is to outline some of these problems to review progress made in solving them in a number of diverse communities and to review some problems that remain open an important contribution of our work is to bring together material from several areas of research and to present results in a unified manner we begin our review by relating the stability problem for switched linear systems and a class of linear differential inclusions closely related to the concept of stability are the notions of exponential growth rates and converse lyapunov theorems both of which are discussed in detail in particular results on common quadratic lyapunov functions and piecewise linear lyapunov functions are presented as they represent constructive methods for proving stability and also represent problems in which significant progress has been made we also comment on the inherent difficulty in determining stability of switched systems in general which is exemplified by np hardness and undecidability results we then proceed by considering the stability of switched systems in which there are constraints on the switching rules through both dwell time requirements and state dependent switching laws also in this case the theory of lyapunov functions and the existence of converse theorems are reviewed we briefly comment on the classical lur e problem and on the theory of stability radii both of which contain many of the features of switched systems and are rich sources of practical results on the topic finally we present a list of questions and open problems which provide motivation for continued research in this area

extension of the fujita solution to air and water movement in soils for specific soil properties the fundamental solution of fujita 1952b is extended to include the effect of air movement on horizontal absorption this exact solution is then used to determine the accuracy and range of applicability of two approximations to the sorptivity with air effects previously presented by parlange et al 1982 and sander and parlange 1984 the optimal sorptivity of parlange et al has very good accuracy while the surface water content s is near saturation but decreases in accuracy as s decreases the approximate sorptivity of sander and parlange however proves to be extremely accurate for all s with a maximum error of only 0 5 sander and parlange additionally give an approximation to the water profile which is also shown by comparison with the exact solution to have a maximum error of only 0 5 the aim of this paper is primarily of a theoretical nature in that we use the exact solution to check the accuracy of the approximate solutions as stated above alternatively the solution may be used for checking the accuracy of numerical models

an analytical solution to richards equation for time varying infiltration an analytical solution is developed for richards equation for time varying infiltration the infiltration rates must be specified as piecewise constants with time the answer is expressed as a sum of two terms the first is a function of the instantaneous infiltration and the second an integral which accounts for the water distribution within the profile for the previous infiltration history the solutions assume specific forms for the soil water diffusivity and hydraulic functions used earlier by broadbridge and white 1988

stability analysis of switched systems using variational principles an introduction many natural and artificial systems and processes encompass several modes of operation with a different dynamical behavior in each mode switched systems provide a suitable mathematical model for such processes and their stability analysis is important for both theoretical and practical reasons we review a specific approach for stability analysis based on using variational principles to characterize the most unstable solution of the switched system we also discuss a link between the variational approach and the stability analysis of switched systems using lie algebraic considerations both approaches require the use of sophisticated tools from many different fields of applied mathematics the purpose of this paper is to provide an accessible and self contained review of these topics emphasizing the intuitive and geometric underlying ideas

reciprocal transformations in 2 1 dimensions reciprocal backlund transformations are introduced for nonlinear integro differential equations in 2 1 dimensions invariance under such transformations is investigated

exact solutions for water infiltration with an arbitrary surface flux or nonlinear solute adsorption several authors have now used the backlund transformation to linearize richard s equation and so obtain analytical solutions for infiltration and redistribution processes for the linearizable class of diffusivities and hydraulic conductivities after transformation a convection diffusion equation is obtained to date all the analytical results making use of the backlund transform have been based on a specified constant flux that is assumed to exist at the surface boundary we generalize the previous work by removing this restriction and obtain a general solution for an arbitrary temporal variation in surface flux the solution contains a function that is specified by a volterra integral equation in practice the volterra equation must be evaluated numerically this task is accomplished using a relatively straightforward algorithm the analytical results for infiltration can be applied to the process of solute transport subject to a nonlinear adsorption isotherm since these two phenomena are equivalent under another mapping our new analytical results therefore also apply to this class of solute transport processes the general form of the adsorption isotherm for which exact results can be obtained is derived this isotherm appears capable of representing a relatively large family of physically relevant cases although this topic remains open for future investigation

a unifying framework for the siso circle criterion and other quadratic stability criteria we present a result on the existence of a common quadratic lyapunov function for a pair of stable linear time invariant systems we show that this result characterizes generalizes and provides new perspectives on several well known stability results

constant rate rainfall infiltration a versatile nonlinear model 1 analytic solution analytic solutions are presented for a nonlinear diffusion convection model describing constant rate rainfall infiltration in uniform soils and other porous materials the model is based on the darcy buckingham approach to unsaturated water flow and assumes simple functional forms for the soil water diffusivity d and hydraulic conductivity k which depend on a single free parameter c and readily measured soil hydraulic properties these d and k yield physically reasonable analytic moisture characteristics the relation between this model and other models which give analytic solutions is explored as c the model reduces to the weakly nonlinear burgers equation which has been applied in certain field situations at the other end of the range as c 1 the model approaches a green ampt like model a wide range of realistic soil hydraulic properties is encompassed by varying the c parameter the general features of the analytic solutions are illustrated for selected c values gradual and steep wetting profiles develop during rainfall aspects seen in the laboratory and field in addition the time dependent surface water content and surface water pressure potential are presented explicitly a simple traveling wave approximation is given which agrees closely with the exact solution at comparatively early infiltration times

tpmsvm a novel twin parametric margin support vector machine for pattern recognition a novel twin parametric margin support vector machine tpmsvm for classification is proposed in this paper this tpmsvm in the spirit of the twin support vector machine twsvm determines indirectly the separating hyperplane through a pair of nonparallel parametric margin hyperplanes solved by two smaller sized support vector machine svm type problems similar to the parametric margin email protected support vector machine email protected svm this tpmsvm is suitable for many cases especially when the data has heteroscedastic error structure that is the noise strongly depends on the input value but there is an advantage in the learning speed compared with the email protected svm the experimental results on several artificial and benchmark datasets indicate that the tpmsvm not only obtains fast learning speed but also shows good generalization

inverse transformations and the reduction of nonlinear dirichlet problems involutory transformations are introduced which allow the solution of a class on nonlinear dirichlet problems the application of coupled involutory and backlund transformations is also described

a note on the stability of bimodal systems in with discontinuous vector fields this paper presents the necessary and sufficient conditions for global asymptotic stability of a class of bimodal piecewise linear systems in r3 the approach being used allows the vector field to be discontinuous on the switching plane it is shown that the discontinuity of the vector field plays a crucial role in global asymptotic stability this point is illustrated by examples where the change in the discontinuity of the vector field without changing the eigenvalues of subsystems can cause the system to be globally asymptotically stable or unstable

analytical model of infiltration under constant concentration boundary conditions 1 known integrable models for 1d flow in unsaturated soil have a rescaled soil water diffusivity that is either constant or proportional to c c 1 c 2 where is the degree of saturation and c 1 is constant with a wider more realistic range of hydraulic conductivity functions than has been used in this context before a formal series solution is developed for infiltration subject to constant concentration boundary conditions a readily programmed iteration algorithm applicable for any value of c is used to construct many coefficients of the infiltration series without requiring any numerical integration in particular for either c 1 small or 1 c small several infiltration series coefficients are constructed as formal power series in c 1 or in 1 c for which we construct a number of terms explicitly in the limit as the diffusivity approaches a delta function the infiltration coefficients are obtained in simpler closed form all but the sorptivity depend on the form of the conductivity function

application of linked backlund transformations to nonlinear boundary value problems a class of nonlinear boundary value problems is reduced to linear canonical form by a combination of backlund transformations

stabilization of second order lti switched systems this paper studies and solves the problem of asymptotic stabilization of switched systems consisting of unstable secondorder linear time invariant lti subsystems necessary and sufficient conditi

towards a stability theory of general hybrid dynamical systems in recent work we proposed a general model for hybrid dynamical systems whose states are defined on arbitrary metric space and evolve along some notion of generalized abstract time for such systems we introduced the usual concepts of lyapunov and lagrange stability we showed that it is always possible to transform this class of hybrid dynamical systems into another class of dynamical systems with equivalent qualitative properties but defined on real time r 0 the motions of this class of systems are in general discontinuous this class of systems may be finite or infinite dimensional for the above discontinuous dynamical systems and hence for the above hybrid dynamical systems we established the principal lyapunov stability theorems as well as lagrange stability theorems for some of these we also established converse theorems we demonstrated the applicability of these results by means of specific classes of hybrid dynamical systems in the present paper we continue the work described above in doing so we first develop a general comparison theory for the class of hybrid dynamical systems resp discontinuous dynamical systems considered herein making use of stability preserving mappings we then show how these results can be applied to establish some of the principal lyaponov stability theorems for the latter we also state and prove a converse theorem not considered previously finally to demonstrate the applicability of our results we consider specific examples throughout the paper

criteria of asymptotic stability of differential and difference inclusions encountered in control theory abstract for a class of differential inclusions to which many of the practically important control systems can be reduced necessary and sufficient conditions for asymptotic stability of the zero solution are established by the method of lyapunov functions similar results are obtained for the corresponding class of difference inclusions

stability tests and stabilization for piecewise linear systems based on poles and zeros of subsystems this paper provides several stability tests for piecewise linear systems and proposes a method of stabilization for bimodal systems in particular we derive an explicit and exact stability test for planar systems which is given in terms of coefficients of transfer functions of subsystems restricting attention to the bimodal and planar case we show simple stability tests in addition we drive a necessary stability condition and a sufficient stability condition for higher order and bimodal systems they are given in terms of the eigenvalue loci and the observability of subsystems all the stability tests provided in this paper are computationally tractable and our results are applied to the stabilizability problem we confirm the exactness and effectiveness of our approach by illustrative examples

group theoretic methods applied to burgers equation abstract in this study the group theoretic methods for calculating the solution of burgers equation with appropriate boundary and initial conditions is presented the application of a one parameter group reduces the number of independent variables by one and consequently the governing partial differential equation with the boundary and initial conditions to an ordinary differential equation with the appropriate corresponding conditions the obtained differential equation is solved analytically and the solution obtained in closed form for a specific choice of boundary condition

a note on stability conditions for planar switched systems this article is concerned with the stability problem for the planar linear switched system where the real matrices a 1 a 2 2 2 are hurwitz and u 0 0 1 is a measurable function we give coordinate invariant necessary and sufficient conditions on a 1 and a 2 under which the system is asymptotically stable for arbitrary switching functions u the new conditions unify those given in previous papers and are simpler to be verified since we reduced to study 4 cases instead of 20 most of the cases are analysed in terms of the function

some non local transformations between nonlinear diffusion equations generalized backlund transformations are applied to derive links between a large number of different types of nonlinear diffusion equations including many which are of physical significance some new exactly linearizable forms are determined

analytical solution to transient richards equation with realistic water profiles for vertical infiltration and parameter estimation a general analytical model for one dimensional transient vertical infiltration is presented the model is based on a combination of the brooks and corey soil water retention function and a generalized hydraulic conductivity function this leads to power law diffusivity and convective term for which the exponents are functions of the inverse of the pore size distribution index accordingly the proposed analytical solution covers many existing realistic models in the literature the general form of the analytical solution is simple and it expresses implicitly the depth as function of water content and time it can be used to model infiltration through semi infinite dry soils with prescribed water content or flux boundary conditions some mathematical expressions of practical importance are also derived the general form solution is useful for comparison between models validation of numerical solutions and for better understanding the effect of some hydraulic parameters based on the analytical expression a complete inverse procedure which allows the estimation of the hydraulic parameters from water content measurements is presented

asymptotic behavior of extremal solutions and structure of extremal norms of linear differential inclusions of order three asymptotic properties of extremal solutions of linear inclusions of order three with zero lyapunov exponent are investigated under certain conditions it is shown that all extremal solutions of such inclusions tend to the same up to a multiplicative factor solution which is central symmetric the structure of the convex set of extremal norm is studied a number of extremal points of this set are described

stability and stabilizability of switched linear systems a survey of recent results during the past several years there have been increasing research activities in the field of stability analysis and switching stabilization for switched systems this paper aims to briefly survey recent results in this field first the stability analysis for switched systems is reviewed we focus on the stability analysis for switched linear systems under arbitrary switching and we highlight necessary and sufficient conditions for asymptotic stability after a brief review of the stability analysis under restricted switching and the multiple lyapunov function theory the switching stabilization problem is studied and a variety of switching stabilization methods found in the literature are outlined then the switching stabilizability problem is investigated that is under what condition it is possible to stabilize a switched system by properly designing switching control laws note that the switching stabilizability problem has been one of the most elusive problems in the switched systems literature a necessary and sufficient condition for asymptotic stabilizability of switched linear systems is described here

non integrability of non linear diffusion convection equations in two spatial dimensions it is generally accepted that integrable partial differential time evolution equations possess lie backlund symmetries of arbitrarily high finite order it has already been established that in 1 1 dimensions the full class of nonlinear fokker planck convection diffusion equations having this property consists of the known integrable examples these are closely related either to the burgers equations or to the fokas yortsos rosen equation both of which have been applied to unsaturated flow in porous media here the author shows that higher order lie backlund symmetries do not exist for any examples of the nonlinear fokker planck equation in 1 2 one time and two space dimensions

reciprocal auto backlund transformations invariant backlund transformations of reciprocal type are derived for a class of n 1 th order conservation laws a new auto backlund transformation for the harry dym equation is presented as a special case of the analysis

characterization of well posedness of piecewise linear systems one of the basic issues in the study of hybrid systems is the well posedness existence and uniqueness of solutions problem of discontinuous dynamical systems the paper addresses this problem for a class of piecewise linear discontinuous systems under the definition of solutions of caratheodory the concepts of jump solutions or of sliding modes are not considered here in this sense the problem to be discussed is one of the most basic problems in the study of well posedness for discontinuous dynamical systems first we derive necessary and sufficient conditions for bimodal systems to be well posed in terms of an analysis based on lexicographic inequalities and the smooth continuation property of solutions next its extensions to the multimodal case are discussed as an application to switching control in the case that two state feedback gains are switched according to a criterion depending on the state we give a characterization of all admissible state feedback gains for which the closed loop system remains well posed

on the exactly solvable equation s_t beta s gamma 2 s_x _x alpha beta s gamma 2 s_x occurring in two phase flow in porous media the symmetry approach of fokas j math phys 21 1980 pp 1318 1325 is used to show that the most general equation of the form s t g s s xx f s s x dg ds ne 0 which is exactly solvable is equivalent to the equation s t beta s gamma 2 s x x alpha beta s gamma 2 s x this equation is then mapped to the linear heat equation through a series of backlund transformations the above result is used to solve an initial boundary value problem modeling the process of waterflooding i e the displacement of oil initially in a porous reservoir by continuously injected water