recovering the initial state of an infinite dimensional system using observers let a be the generator of a strongly continuous semigroup t on the hilbert space x and let c be a linear operator from d a to another hilbert space y possibly unbounded with respect to x not necessarily admissible we consider the problem of estimating the initial state z 0 d a with respect to the norm of x from the output function y t ct tz 0 given for all t in a bounded interval 0 t we introduce the concepts of estimatability and backward estimatability for a c in a more general way than currently available in the literature we introduce forward and backward observers and we provide an iterative algorithm for estimating z 0 from y this algorithm generalizes various algorithms proposed recently for specific classes of systems and it is an attractive alternative to methods based on inverting the gramian our results lead also to a very general formulation of russell s principle i e estimatability and backward estimatability imply exact observability this general formulation of the principle does not require t to be invertible we illustrate our estimation algorithms on systems described by wave and schrodinger equations and we provide results from numerical simulations

strong stability of an unstable wave equation by boundary feedback with only displacement observation the stabilization of a 1 d wave equation that contains instability at its free end and control at the other end is considered the controller is designed through the estimated state that is designed in the case that only displacement is available the method of backstepping is adopted in investigation the c0 semigroup theory and lyapunov method are used to show that the closed loop system is asymptotically stable

blow up and instability of solitary wave solutions to a generalized kadomtsev petviashvili equation in this paper we consider a generalized kadomtsev petviashvili equation in the form ut uxxx upux uyy x y e r2 t 0 it is shown that the solutions blow up in finite time for the supercritical power of nonlinearity p 4 3 with p the ratio of an even to an odd integer moreover it is shown that the solitary waves are strongly unstable if 2 p 4 that is the solutions blow up in finite time provided they start near an unstable solitary wave

identification of unknown temporal and spatial load distributions in a vibrating euler bernoulli beam from dirichlet boundary measured data source identification problems in a system governed by euler bernoulli beam equation x u t t r x u x x x x f x h t x t 0 l 0 t from available boundary observation measured data namely from measured slope t u x 0 t at x 0 are considered we propose a new approach to identifying the unknown temporal h t and spatial f x loads this novel approach is based on weak solution theory for pdes and quasi solution method for inverse problems combined with the adjoint method it allows to construct not only a mathematical theory of inverse source problems for euler bernoulli beam but also an effective numerical algorithm for reconstruction of unknown loads introducing the input output operators h t u x 0 t h and f t u x 0 t f t 0 t we show that both operators are compact based on this result and general regularization theory we prove an existence of unique solutions of the regularized normal equations i h and i f then we develop the adjoint problem approach to prove frechet differentiability of the corresponding cost functionals and lipschitz continuity of the frechet gradients derived explicit gradient formulas via the adjoint problem solution and known load allow use of gradient type convergent iterative algorithms results of numerical simulations for benchmark problems illustrate robustness and high accuracy of the algorithm based on the proposed approach

adaptive output feedback stabilization of non local hyperbolic pdes we address the problem of adaptive output feedback stabilization of general first order hyperbolic partial integro differential equations pide such systems are also referred to as pdes with non local in space terms we apply control at one boundary take measurements on the other boundary and allow the system s functional coefficients to be unknown to deal with the absence of both full state measurement and parameter knowledge we introduce a pre transformation which happens to be based on backstepping of the system into an observer canonical form in that form the problem of adaptive observer design becomes tractable both the parameter estimator and the control law employ only the input and output signals and their histories over one unit of time prior to presenting the adaptive design we present the non adaptive baseline controller which is novel in its own right and facilitates the understanding of the more complex adaptive system the parameter estimator is of the gradient type based on a parametric model in the form of an integral equation relating delayed values of the input and output for the closed loop system we establish boundedness of all signals pointwise in space and time and convergence of the pde state to zero pointwise in space we illustrate our result with a simulation

interactions among periodic waves and solitary waves for a higher dimensional system the general variable separated approach is successfully extended to the 2 1 dimensional physical model and we obtain a universal formula involving arbitrary number of variable separated functions because of the existence of the arbitrary functions in the universal formula many new types of structures of periodic waves for example the periodic periodic interaction waves periodic kink interaction waves periodic peakon interaction waves periodic compacton interaction waves periodic foldon interaction waves etc are investigated both analytically and graphically some novel features or interesting behaviours are revealed

singularity analysis and localized coherent structures in 2 1 dimensional generalized korteweg de vries equations in this article a singularity structure analysis of a 2 1 dimensional generalized korteweg de vries equation studied originally by boiti et al admitting a weak lax pair is carried out and it is proven that the system satisfies the painleve property its bilinear form is constructed in a natural way from the p analysis and then it is used to generate multidromion solutions exponentially decaying solutions in all directions the same analysis can be extended to construct the multidromion solutions of the generalized nizhnik novikov veselov nnv equation from which the nnv equation follows as a special case

control of a tip force destabilized shear beam by observer based boundary feedback we consider a model of the undamped shear beam with a destabilizing boundary condition the motivation for this model comes from atomic force microscopy where the tip of the cantilever beam is destabilized by van der waals forces acting between the tip and the material surface previous research efforts relied on collocated actuation and sensing at the tip exploiting the passivity property between the corresponding input and output in the beam model in this paper we design a stabilizing output feedback controller in a noncollocated setting with measurements at the free end tip of the beam and actuation at the beam base our control design is a novel combination of the classical damping boundary feedback idea with a recently developed backstepping approach a change of variables is constructed which converts the beam model into a wave equation for a very short string with boundary damping this approach is physically intuitive and allows both an elegant stability analysis and an easy selection of design parameters for achieving desired performance our observer design is a dual of the similar ideas combining the damping feedback with backstepping adapted to the observer error system both stability and well posedness of the closed loop system are proved the simulation results are presented

capacities and 1 strict subsets in metric spaces abstract in a complete metric space that is equipped with a doubling measure and supports a poincare inequality we study strict subsets i e sets whose variational capacity with respect to a larger reference set is finite in the case p 1 relying on the concept of fine topology we give a characterization of those strict subsets that are also sets of finite perimeter and then we apply this to the study of condensers as well as bv capacities we also apply the theory to prove a pointwise approximation result for functions of bounded variation

characterizations of sobolev inequalities on metric spaces we discuss maz ya type isocapacitary characterizations of sobolev inequalities on metric measure spaces

robust boundary control of an axially moving string by using a pr transfer function in this note a scheme for the vibration suppression of a translating string using a positive real pr transfer function is investigated the transverse vibration of the string is controlled by hydraulic touch rolls located at the right end of the string the mathematical model of the system which consists of a hyperbolic partial differential equation pde describing the dynamics of the moving string and an ordinary differential equation ode for the actuator dynamics is derived by using hamilton s principle for translating continua the transfer function of the proposed boundary controller is a nonproper but pr function the asymptotic stability of the closed loop system in the presence of output disturbance is proved

comparisons of relative bv capacities and sobolev capacity in metric spaces abstract we study relations between the variational sobolev 1 capacity and versions of variational bv capacity in a complete metric space equipped with a doubling measure and supporting a weak 1 1 poincare inequality we prove the equality of 1 modulus and the continuous 1 capacity extending the known results for 1 p to also cover the more geometric case p 1 then we give alternative definitions for variational bv capacities and obtain equivalence results between them finally we study relations between total 1 capacity and versions of bv capacity

the regularity of the wave equation with partial dirichlet control and colocated observation in this paper we analyze a multidimensional controlled wave equation on a bounded domain subject to partial dirichlet control and colocated observation by means of a partial fourier transform it is shown that the system is well posed and regular in the sense of d salamon and g weiss the corresponding feedthrough operator is found to be the identity operator on the input space

quasiperiodic waves and asymptotic behavior for bogoyavlenskii s breaking soliton equation in 2 1 dimensions based on a multidimensional riemann theta function the hirota bilinear method is extended to explicitly construct multiperiodic quasiperiodic wave solutions for the 2 1 dimensional bogoyavlenskii breaking soliton equation among these periodic waves the one periodic waves are well known cnoidal waves their surface pattern is one dimensional and often they are used as one dimensional models of periodic waves in shallow water the two periodic biperiodic waves are a direct generalization of one periodic waves their surface pattern is two dimensional that is they have two independent spatial periods in two independent horizontal directions the two periodic waves may be considered to represent periodic waves in shallow water without the assumption of one dimensionality a limiting procedure is presented to analyze asymptotic behavior of the one and two periodic waves in details the exact relations between the periodic wave solutions and the well known soliton solutions are established it is rigorously shown that the periodic wave solutions tend to the soliton solutions under a small amplitude limit

boundary stabilization of a 1 d wave equation with in domain antidamping we consider the problem of boundary stabilization of a 1 d one dimensional wave equation with an internal spatially varying antidamping term this term puts all the eigenvalues of the open loop system in the right half of the complex plane we design a feedback law based on the backstepping method and prove exponential stability of the closed loop system with a desired decay rate for plants with constant parameters the control gains are found in closed form our design also produces a new lyapunov function for the classical wave equation with passive boundary damping

a computational approach to soliton solutions of the kadomtsev petviashvili equation in this paper we present a computational approach to develop soliton solutions of the nonlinear kadomtsev petviashvili equation our approach rests mainly on the adomian decomposition method to include few components of the decomposition series the proposed framework is presented in a general way so that it can be used in nonlinear evolution equations of the same type numerical examples are tested to illustrate the proposed scheme

exact solutions of a generalized 3 1 dimensional kadomtsev petviashvili equation using lie symmetry analysis abstract in this paper lie group analysis is employed to derive some exact solutions of a generalized 3 1 dimensional kadomtsev petviashvili equation which describes the dynamics of solitons and nonlinear waves in plasmas and superfluids

output feedback stabilization of an unstable wave equation we consider the problem of stabilization of a one dimensional wave equation that contains instability at its free end and control on the opposite end in contrast to classical collocated boundary damper feedbacks for the neutrally stable wave equations with one end satisfying a homogeneous boundary condition the controllers and the associated observers designed in the paper are more complex due to the open loop instability of the plant the controller and observer gains are designed using the method of backstepping which results in explicit formulae for the gain functions we prove exponential stability and the existence and uniqueness of classical solutions for the closed loop system we also derive the explicit compensators in frequency domain the results are illustrated with simulations

boundary state feedback exponential stabilization for a one dimensional wave equation with velocity recirculation abstract in this paper we consider boundary state feedback stabilization of a one dimensional wave equation with in domain feedback recirculation of an intermediate point velocity we firstly construct an auxiliary control system which has a nonlocal term of the displacement at the same intermediate point then by choosing a well known exponentially stable wave equation as its target system we find one backstepping transformation from which a state feedback law for this auxiliary system is proposed finally taking the resulting closed loop of the auxiliary system as a new target system we obtain another backstepping transformation from which a boundary state feedback controller for the original system is designed by the equivalence of three systems the closed loop of original system is proved to be well posed and exponentially stable some numerical simulations are presented to validate the theoretical results

adaptive output feedback stabilization for one dimensional wave equation with corrupted observation by harmonic disturbance in this paper we are concerned with the output feedback stabilization of a one dimensional wave equation with an unstable term at one end and the observation suffered by a general harmonic disturbance with unknown magnitudes at the other end an adaptive observer is designed in terms of the corrupted observation the backstepping method for infinite dimensional systems is adopted in the design of the feedback law it is shown that the resulting closed loop system is asymptotically stable meanwhile the estimated parameters are shown to be convergent to the unknown parameters as time goes to infinity

a unified lyapunov based design for a dynamic compensator of linear parabolic mimo pdes this paper addresses the problem of dynamic compensator design for exponential stabilisation of linear parabolic partial differential equations pdes with multiple actuation control inputs and mul

2 1 dimensional compacton solutions with and without completely elastic interaction properties taking the 2 1 dimensional broer kaup kupershmidt system as a simple example some special types of 2 1 dimensional compacton solutions are constructed it is shown that there is quite rich interaction behaviour between two travelling compactons for some types of compactons the interactions among them may not be completely elastic for some others the interactions are completely elastic there is no phase shift for the interactions of the 2 1 dimensional compactons discussed in this paper

localized solutions with chaotic and fractal behaviours in a 2 1 dimensional dispersive long wave system it is known that in high dimensions there exist abundant localized excitations such as dromions lumps ring soliton solutions and so on in this paper the possible chaotic and fractal localized structures are revealed for the 2 1 dimensional dispersive long wave equation the chaotic and fractal dromion and lump patterns of the model are constructed by some types of lower dimensional chaotic and fractal patterns

stability and instability of the kdv solitary wave under the kp i flow we consider the kp i and gkp i equations in mathbb r times mathbb r 2 pi mathbb z we prove that the kdv soliton with subcritical speed 0 c in the spirit of the work by duyckaerts and merle gafa 18 1787 1840 2009 we sharpen our previous instability result and construct a global solution which is different from the solitary wave and its translates and which converges to the solitary wave as time goes to infinity this last result also holds for the gkp i equation

a new general type of solitary wave of a 2 1 dimensional system using the extended homogeneous balance method ehbm and variable separation approach vsa an exact variable separation excitation of a 2 1 dimensional system is derived based on the derived excitation a new general type of solitary wave i e semifolded solitary waves and semifoldons sfsws is defined and studied we investigate the behaviours of the interactions for the semifolded localized structures and find that the interactions possess some novel and interesting features

observers and initial state recovering for a class of hyperbolic systems via lyapunov method a b s t r a c t recently the problem of estimating the initial state of some linear infinite dimensional systems from measurements on a finite interval was solved by using the sequence of forward and backward observers ramdani tucsnak and weiss 2010 in the present paper we introduce a direct lyapunov approach to the problem and extend the results to the class of semilinear systems governed by wave and beam equations with boundary measurements from a finite interval we first design forward observers and derive linear matrix inequalities lmis for the exponential stability of the estimation errors further we obtain simple finite dimensional conditions in terms of lmis for an upper bound t on the minimal time that guarantees the convergence of the sequence of forward and backward observers on 0 t for the initial state recovering this t represents also an upper bound on the observability time for observation times bigger than t these lmis give upper bounds on the convergence rate of the iterative algorithm in the norm defined by the lyapunov functions in our approach t is found as the minimal dwelling time for the switched exponentially stable forward and backward estimation error systems with the different lyapunov functions liberzon 2003 the efficiency of the results is illustrated by numerical examples 2013 elsevier ltd all rights reserved

performance output tracking for a wave equation subject to unmatched general boundary harmonic disturbance in this paper we consider performance output reference tracking for a wave equation with general harmonic disturbance at one end and the control at the other end we first design a state reference system determined completely by the measured output and the reference signal where the parameter update law is presented an adaptive servomechanism output feedback control is then designed it is shown that the closed loop system is well posed three control objectives are achieved a the performance output is tracking the reference signal b all the internal loops are bounded c when the disturbance and reference are disconnected the closed loop is exponentially stable finally the state of system is shown to be tracking the reference state and the updated parameters are convergent to their true values

localized excitations of the 2 1 dimensional sine gordon system the 2 1 dimensional sine gordon system is investigated via a generalized bilinear operator representation plateau basin bowl and saddle type ring solitons are thereby constructed it is indicated how iterated moutard transformations may be employed to extend the range of the method

exact solutions of compact and noncompact structures for the kp bbm equation the tanh and the sine cosine methods are used for a reliable treatment of the 2 1 dimensional kp bbm equation generalized forms of the kp bbm equation are investigated as well a variety of exact travelling wave solutions of distinct physical structures compactons solitons solitary patterns and periodic solutions are formally derived

interactions of solitary waves under the conditions of the 3 1 dimensional kadomtsev petviashvilli equation using an extended mapping method with a linear variable separation process a new family of the exact solutions of the 3 1 dimensional kadomtsev petviashvilli kp equation was derived by applying the solitary wave solutions this paper studied some newly localized excitations and the interactions of various solitary waves under the conditions of the 3 1 dimensional kp equation

numerical study of the regularizing effect of the 3d weakly transverse bbm equations for long times from a spectral method combined with a predictor corrector scheme we numerically study the behavior in time of solutions of the three dimensional generalized kadomtsev petviashvili equations regularized using the bbm trick the solution no longer blows up and the solitonic behavior is observed

decay of solutions of wave equations in a bounded region with boundary dissipation abstract an energy decay rate is obtained for solutions of wave type equations in a bounded region in rn whose boundary consists partly of a nontrapping reflecting surface and partly of an energy absorbing surface

restricted flows of soliton hierarchies coupled kdv and harry dym case restricted flows of soliton hierarchies associated with the energy dependent schrodinger spectral problem are determined explicitly a remarkable connection with separable potentials is used for proving complete integrability of the restricted flows a previously unknown lagrangian and hamiltonian formulation of the neumann system is found whole families of generalizations of the neumann and garnier systems are given

boundary feedback stabilization of a class of coupled hyperbolic equations with nonlocal terms this paper solves the problem of boundary feedback stabilization of a class of coupled ordinary differential equations hyperbolic equations with boundary trace and integral nonlocal terms using the backstepping approach the controller is designed by formulating an integral operator whose kernel is required to satisfy a coupled hyperbolic partial integral differential equation by applying the method of successive approximations the kernel s well posedness is given we prove the exponential stability of the origin of the system in a suitable hilbert space moreover a wave system with nonlocal terms is stabilized by applying the above result

adaptive output feedback stabilisation for wave equations with dynamic boundary condition and corrupted boundary measurement this paper is devoted to the output feedback stabilisation for a class of wave equations remarkably only one boundary measurement corrupted by disturbance is available for feedback and the bound

output feedback stabilisation of a tree shaped network of vibrating strings with non collocated observation in this article the authors consider the simple tree shaped network of strings with non collocated observation suppose that the network consists of three strings and the displacements of these strings are continuous at the common vertex the sensors and actuators are required to be set at different vertices we propose an observer based output feedback controller to stabilise this network the exponential stability of the closed loop network is obtained based on the riesz basis approach together with estimates of the spectral distribution finally a numerical simulation is given to support these results

the cauchy problem and stability of solitary wave solutions for rlw kp type equations abstract the kadomtsev petviashvilli kp equation ut ux uux uxxx x euyy 0 arises in various contexts where nonlinear dispersive waves propagate principally along the x axis but with weak dispersive effects being felt in the direction parallel to the y axis perpendicular to the main direction of propagation we propose and analyze here a class of evolution equations of the form ut ux upux lut x euyy 0 which provides an alternative to eq in the same way the regularized long wave equation is related to the classical korteweg de vries kdv equation the operator l is a pseudo differential operator in the x variable p 1 is an integer and e 1 after discussing the underlying motivation for the class a local well posedness theory for the initial value problem is developed with assumptions on l and p that include conditions appertaining to models of interesting physical phenomenon the solutions defined locally in time t are shown to be smoothly extendable to the entire time axis in the particularly interesting case where l x2 and e 1 possesses travelling wave solutions u x y t c x ct y provided c 1 and 0 0 4 3 and c 1 and for 4 3 if c 4p 4 p the paper concludes with commentary on extensions of the present theory to more than two space dimensions

stabilization of an abstract second order system with application to wave equations under non collocated control and observations abstract the objective of this paper is to deal with the stabilization of multi dimensional wave equations under non collocated control and observation with the following cases a locally internal distributed control and observation b internal distributed control and boundary observation c boundary control and internal distributed observation d locally internal distributed control and boundary observation we first discuss the stability of an abstract second order infinite dimensional system under the estimated state feedback control the well posedness in the sense of d salamon is specially emphasized the result is then applied to the multi dimensional wave equations with the aforementioned three different non collocated control and observations a c finally we prove the stabilizability of a wave equation with non collocated control and observation d by using the same idea although it cannot be cast into the abstract setting

parameter estimation and stabilisation for a one dimensional wave equation with boundary output constant disturbance and non collocated control we are concerned with the parameter estimation and stabilisation of a one dimensional wave equation with an unknown constant disturbance suffered by boundary observation at one end and the non collocated control at another end an adaptive observer is designed in terms of measured velocity with unknown constant disturbance the backstepping method for infinite dimensional system is adopted in the design of the feedback law it is shown that the resulting closed loop system is asymptotically stable meanwhile the estimated parameter is shown to be convergent to the unknown parameter as time goes to infinity

parameter estimation and non collocated adaptive stabilization for a wave equation subject to general boundary harmonic disturbance this paper is concerned with the parameter estimation and asymptotic stabilization of a 1 d wave equation that is subject to general harmonic disturbances at the controlled end and suffers from instability at the other end first we design an adaptive observer in terms of measured position and velocity we then adopt the backstepping method for infinite dimensional systems to design an observer based output feedback law the resulting closed loop system is shown to be asymptotically stable and the estimates of the parameters converge to the unknown parameters

on the inverse scattering problem on branching graphs the inverse scattering problem on branching graphs is studied the definition of the schrodinger operator on such graphs is discussed the operator is defined with real potentials with finite first momentum and using special boundary conditions connecting values of the functions at the vertices it is shown that in general the scattering matrix does not determine the topology of the graph the potentials on the edges and the boundary conditions uniquely

boundary stabilization of wave equation with velocity recirculation nonlocal terms have been the mainstay of the applications of partial differential equation pde backstepping methods to parabolic pdes the problem of similar nonlocal terms for wave equations is still open for wave equations similar nonlocal terms have not been studied in this paper we open the topic of exploration of control of wave pdes with nonlocal terms this paper is concerned with the wave equation with in domain feedback recirculation of a boundary velocity with a spatially constant coefficient due to this nonlocal term the passivity of the wave equation is destroyed we first design an explicit state feedback controller to achieve exponential stability for the closed loop system then we design the output feedback by using infinite dimensional observer the backstepping approach is adopted in investigation it is shown that by using two measurements only the output feedback makes the closed loop system exponentially stable

the stabilization of a one dimensional wave equation by boundary feedback with noncollocated observation this note addresses the stabilization of a one dimensional wave equation with control at one end and noncollocated observation at another end a simple exponentially convergent observer is constructed the dynamical stabilizing boundary output feedback is designed via the observed state while the closed loop system is nondissipative we show its exponential stability using riesz basis approach

asymptotic behavior of the periodic wave solution for the 3 1 dimensional kadomtsev petviashvili equation abstract in this paper the one and two periodic wave solutions for the 3 1 dimensional kadomtsev petviashvili equation are presented by means of the hirota s bilinear method and the riemann theta function the rigorous proofs on asymptotic behaviors of these two solutions are given that soliton solution can be obtained from the periodic wave solution in an appropriate limiting procedure

c neumann and bargmann systems associated with the coupled kdv soliton hierarchy under two different constraints between the potentials and the eigenfunctions the eigenvalue problem associated with the coupled kdv hierarchy is nonlinearised to be a completely integrable c neumann system on the tangent bundle of sphere tsn 1 with the hamiltonian h 1 2 lambda q p 1 2 q q p p 1 2 q p 2 1 2 q p lambda p p and a completely integrable bargmann system r2n dpdq h with the hamiltonian h 1 2 lambda q p 1 2 p p q p 1 2 q q respectively the involutive solutions of the coupled kdv equation associated with the two systems are given

localized excitations in 2 1 dimensional systems by means of a special variable separation approach a common formula with some arbitrary functions has been obtained for some suitable physical quantities of various 2 1 dimensional models such as the davey stewartson ds model the nizhnik novikov veselov nnv system asymmetric nnv equation asymmetric ds equation dispersive long wave equation broer kaup kupershmidt system long wave short wave interaction model maccari system and a general n m component ablowitz kaup newell segur akns system selecting the arbitrary functions appropriately one may obtain abundant stable localized interesting excitations such as the multidromions lumps ring soliton solutions breathers instantons etc it is shown that some types of lower dimensional chaotic patterns such as the chaotic chaotic patterns periodic chaotic patterns chaotic line soliton patterns chaotic dromion patterns fractal lump patterns and fractal dromion patterns may be found in higher dimensional soliton systems the interactions between the traveling ring type soliton solutions are completely elastic the traveling ring solitons pass through each other and preserve their shapes velocities and phases some types of localized weak solutions peakons are also discussed especially the interactions between two peakons are not completely elastic after the interactions the traveling peakons also pass through each other and preserve their velocities and phases however they completely exchange their shapes

kp lumps in ferromagnets a three dimensional kdv burgers model the three dimensional long wave approximation for electromagnetic waves in saturated ferromagnetic media is considered taking into account damping and inhomogeneous exchange the wave evolution is governed by a 3 1 dimensional generalization of the korteweg de vries burgers kadomtsev petviashvili and zabolotskaya khokhlov equations neglecting the damping we give plane soliton and line lump solutions and show that they are unstable

four 2 1 dimensional integrable extensions of the kdv equation multiple soliton and multiple singular soliton solutions in this work four 2 1 dimensional nonlinear completely integrable equations generated by extending the kdv equation are developed the necessary condition for the complete integrability of these equation are formally derived multiple soliton solutions and multiple singular soliton solutions are determined to emphasize the compatability of these models the dispersion relations of these models are characterized by distinct physical structures the resonance phenomenon for these equations does not exist for any model

the finite band solution of the jaulent miodek equation a method to construct the finite band solution to the soliton equation is presented we take the jaulent miodek equation as an example with the use of the nonlinearization of lax pair the jaulent miodek equation is decomposed into two finite dimensional integrable systems whose properties are studied from the view of an r matrix then through solving these finite dimensional integrable systems the finite band solution of the jaulent miodek equation is obtained

backstepping observer design for parabolic pdes with measurement of weighted spatial averages this paper is concerned with the observer design for one dimensional linear parabolic partial differential equations whose output is a weighted spatial average of the state over the entire spatial domain we focus on the backstepping approach which provides a systematic procedure to design an observer gain for systems with boundary measurement if the output is not a boundary value of the state the backstepping approach is not directly applicable to obtaining an observer gain that stabilizes the error dynamics therefore we attempt to convert the error system into another system to which backstepping is applicable the conversion is successfully achieved for a class of weighting functions and the resultant observer realizes exponential convergence of the estimation error with an arbitrary decay rate in terms of the l 2 norm in addition an explicit expression of the observer gain is available in a special case the effectiveness of the proposed observer is also confirmed by numerical simulations

quasiconformal maps in metric spaces with controlled geometry this paper develops the foundations of the theory of quasiconformal maps in metric spaces that satisfy certain bounds on their mass and geometry the principal message is that such a theory is both relevant and viable the first main issue is the problem of definition which we next describe quasiconformal maps are commonly understood as homeomorphisms that distort the shape of infinitesimal balls by a uniformly bounded amount this requirement makes sense in every metric space given a homeomorphism f from a metric space x to a metric space y then for x x and r 0 set

the decomposition method for solving 2 1 dimensional boussinesq equation and 3 1 dimensional kp equation we study the solitary wave solutions of the 2 1 dimensional boussinesq equation u t t u x x u y y u 2 x x u x x x x 0 and 3 1 dimensional kp equation u x t 6u x 2 6uu x x u x x x x u y y u z z 0 in this paper by considering the decomposition scheme we first obtain the exact solitary wave solutions of the 2 1 dimensional boussinesq equation and 3 1 dimensional kp equation for the initial conditions without using any classical transformations and then its numerical solutions are constructed without using any discretization technique the numerical solutions are compared with the known analytical solutions its remarkable accuracy is finally demonstrated in the study of 2 1 dimensional boussinesq equation and 3 1 dimensional kp equation

the camassa holm hierarchy n dimensional integrable systems and algebro geometric solution on a symplectic submanifold this paper shows that the camassa holm ch spectral problem yields two different integrable hierarchies of nonlinear evolution equations nlees one is of negative order ch hierachy while the other one is of positive order ch hierarchy the two ch hierarchies possess the zero curvature representations through solving a key matrix equation we see that the well known ch equation is included in the negative order ch hierarchy while the dym type equation is included in the positive order ch hierarchy furthermore under two constraint conditions between the potentials and the eigenfunctions the ch spectral problem is cast in 1 a new neumann like n dimensional system when it is restricted into a symplectic submanifold of 2n which is proven to be integrable by using the dirac poisson bracket and the r matrix process and 2 a new bargmann like n dimensional system when it is considered in the whole 2n which is proven to be integrable by using the standard poisson bracket and the r matrix process

adaptive output feedback stabilisation for hybrid pde ode systems with uncertain input disturbances abstractthis paper is concerned with the adaptive output feedback stabilisation for a class of hybrid partial differential equation pde ordinary differential equation ode systems with uncertain input disturbances remarkably in this paper only two boundary measurements of the considered system are available for feedback moreover the unknown parameters involved in the system are allowed to belong to an unknown interval these two aspects make the considered system essentially different from those in the closely related literature inspired by the existing results an observer is first introduced to estimate the unmeasured states of the original system then by adaptive technique and backstepping method an adaptive output feedback controller is successfully constructed which guarantees that the entire closed loop system is asymptotically stable and moreover the parameter estimates converge to their own real values ultimately besides by the semigroup approach the well posedness of the entire

output feedback stabilization of an unstable wave equation with general corrupted boundary observation we consider boundary output feedback stabilization for an unstable wave equation with boundary observation subject to a general disturbance we adopt for the first time the active disturbance rejection control approach to stabilization for a system described by the partial differential equation with corrupted output feedback by the approach the disturbance is first estimated by a relatively independent estimator it is then canceled in the feedback loop as a result the control law can be designed almost as that for the system without disturbance we show that with a time varying gain properly designed the observer driven by the disturbance estimator is convergent and that all subsystems in the closed loop are asymptotically stable in the energy state space we also provide numerical simulations which demonstrate the convergence results and underline the effect of the time varying gain estimator on peaking value reduction

disturbance estimator and servomechanism based performance output tracking for a 1 d euler bernoulli beam equation in this paper we consider performance output tracking for a one dimensional euler bernoulli beam equation with boundary moment control and unmatched internal uncertainty and external disturbance for such system the reference feng and guo 2017 firstly studied the active disturbance rejection control adrc problem we design an infinite dimensional disturbance estimator to estimate the total disturbance internal uncertainty and external disturbance and the system state simultaneously a servomechanism generated by the reference signal is designed based on the estimated total disturbance we design an output feedback control law such that the performance output signal tracks the reference signal and all the signals of closed loop system are bounded some illustrative simulations are presented

transverse instability of the line solitary water waves we prove the linear and nonlinear instability of the line solitary water waves with respect to transverse perturbations