containment control for partial differential multi agent systems abstract this paper investigates the containment control problem for a class of partial differential multi agent systems which are governed by the parabolic equations heat equations or the second order hyperbolic equations wave equations the distributed containment based feedback control laws are obtained by constructing an appropriate lyapunov functional and under the action of the feedback control laws the containment errors between the followers states and the convex hull spanned by the leaders states on an appropriate sobolev space can converge to zero as time tends to infinity simulation examples are finally used to show the effectiveness of the theoretical analysis

the virtual spivak fiber duality on fibrations and gorenstein spaces in this paper we study a generalization of the homology of the spivak fiber of a 1 connected space over any field and deduce consequences concerning poincare complexes gorenstein spaces and finiteness properties on fibrations

variational properties and orbital stability of standing waves for nls equation on a star graph we study standing waves for a model of nonlinear schr odinger equation on a graph the graph is obtained joining n halflines at a vertex i e it is a star graph at the vertex an interaction occurs due to a boundary condition of delta type with strength alpha leqslant 0 which includes the free or kirchhoff case alpha 0 there imposed the nonlinearity is of focusing power type the dynamics is given by an equation of the form i frac d dt psi t h psi t psi t 2 mu psi t where h is the s a operator which generates the linear schr odinger dynamics on the graph in the hilbert space l 2 gg we show the existence of several families of standing waves solutions of the form phi t e i omega t psi omega where psi omega is a vector amplitude parametrized by a frequency omega they exist for every sign of the coupling at the vertex attractive alpha 0 and for every omega frac alpha 2 n 2 the number of the families depends on the number n of the edges of the graph we moreover determine the ground state which is the standing wave the amplitude of which minimizes the action on the natural nehari constraint and order the various families of standing waves according to their increasing action thus determining a nonlinear spectrum of the problem finally we show that the ground state is orbitally stable for every allowed omega if the nonlinearity is subcritical or critical for supercritical nonlinearities there exists an omega frac alpha 2 n 2 such that the ground state is orbitally stable for frac alpha 2 n 2 omega omega action minimization requires technical subtleties in particular an adaptation of decreasing rearrangements on a star graph is developed with a generalization of the p olya szeg h o inequality

sliding mode control in indefinite dimensional systems abstract a mathematical description of motion in a sliding mode is considered for systems with discontinuous controls described by differential equations in banach space applying the regularization principle to such systems the sliding mode equations are obtained for a set of interconnected heat processes the control system with the prescribed dynamics is designed through deliberate introduction of sliding modes

disturbance estimator based output feedback exponential stabilization for euler bernoulli beam equation with boundary control in this paper we are concerned with the output feedback exponential stabilization problem for a system plant described by a one dimensional euler bernoulli beam equation the measurements are only the displacement and the angular velocity at the right end an infinite dimensional estimator is designed to estimate the disturbance with the estimated disturbance we propose a state observer that is exponentially convergent to the original system then design two different kinds of stabilizing controllers one is based on the velocity feedback the other is based on the angular velocity feedback in both cases by adopting the riesz basis approach the exponential stability of the closed loop systems is built with guaranteeing that all internal systems are uniformly bounded the numerical experiments are carried out to illustrate the theoretical results

fuzzy output tracking control of semi linear first order hyperbolic pde systems with matched perturbations abstract this paper considers the design problem of a fuzzy output tracking controller via a finite number of actuators for a class of nonlinear distributed parameter systems with unknown matched perturbations described by semi linear first order hyperbolic partial differential equations pdes firstly a t s fuzzy hyperbolic pde model is assumed to be used to accurately represent the semi linear first order hyperbolic pde system subsequently based on the t s fuzzy pde model the classic variable structure control technique is adopted to develop a fuzzy output tracking controller such that the output of the semi linear pde system exponentially tracks a desired reference and the state of the system is uniformly bounded in the filippov sense by constructing a strict lyapunov function the sufficient condition on the existence of the fuzzy output tracking controller is derived and presented in terms of algebraic linear matrix inequalities lmis in space with an equality constraint it has been shown that when these algebraic lmis with an equality constraint are feasible fuzzy tracking control design can be obtained via the existing convex optimization techniques finally the developed design methodology is successfully applied to the control of a semi linear lotka volterra pde system and the achieved results illustrate its effectiveness

linear pdes and numerical methods that preserve a multisymplectic conservation law multisymplectic methods have recently been proposed as a generalization of symplectic ode methods to the case of hamiltonian pdes their excellent long time behavior for a variety of hamiltonian wave equations has been demonstrated in a number of numerical studies a theoretical investigation and justification of multisymplectic methods is still largely missing in this paper we study linear multisymplectic pdes and their discretization by means of numerical dispersion relations it is found that multisymplectic methods in the sense of bridges and reich phys lett a 284 2001 pp 184 193 and reich j comput phys 157 2000 pp 473 499 such as gauss legendre runge kutta methods possess a number of desirable properties such as nonexistence of spurious roots and conservation of the sign of the group velocity a certain cfl type restriction on delta t delta x might be required for methods higher than second order in time it is also demonstrated by means of the explicit midpoint method that multistep methods may exhibit spurious roots in the numerical dispersion relation for any value of delta t delta x despite being multisymplectic in the sense of discrete variational mechanics j e marsden g p patrick and s shkoller commun math phys 199 1999 pp 351 395

boundary control of an unstable heat equation via measurement of domain averaged temperature in this note a feedback boundary controller for an unstable heat equation is designed the equation can be viewed as a model of a thin rod with not only the heat loss to a surrounding medium stabilizing but also the heat generation inside the rod destabilizing the heat generation adds a destabilizing linear term on the right hand side of the equation the boundary control law designed is in the form of an integral operator with a known continuous kernel function but can be interpreted as a backstepping control law this interpretation provides a lyapunov function for proving stability of the system the control is applied by insulating one end of the rod and applying either dirichlet or neumann boundary actuation on the other

solitary wave dynamics in an external potential we study the behavior of solitary wave solutions of some generalized nonlinear schrodinger equations with an external potential the equations have the feature that in the absence of the external potential they have solutions describing inertial motions of stable solitary waves we consider solutions of the equations with a non vanishing external potential corresponding to initial conditions close to one of these solitary wave solutions and show that over a large interval of time they describe a solitary wave whose center of mass motion is a solution of newton s equations of motion for a point particle in the given external potential up to small corrections corresponding to radiation damping

sharp upper bound on the blow up rate for the critical nonlinear schrodinger equation we consider the critical nonlinear schrodinger equation iu t delta u u 4 n with initial condition u 0 x u0

boundary observer design for a class of semi linear hyperbolic pde systems with recycle loop abstractthe present manuscript considers the observer design problem for a class of scalar semi linear hyperbolic partial differential equation pde systems with a recycle loop through the boundar

gorenstein space with nonzero evaluation map let a d be a differential graded algebra of finite type if h a is a gorenstein graded algebra then so is a the purpose of this paper is to prove the converse under some mild hypotheses we deduce a new characterization of poincare duality spaces as well as spaces with a nonzero evaluation map

sliding mode control of schrodinger equation ode in the presence of unmatched disturbances abstract in this paper we consider boundary stabilization for a cascade of schrodinger equation ode system with both matched and unmatched disturbances the backstepping method is first applied to transform the system into an equivalent target system where the target system is input to state stable to reject the matched disturbance the sliding mode control smc law is designed for the target system the well posedness of the closed loop system is proved and the reachability of the sliding manifold in finite time is justified by infinite dimensional system theory it is shown that the resulting closed loop system is input to state stable a numerical example illustrates the efficiency of the sliding mode design that reduces the ultimate bound of the closed loop system by rejecting the matched disturbance

unstable eigenvalues and the linearization about solitary waves and fronts with symmetry the linear stability of solitary wave or front solutions of hamiltonian evolutionary equations which are equivariant with respect to a lie group is studied the organizing centre for the analysis is a multisymplectic formulation of hamiltonian pdes where a distinct symplectic operator is assigned for time and space this separation of symplectic structures leads to new characterizations of the following components of the analysis the states at infinity are characterized as manifolds of relative equilibria associated with the spatial symplectic structure the momentum of the connecting orbit or shape of the solitary wave considered as a heteroclinic orbit in a phase space representation is given a new characterization as a one form on the tangent space to the heteroclinic manifold and this one form is a restriction of the temporal symplectic structure for the linear stability analysis a new symplectic characterization of the evans function and its derivatives are obtained leading to an abstract geometric proof of instability for a large class of solitary wave states of equivariant hamiltonian evolutionary pdes the theory sheds new light on several well known models the gkdv equation a boussinesq system and a nonlinear wave equation the generalization to solitary waves associated with multidimensional heteroclinic manifolds and the implications for solitary waves or fronts which are biasymptotic to invariant manifolds such as periodic states are also discussed

on global stabilization of burgers equation by boundary control abstract although often referred to as a one dimensional cartoon of navier stokes equation because it does not exhibit turbulence the burgers equation is a natural first step towards developing methods for control of flows recent references include burns and kang nonlinear dynamics 2 1991 235 262 choi et al j fluid mech 253 1993 509 543 ito and kang siam j control optim 32 1994 831 854 ito and yan j math anal appl 227 1998 271 299 byrnes et al j dynam control systems 4 1998 457 519 and van ly et al numer funct anal optim 18 1997 143 188 while these papers have achieved tremendous progress in local stabilization and global analysis of attractors the problem of global asymptotic stabilization has remained open this problem is non trivial because for large initial conditions the quadratic convective term which is negligible in a linear local analysis dominates the dynamics we derive nonlinear boundary control laws that achieve global asymptotic stability we consider both the viscous and the inviscid burgers equation using both neumann and dirichlet boundary control we also study the case where the viscosity parameter is uncertain as well as the case of stochastic burgers equation for some of the control laws that would require the measurement in the interior of the domain we develop the observer based versions

closed form boundary state feedbacks for a class of 1 d partial integro differential equations in this paper a problem of boundary stabilization of a class of linear parabolic partial integro differential equations p i des in one dimension is considered using the method of backstepping avoiding spatial discretization required in previous efforts the problem is formulated as a design of an integral operator whose kernel is required to satisfy a hyperbolic p i de the kernel p i de is then converted into an equivalent integral equation and by applying the method of successive approximations the equation s well posedness and the kernel s smoothness are established it is shown how to extend this approach to design optimally stabilizing controllers an adaptation mechanism is developed to reduce the conservativeness of the inverse optimal controller and the performance bounds are derived for a broad range of physically motivated special cases feedback laws are constructed explicitly and the closed loop solutions are found in closed form a numerical scheme for the kernel p i de is proposed its numerical effort compares favorably with that associated with operator riccati equations

asymptotic stability of small gap solitons in nonlinear dirac equations we prove dispersive decay estimates for the one dimensional dirac operator and use them to prove asymptotic stability of small gap solitons in the nonlinear dirac equations with quintic and higher order nonlinear terms

stabilisation of a thermo elastic system under nonuniformly bounded disturbances in the boundary input abstractwe consider the exponential stabilisation for a thermo elastic system of type ii with nonuniformly bounded external disturbances in the boundary control we begin with finding feedback controls for the system without disturbances by lyapunov function approach then based on the resulting feedback control we give new dynamic feedback control laws that make the system exponentially stable by tracking disturbances exponentially through a time varying high gain observer we apply the linear operator semi group theory to prove the well posedness of the resulting closed system finally we prove that the solution of the closed loop system can approach zero polynomially or exponentially by the lyapunov function approach

symplectic and multi symplectic methods for the nonlinear schrodinger equation the hamiltonian and the multi symplectic formulations of the nonlinear schrodinger equation are considered for the multi symplectic formulation a new six point difference scheme which is equivalent to the multi symplectic preissman integrator is derived numerical experiments are also reported c 2002 elsevier science ltd all rights reserved

tracking control for boundary controlled parabolic pdes with varying parameters combining backstepping and differential flatness the combination of backstepping based state feedback control and flatness based trajectory planning and feedforward control is considered for the design of an exponentially stabilizing tracking controller for a linear diffusion convection reaction system with spatially and temporally varying parameters and nonlinear boundary input for this in a first step the backstepping transformation is utilized to determine a state feedback controller which transforms the original distributed parameter system into an appropriately chosen exponentially stable distributed parameter target system of a significantly simpler structure in a second step the flatness property of the target system is exploited in order to determine the feedforward controller which allows us to realize the tracking of suitably prescribed trajectories for the system output this results in a systematic procedure for the design of an exponentially stabilizing tracking controller for the considered general linear diffusion convection reaction system with varying parameters whose applicability and tracking performance is evaluated in simulation studies

output feedback exponential stabilization for a one dimensional wave equation with control matched nonlinear disturbance in this paper we study the output feedback exponential stabilization of a one dimensional wave equation with control matched nonlinear disturbance the well posedness of the open loop system which is essentially important for the designs of observer and controller is verified by designing an infinite dimensional unknown input state observer in place of the conventional extended state observer we estimate the total disturbance in real time moreover we prove the convergence of the disturbance estimator based on the disturbance estimator we design a state observer by compensating the total disturbance and an estimated state feedback controller to exponentially stabilize the original system by means of the riesz basis approach some numerical simulations are presented

multisymplectic geometry variational integrators and nonlinear pdes this paper presents a geometric variational approach to continuous and discrete mechanics and field theories using multisymplectic geometry we show that the existence of the fundamental geometric structures as well as their preservation along solutions can be obtained directly from the variational principle in particular we prove that a unique multisymplectic structure is obtained by taking the derivative of an action function and use this structure to prove covariant generalizations of conservation of symplecticity and noether s theorem natural discretization schemes for pdes which have these important preservation properties then follow by choosing a discrete action functional in the case of mechanics we recover the variational symplectic integrators of veselov type while for pdes we obtain covariant spacetime integrators which conserve the corresponding discrete multisymplectic form as well as the discrete momentum mappings corresponding to symmetries we show that the usual notion of symplecticity along an infinite dimensional space of fields can be naturally obtained by making a spacetime split all of the aspects of our method are demonstrated with a nonlinear sine gordon equation including computational results and a comparison with other discretization schemes

stable standing waves of nonlinear klein gordon equations in this paper we give sufficient conditions for the stability of the standing waves of least energy for nonlinear klein gordon equations

the active disturbance rejection control to stabilization for multi dimensional wave equation with boundary control matched disturbance in this paper we consider boundary stabilization for a multi dimensional wave equation with boundary control matched disturbance that depends on both time and spatial variables the active disturbance rejection control adrc approach is adopted in investigation an extended state observer is designed to estimate the disturbance based on an infinite number of ordinary differential equations obtained from the original multi dimensional system by infinitely many test functions the disturbance is canceled in the feedback loop together with a collocated stabilizing controller all subsystems in the closed loop are shown to be asymptotically stable in particular the time varying high gain is first time applied to a system described by the partial differential equation for complete disturbance rejection purpose and the peaking value reduction caused by the constant high gain in literature the overall picture of the adrc in dealing with the disturbance for multi dimensional partial differential equation is presented through this system the numerical experiments are carried out to illustrate the convergence and effect of peaking value reduction

rational fibrations in differential homological algebra in this paper a result of 6 is generalized as follows given a fibration f e b of simply connected spaces in which either the fibre has finite dimensional rational cohomology or it has finite dimensional rational homotopy and induces a surjection in rational homotopy we construct an explicit isomorphism ext c b q q c b q 6aext c f q q c f q ext c e q q c e q this is deduced from its algebraic translation a more general result in the environment of graded differential homological algebra

on the dynamics of solitons in the nonlinear schrodinger equation we study the behavior of the soliton solutions of the equation i frac partial psi partial t frac 1 2m delta psi frac 1 2 w varepsilon prime psi v x psi where w varepsilon prime is a suitable nonlinear term which is singular for varepsilon 0 we use the strong nonlinearity to obtain results on existence shape stability and dynamics of the soliton the main result of this paper theorem 1 shows that for varepsilon rightarrow 0 the orbit of our soliton approaches the orbit of a classical particle in a potential v x

discontinuous unit feedback control of uncertain infinite dimensional systems control systems driven by a discontinuous unit feedback in a hilbert space are studied the equation which describes a system motion taking place in the discontinuity manifold and further referred to as a sliding mode is derived by means of a special regularization technique based on the sliding mode equation the procedure of synthesis of a discontinuous unit control signal is developed restricted to a class of infinite dimensional systems with finite dimensional unstable part this procedure generates the control law which ensures desired dynamic properties as well as robustness of the closed loop system with respect to matched disturbances as an illustration of the capabilities of the procedure proposed a scalar unit controller of an uncertain exponentially minimum phase dynamic system is constructed and applied to heat processes and distributed mechanical oscillators

stable solutions of the logarithmic schrodinger equation on considere l equation de schrodinger suivante u t u ulog u 2 0 dans r r n on donne des resultats sur le comportement asymptotique des solutions

a solution to the focusing 3d nls that blows up on a contracting sphere we rigorously construct radial h 1 solutions to the 3d cubic focusing nls equation i t 2j j 2 0 that blow up along a contracting sphere with blow up time set to t 0 the solutions concentrate on a sphere at radius t 1 3 but focus towards this sphere at the faster rate t 2 3 such dynamics were originally proposed heuristically by degtyarev zakharov rudakov 2 in 1975 and independently later in holmer roudenko 5 in 2007 where it was demonstrated to be consistent with all conservation laws of this equation in the latter paper it was proposed as a solution that would yield divergence of the l 3 norm within the wide radius kr u t k 1 2 l2 but not within the tight radius kr u t k 2 l2

nonlinear schrodinger equations and sharp interpolation estimates a sharp sufficient condition for global existence is obtained for the nonlinear schrodinger equation begin array 20 c nls 2i phi t delta phi left phi right 2 sigma phi 0 x in mathbb r n t in mathbb r end array in the case 2 n this condition is in terms of an exact stationary solution nonlinear ground state of nls it is derived by solving a variational problem to obtain the best constant for classical interpolation estimates of nirenberg and gagliardo

lyapunov approach to output feedback stabilization for the euler bernoulli beam equation with boundary input disturbance we propose a boundary output feedback control law for a one dimensional euler bernoulli beam equation with general external disturbance entering the control end a galerkin approximation scheme is constructed to show the existence of solution to the closed loop system the exponential stability of the closed loop system is obtained by the lyapunov functional method numerical simulations are presented for illustration

stabilization of one dimensional wave equation with nonlinear boundary condition subject to boundary control matched disturbance in this paper we consider the stabilization of a one dimensional wave equation with nonlinear van der pol type boundary condition that covers the antistable boundary and subject to boundary control matched disturbance on the other side due to the nonlinear boundary condition and disturbance the uncontrolled system may present spatiotemporal chaotic period doubling bifurcation and some other dynamical behaviors we will deal with this disturbance which is supposed to be bounded only by the integral sliding mode control the well posedness of the system for the closed loop system is proved and the reaching condition is obtained finally we provide some numerical simulations to illustrate the theoretical outcomes

relaxation of solitons in nonlinear schrodinger equations with potential abstract in this paper we study dynamics of solitons in the generalized nonlinear schrodinger equation nls with an external potential in all dimensions except for 2 for a certain class of nonlinearities such an equation has solutions which are periodic in time and exponentially decaying in space centered near different critical points of the potential we call those solutions which are centered near the minima of the potential and which minimize energy restricted to l 2 unit sphere trapped solitons or just solitons in this paper we prove under certain conditions on the potentials and initial conditions that trapped solitons are asymptotically stable moreover if an initial condition is close to a trapped soliton then the solution looks like a moving soliton relaxing to its equilibrium position the dynamical law of motion of the soliton i e effective equations of motion for the soliton s center and momentum is close to newton s equation but with a dissipative term due to radiation of the energy to infinity

stabilization of ode with hyperbolic equation actuator subject to boundary control matched disturbance in this paper we consider stabilisation for a cascade of ode and first order hyperbolic equation with external disturbance flowing to the control end the active disturbance rejection control adr

self focusing on bounded domains the critical nonlinear schrodinger equation nls on bounded domains models the propagation of cw laser beams in hollow core fibers unlike the nls on unbounded domains which models propagation in bulk media the ground state waveguide solutions are stable and the condition of critical power for singularity formation is generically sharp

multisymplectic geometry and multisymplectic preissmann scheme for the kdv equation the multisymplectic structure of the kdv equation is presented directly from the variational principle from the numerical view point we give a multisymplectic twelve points scheme which is equivalent to the multisymplectic preissmann scheme finally we test the twelve points scheme on solitary waves over long time intervals

instability of nonlinear bound states we establish a sharp instability theorem for the bound states of lowest energy of the nonlinear klein gordon equation utt u f u 0 and the nonlinear schrodinger equation iut u f u 0

on control design for pdes with space dependent diffusivity or time dependent reactivity in this paper the recently introduced backstepping method for boundary control of linear partial differential equations pdes is extended to plants with non constant diffusivity thermal conductivity and time varying coefficients the boundary stabilization problem is converted to a problem of solving a specific klein gordon type linear hyperbolic pde this pde is then solved for a family of system parameters resulting in closed form boundary controllers the results of a numerical simulation are presented for the case when an explicit solution is not available

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 centre stable manifold for the focussing cubic nls in mathbb r 1 3 consider the focussing cubic nonlinear schrodinger equation in mathbb r 3 i psi t delta psi psi 2 psi quad 0 1 it admits special solutions of the form eit where phi in mathcal s mathbb r 3 is a positive 0 solution of delta phi alpha phi phi 3 quad 0 2 the space of all such solutions together with those obtained from them by rescaling and applying phase and galilean coordinate changes called standing waves is the 8 dimensional manifold that consists of functions of the form e i v cdot gamma phi cdot y alpha we prove that any solution starting sufficiently close to a standing wave in the sigma w 1 2 mathbb r 3 cap x 1 l 2 mathbb r 3 norm and situated on a certain codimension one local lipschitz manifold exists globally in time and converges to a point on the manifold of standing waves furthermore we show that mathcal n is invariant under the hamiltonian flow locally in time and is a centre stable manifold in the sense of bates jones batjon the proof is based on the modulation method introduced by soffer and weinstein for the l2 subcritical case and adapted by schlag to the l2 supercritical case an important part of the proof is the keel tao endpoint strichartz estimate in mathbb r 3 for the nonselfadjoint schrodinger operator obtained by linearizing 0 1 around a standing wave solution all results in this paper depend on the standard spectral assumption that the hamiltonian mathcal h left begin array cc delta 2 phi cdot alpha 2 alpha quad phi cdot alpha 2 phi cdot alpha 2 quad delta 2 phi cdot alpha 2 alpha end array right quad 0 3 has no embedded eigenvalues in the interior of its essential spectrum infty alpha cup alpha infty

solitary waves of nonlinear dispersive evolution equations with critical power nonlinearities we study the nonlinear schrodinger nls generalized korteweg de vries gkdv and generalized benjamin ono equations gbo with critical power nonlinearities the ground state solitary wave is conditionally stable modulo the group of translations in space and phase and dilations this is a natural extension of the result for subcritical nonlinearities where solitary waves are stable modulo translations alone to a case where solutions may blow up we also present results on global existence for gbo and the structure of singularities for certain solutions of gbo which are analogues of previous results by the author for nls and gkdv

exponential stabilization of conservation systems with interior disturbance abstract in this paper we consider the stabilization problem of the conservation systems with interior disturbance employing the idea of sliding mode control we design a nonlinear distributed feedback controller we prove the solvability of the resulted closed loop system by the maximal monotone operator theory further we prove the exponential stability of the closed loop system in particular we prove that the requirement of a classical solution in the lyapunov approach is unnecessary

on the evaluation map the evaluation map of a differential graded algebra or of a space is described under two different approaches this concept turns out to have geo metric implications i a 1 connected topological space with finite dimensional rational homotopy has finite dimensional rational cohomology if and only if it has nontrivial evaluation map ii let e b be a fibration of simply connected spaces if the rational cohomology of the fibre is finite dimensional and the evaluation map of the base is different from zero then the evaluation map of the total space is nonzero also if p is surjective in rational homotopy and the evaluation map of e is nontrivial then the evaluation map of the fibre is different from zero

the multi symplectic fourier pseudospectral method for solving two dimensional hamiltonian pdes in this paper the multi symplectic fourier pseudospectral msfp method is generalized to solve two dimensional hamiltonian pdes with periodic boundary conditions using the fourier pseudospectral method in the space of the two dimensional hamiltonian pde 2d hpde the semi discrete system obtained is proved to have semi discrete multi symplectic conservation laws and a global symplecticity conservation law then the implicit midpoint rule is employed for time integration to obtain the msfp method for the 2d hpde the fully discrete multi symplectic conservation laws are also obtained in addition the proposed method is applied to solve the zakharov kuznetsov zk equation and the kadomtsev petviashvili kp equation numerical experiments on soliton solutions of the zk equation and the kp equation show the high accuracy and effectiveness of the proposed method

stability theory of solitary waves in the presence of symmetry ii abstract consider an abstract hamiltonian system which is invariant under a group of operators we continue to study the effect of the group invariance on the stability of solitary waves applications are given to bound states and traveling wave solutions of nonlinear wave equations

long wave approximations for water waves in this paper we obtain new nonlinear systems describing the interaction of long water waves in both two and three dimensions these systems are symmetric and conservative rigorous convergence results are provided showing that solutions of the complete free surface euler equations tend to associated solutions of these systems as the amplitude becomes small and the wavelength large using this result as a tool a rigorous justification of all the two dimensional approximate systems recently put forward and analysed by bona chen and saut is obtained in the two dimensional context our methods also allows a significant improvement of the convergence estimate obtained by schneider and wayne in their justification of the decoupled korteweg de vries approximation of the two dimensional euler equations it also follows from our theory that coupled models provide a better description than the decoupled ones over short time scales results are obtained both on an unbounded domain for solutions that evanesce at infinity as well as for solutions that are spatially periodic

a sampled data singularly perturbed boundary control for a heat conduction system with noncollocated observation this note presents a sampled data strategy for a boundary control problem of a heat conduction system modeled by a parabolic partial differential equation pde using the zero order hold the control law becomes a piecewise constant signal in which a step change of value occurs at each sampling instant through the dasialiftingpsila technique the pde is converted into a sequence of constant input problems to be solved individually for a sampled data formulation the eigenspectrum of the parabolic system can be partitioned into two groups a finite number of slow modes and an infinite number of fast modes which is studied via the theory of singular perturbations controllability and observability of the sampled data system are preserved irrelevant to the sampling period a noncollocated output feedback design based upon the state observer is employed for set point regulation the state observer serves as an output feedback compensator with no static feedback directly from the output satisfying the so called dasialow pass propertypsila the feedback controller is thus robust against the observation error due to the neglected fast modes

the active disturbance rejection control approach to stabilisation of coupled heat and ode system subject to boundary control matched disturbance we consider stabilisation for a linear ordinary differential equation system with input dynamics governed by a heat equation subject to boundary control matched disturbance the active disturbance rejection control approach is applied to estimate in real time the disturbance with both constant high gain and time varying high gain the disturbance is cancelled in the feedback loop the closed loop systems with constant high gain and time varying high gain are shown respectively to be practically stable and asymptotically stable

sliding mode control to stabilization of cascaded heat pde ode systems subject to boundary control matched disturbance in this paper we are concerned with the boundary feedback stabilization of a cascade of heat pde ode system with dirichlet neumann interconnection and with the external disturbance flowing the control end in order to deal with the disturbance the sliding model control smc is integrated with the backstepping approach where the disturbance is supposed to be bounded only the existence and uniqueness of the solution for the closed loop system are proved and the monotonicity of the reaching condition is presented without differentiation of the sliding mode function for which it may not always exist for the weak solution of the closed loop system finally the numerical simulations validate the effectiveness of this method for the system with periodic and normal random disturbances respectively

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

brief paper sliding mode boundary control of a parabolic pde system with parameter variations and boundary uncertainties this paper considers the stabilization problem of a one dimensional unstable heat conduction system rod modeled by a parabolic partial differential equation pde powered with a dirichlet type actuator from one of the boundaries by applying the volterra integral transformation a stabilizing boundary control law is obtained to achieve exponential stability in the ideal situation when there are no system uncertainties the associated lyapunov function is used for designing an infinite dimensional sliding manifold on which the system exhibits the same type of stability and robustness against certain types of parameter variations and boundary disturbances it is observed that the relative degree of the chosen sliding function with respect to the boundary control input is zero a continuous control law satisfying the reaching condition is obtained by passing a discontinuous signum signal through an integrator

active disturbance rejection control for rejecting boundary disturbance from multidimensional kirchhoff plate via boundary control in this paper an algorithm is developed to reject time and spatially varying boundary disturbances from a multidimensional kirchhoff plate via boundary control the disturbance and control input are assumed to be matched the active disturbance rejection control approach is adopted for developing the algorithm a state feedback scheme is designed to estimate the disturbance based on an infinite number of ordinary differential equations obtained from the original multidimensional system using infinitely many time dependent test functions the proposed control law cancels the disturbance using its estimated value all subsystems in the closed loop are shown to be asymptotically stable simulation results are presented to validate the theoretical conclusions and to exhibit the reduction in the peaking phenomenon due to the use of time varying gains instead of constant high gains

homoclinic orbits in reversible systems and their applications in mechanics fluids and optics abstract this survey article reviews the theory and application of homoclinic orbits to equilibria in reversible continuous time dynamical systems where the homoclinic orbit and the equilibrium are both reversible the focus is on even order reversible systems in four or more dimensions local theory generic argument and global existence theories are examined for each qualitatively distinct linearisation several recent results such as coalescence caused by non transversality and the reversible orbit flip bifurcation are covered a number of open problems are highlighted applications are reviewed to systems arising from a variety of disciplines with the aid of numerical methods three examples are presented in detail one of which is infinite dimensional

the active disturbance rejection and sliding mode control approach to the stabilization of the euler bernoulli beam equation with boundary input disturbance in this paper we are concerned with the boundary feedback stabilization of a one dimensional euler bernoulli beam equation with the external disturbance flowing to the control end the active disturbance rejection control adrc and sliding mode control smc are adopted in investigation by the adrc approach the disturbance is estimated through an extended state observer and canceled online by the approximated one in the closed loop it is shown that the external disturbance can be attenuated in the sense that the resulting closed loop system under the extended state feedback tends to any arbitrary given vicinity of zero as the time goes to infinity in the second part we use the smc to reject the disturbance by removing the condition in adrc that the derivative of the disturbance is supposed to be bounded the existence and uniqueness of the solution for the closed loop via smc are proved and the monotonicity of the reaching condition is presented without the differentiation of the sliding mode function for which it may not always exist for the weak solution of the closed loop system the numerical simulations validate the effectiveness of both methods

constructing discontinuity surfaces for variable structure systems a lyapunov approach the problem of constructing discontinuity surfaces in variable structure systems is studied from a lyapunov point of view a switching surface determined by the control coefficient matrix and the associated lyapunov function is able to ensure asymptotic stability for the system in sliding mode the proposed method may also be used for systems with nonlinear dynamics and for linear systems with delays

on the multi symplectic structure of boussinesq type systems i derivation and mathematical properties the boussinesq equations are known since the end of the xixst century however the proliferation of various boussinesq type systems started only in the second half of the xxst century today they come under various flavours depending on the goals of the modeller at the beginning of the xxist century an effort to classify such systems at least for even bottoms was undertaken and developed according to both different physical regimes and mathematical properties with special emphasis in this last sense on the existence of symmetry groups and their connection to conserved quantities of particular interest are those systems admitting a symplectic structure with the subsequent preservation of the total energy represented by the hamiltonian in the present paper a family of boussinesq type systems with multi symplectic structure is introduced some properties of the new systems are analyzed their relation with already known boussinesq models the identification of those systems with additional hamiltonian structure as well as other mathematical features like well posedness and existence of different types of solitary wave solutions the consistency of multi symplectic systems with the full euler equations is also discussed

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

stabilisation of timoshenko beam system with a tip payload under the unknown boundary external disturbances this paper is concerned with the boundary stabilisation of a timoshenko beam with a tip payload under the boundary external disturbances nonlinear feedback control laws are designed to reduce the effects of the external disturbances since the controlled system is nonlinear the well posedness of the nonlinear closed loop system is investigated using the theory of nonlinear maximal monotone operators and the variational principle in addition we show the exponential stability of the nonlinear controlled system using the lyapunov direct method in the end the numerical simulations are displayed to illustrate the effectiveness of the proposed boundary controls

sliding mode control in dynamic systems the concept of sliding modes in abstract dynamic systems described by a semigroup of state space transformations is introduced the sliding mode design procedure is used for designing finite observers sliding mode control for systems with delays and differential difference systems which is illustrated by sliding mode control of longitudinal oscillations

orbital stability of solitary waves of a 3 coupled nonlinear schrodinger system abstract in this paper consideration is given to the 3 coupled nonlinear schrodinger system i t u j 2 x 2 u j i 1 3 b i j u i 2 u j 0 where u j are complex valued functions of x t r 2 j 1 2 3 and b i j are positive constants satisfying b i j b j i it will be shown first that if the symmetric matrix b b i j satisfies certain conditions then ground state solutions of the 3 coupled nonlinear schrodinger system exist and moreover they are orbitally stable the theory is then extended to include solitary waves as well in particular it will be shown that when a solitary wave is perturbed the perturbed solution must stay close to a solitary wave profile in which the translation and phase parameters are prescribed functions of time properties of these functions are then studied this is a continuous work of our previous paper where the 2 coupled nonlinear schrodinger system was considered

the regularized benjamin ono and bbm equations well posedness and nonlinear stability abstract nonlinear stability of nonlinear periodic solutions of the regularized benjamin ono equation and the benjamin bona mahony equation with respect to perturbations of the same wavelength is analytically studied these perturbations are shown to be stable we also develop a global well posedness theory for the regularized benjamin ono equation in the periodic and in the line setting in particular we show that the cauchy problem in both periodic and nonperiodic case cannot be solved by an iteration scheme based on the duhamel formula for negative sobolev indices

lyapunov stability of ground states of nonlinear dispersive evolution equations on presente une nouvelle demonstration de la stabilite orbitale des solitons d etat fondamental de l equation de schrodinger non lineaire pour une grande classe de non linearites on demontre la stabilite de l onde solitaire pour l equation de korteweg de vries generalisee

ude based robust boundary control for an unstable parabolic pde with unknown input disturbance a novel control strategy named uncertainty and disturbance estimator ude based robust control is applied to the stabilization of an unstable parabolic partial differential equation pde with a dirichlet type boundary actuator and an unknown time varying input disturbance the unstable pde is stabilized by the backstepping approach and the unknown input disturbance is compensated by the ude which constructs an estimation of the disturbance by filtering the system input and measurable state signals the proposed ude based robust boundary control only requires the spectrum information of the disturbance signal and one boundary output measurement stability analysis of the closed loop systems is carried out and a numerical example is provided to validate the proposed method

long time oscillation of solutions of nonlinear schrodinger equations near minimal mass ground state abstract in this paper we consider the long time dynamics of radially symmetric solutions of nonlinear schrodinger equations nls having a minimal mass ground state in particular we show that there exist solutions with initial data near the minimal mass ground state that oscillate for long time more precisely we introduce a coordinate defined near the minimal mass ground state which consists of finite and infinite dimensional part associated to the discrete and continuous part of the linearized operator then we show that the finite dimensional 2 d part approximately obeys newton s equation of motion for a particle in an anharmonic potential well showing that the infinite dimensional part is well separated from the finite dimensional part we will have long time oscillation

periodic finite genus solutions of the kdv equation are orbitally stable abstract the stability of periodic solutions of partial differential equations has been an area of increasing interest in the last decade the kdv equation is known to have large families of periodic solutions that are parameterized by hyperelliptic riemann surfaces they are generalizations of the famous multi soliton solutions we show that all such periodic solutions are orbitally stable with respect to subharmonic perturbations perturbations that are periodic with period equal to an integer multiple of the period of the underlying solution

sliding mode boundary control of an euler bernoulli beam subject to disturbances in this paper a control technique is presented for an undamped pinned pinned euler bernoulli beam with control inputs and bounded disturbances on one boundary the control strategy drives the system to its origin at an arbitrary exponential rate in the presence of the disturbances this is achieved in two main steps first a backstepping transformation is used to convert the marginally stable euler bernoulli beam system to a new form that has an exponentially stable homogeneous form control inputs are needed to fully convert the system to this form however since they are distorted by unknown bounded disturbances the next step implements a sliding mode controller to account for them the proposed sliding manifolds require a combination of classical and second order techniques in order to avoid discontinuous chattering on the physical system therefore the continuous sliding mode controllers developed return the beam to its origin at an arbitrary exponential rate and do so in the presence of unknown bounded disturbances on the boundary the main contributions of this paper with respect to previous backstepping designs for the euler bernoulli beam are that all three of the following goals are accomplished together i steady state position is the origin ii decay rate has no theoretical restrictions and iii is robust to bounded disturbances

technical communique an exponential stability result for the wave equation we consider a system described by the one dimensional linear wave equation in a bounded domain with appropriate boundary conditions to stabilize this system we propose a dynamic boundary controller applied at the free end of the system the transfer function of the proposed controller is a proper rational function which consists of a strictly positive real function and some poles on the imaginary axis we then show that under some conditions the closed loop system is exponentially stable

boundary feedback stabilization of an unstable heat equation in this paper we study the problem of boundary feedback stabilization for the unstable heat equation ut x t uxx x t a x u x t this equation can be viewed as a model of a heat conducting rod in which not only is the heat being diffused mathematically due to the diffusive term uxx but also the destabilizing heat is generating mathematically due to the term a u with a 0 we show that for any given continuously differentiable function a and any given positive constant l we can explicitly construct a boundary feedback control law such that the solution of the equation with the control law converges to zero exponentially at the rate of l this is a continuation of the recent work of boskovic krstic and liu ieee trans automat control 46 2001 pp 2022 2028 and balogh and krstic european j control 8 2002 pp 165 176

on the solitary wave dynamics under slowly varying medium for nonlinear schrodinger equations we consider the problem of a solitary wave propagation in a slowly varying medium for a variable coefficients nonlinear schrodinger equation we prove global existence and uniqueness of solitary wave solutions for a large class of slowly varying media moreover we describe for all time the behavior of these solutions which include refracted and reflected solitary waves depending on the initial energy

rational l s category and its applications let s be a 1 connected cw complex of finite type and put cat0 s lusternik schnirelmann category of the localization sq this invariant is char acterized in terms of the minimal model of s it is shown that if is the fibre inclusion of a fibration it is also shown that if dim h s q oo then either dim it s q oo or the groups nk s q grow exponentially with k introduction in this paper topological space means a pointed path connected normal space s1 such that the inclusion of the basepoint is a cofibration and the rational singular homology is finite dimensional in each degree all vector spaces and algebras have q as ground field homology and cohomology of spaces is singular with rational coefficients we shall be dealing with category of spaces cat s in the sense of lusternik and schnirelmann 18 a recent survey is given in 15 but our definition differs by one from that of 15 so that for us spheres have category one not two our purpose is to provide a computationally useful characterization of the rational category cat0 5 of space 5 and to apply this in a variety of situations the definition of cat0 5 is sketched below and given precisely in 4 it satisfies cat0 s cauls when 5 is 1 connected cat0 5 coincides with the category of the localization sq which has been studied by toomer in 26 and 27 and lemaire and sigrist in 17 the characterization problem was posed by berstein 26 most of the applications depend on the following fundamental mapping theorem which we derive as a consequence of our characterization