exponential power series expansion for the propagator of general diffusion processes abstract the propagator of a general diffusion process is determined by expanding its exponent in a power series of a time increment t the expansion coefficients can be analytically evaluated from recursive relations we are thus able to construct a covariant short time approximation for the propagator valid to any desired precision in t attention is given both to the mathematics and its physical interpretation this allows us to shed further light on the results already known in the literature on quantum mechanics and theory of continuous markov processes

nondispersive solutions to the l 2 critical half wave equation we consider the focusing l2 critical half wave equation in one space dimension i partial t u d u u 2 u where d denotes the first order fractional derivative standard arguments show that there is a critical threshold m 0 such that all h1 2 solutions with u l 2 0 and the linear momentum p 0 in mathbb r in particular our main result and its proof can be seen as a model scenario of minimal mass blowup for l2 critical nonlinear pdes with nonlocal dispersion

conditional probabilities in multiplicative noise processes we address the calculation of transition probabilities in multiplicative noise stochastic differential equations using a path integral approach we show the equivalence between the conditional probability and the propagator of a quantum particle with variable mass introducing a time reparametrization we are able to transform the problem of multiplicative noise fluctuations into an equivalent additive one we illustrate the method by showing the explicit analytic computation of the conditional probability of a harmonic oscillator in a nonlinear multiplicative environment

surmounting oscillating barriers path integral approach for weak noise we consider the thermally activated escape of an overdamped brownian particle over a potential barrier in the presence of periodic driving a time dependent path integral formalism is developed which allows us to derive asymptotically exact weak noise expressions for both the instantaneous and the time averaged escape rate our results comprise a conceptually different systematic treatment of the rate prefactor multiplying the exponentially leading arrhenius factor moreover an estimate for the deviations at finite noise strengths is provided and a supersymmetry type property of the time averaged escape rate is verified for piecewise parabolic potentials the rate expression can be evaluated in closed analytical form while in more general cases as exemplified by a cubic potential an action integral remains to be minimized numerically our comparison with very accurate numerical results demonstrates an excellent agreement with the theoretical predictions over a wide range of driving strengths and driving frequencies

uniqueness of non linear ground states for fractional laplacians in mathbb r we prove uniqueness of ground state solutions q q x 0 of the non linear equation sq q q 1 0inr s q q q 1 0 i n r where 0 s 1 and 0 4s 1 2s for s 12 s 1 2 and 0 for s 12 s 1 2 here s denotes the fractional laplacian in one dimension in particular we answer affirmatively an open question recently raised by kenig martel robbiano and we generalize by completely different techniques the specific uniqueness result obtained by amick and toland for s 12 s 1 2 and 1 in 5 for the benjamin ono equation as a technical key result in this paper we show that the associated linearized operator l s 1 1 q is non degenerate i e its kernel satisfies ker l span q this result about l proves a spectral assumption which plays a central role for the stability of solitary waves and blowup analysis for non linear dispersive pdes with fractional laplacians such as the generalized benjamin ono bo and benjamin bona mahony bbm water wave equations

stability and instability of solitary waves of korteweg de vries type considered herein are the stability and instability properties of solitary wave solutions of a general class of equations that arise as mathematical models for the unidirectional propagation of weakly nonlinear dispersive long waves special cases for which our analysis is decisive include equations of the korteweg de vries and benjamin ono type necessary and sufficient conditions are formulated in terms of the linearized dispersion relation and the nonlinearity for the solitary waves to be stable

fuzzy guaranteed cost sampled data control of nonlinear systems coupled with a scalar reaction diffusion process a fuzzy guaranteed cost sampled data control problem is addressed in this paper for a class of nonlinear coupled systems of an n dimensional lumped parameter system modeled by ordinary differential equation ode with a scalar reaction diffusion process represented by parabolic partial differential equation pde a takagi sugeno t s fuzzy coupled ode pde model is initially proposed to accurately represent the nonlinear coupled ode pde system based on the t s fuzzy coupled ode pde model a fuzzy sampled data controller is subsequently developed via the lyapunov s direct method to not only locally exponentially stabilize the nonlinear coupled system but also provide an upper bound of the given cost function moreover a suboptimal fuzzy sampled data control design is also addressed in the sense of minimizing the upper bound of the cost function the main contribution of this study lies in that a new parameterized linear matrix inequality lmi technique is proposed to reduce the conservativeness of the method and an lmi based fuzzy suboptimal sampled data control design is developed for the nonlinear coupled ode pde system based on this parameterized lmi technique simulation results on the sampled data control of hypersonic rocket car are provided to illustrate the effectiveness and merit of the design method

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

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

role of an intermediate state in homogeneous nucleation we explore the role of an intermediate state phase in homogeneous nucleation by examining the decay process through a double humped potential barrier we analyze the one dimensional fokker planck fp equations with the fourth and sixth order landau potentials in the low temperature case we apply the wkb method to the fp equation and obtain an analytic expression for the decay rate which is accurate for a wide range of depth and curvature of the middle well in the case of a deep middle well it reduces to an extended kramers formula in which the barrier height in the original formula is replaced by the arithmetic mean height of the higher outer and lower inner barriers and the curvature of the initial well in the original one is replaced by the geometric mean curvature of the initial and intermediate wells in the case of a shallow middle well the kramers escape rate is evaluated also within the standard framework of the mean first passage time problem whose result is consistent with our wkb analysis criteria for enhancement of the decay rate are revealed

dynamical analysis in a bioeconomic phytoplankton zooplankton system with double time delays and environmental stochasticity abstract this paper presents a double delayed bioeconomic phytoplankton zooplankton system with commercial harvesting on zooplankton and environmental stochasticity maturation delay for toxin producing phytoplankton and gestation delay for zooplankton are considered environmental stochasticity is incorporated into the proposed system in form of gaussian white noises some sufficient conditions are derived to show that the proposed system has a unique global positive solution in absence of double time delays stochastic stability and existence of stochastic hopf bifurcation are studied based on invariant measure theory and singular boundary theory of diffusion process for the proposed system in presence of double time delays asymptotic behaviors of the interior equilibrium are discussed by constructing some appropriate lyapunov functions

investigation of a class of stochastic processes using path integral techniques a study is made of the langevin equation x v chi g chi zeta t eta t where the noises eta t and zeta t are gaussian with zero mean and with eta t eta t 2r delta t t zeta t zeta t d tau exp mod t t mod tau path integral representations for conditional probability distributions are given for the two cases tau 0 and tau not 0 for r and d small but of the same order the appropriate path integrals are evaluated to leading order using the method of steepest descents in order to find the stationary probability distribution pst x and the mean relaxation time t for escape from a potential well analytical expressions are given for t when tau 0 and when tau is small for general tau the authors present numerical results for the stationary probability distribution using the particular forms v x 1 2ax2 1 4ax4 r 1n x and g x x which are appropriate to the dye laser

mixed h2 h sampled data output feedback control design for a semi linear parabolic pde in the sense of spatial l norm abstract this paper addresses distributed mixed h 2 h sampled data output feedback control design for a semi linear parabolic partial differential equation pde with external disturbances in the sense of spatial l norm under the assumption that a finite number of local piecewise measurements in space are available at sampling instants a static sampled data output feedback controller is suggested where the sampling interval in time is bounded the local piecewise measurements bring additional difficulty for the exponential stability and performance analysis since the existing poincare wirtinger inequality in 1d spatial domain is not applicable by constructing an appropriate lyapunov krasovskii functional candidate and employing wirtinger s inequalities a variant of poincare wirtinger inequality in 1d spatial domain as well as agmon s inequality it is shown that the suggested static sampled data output feedback controller not only guarantees the output exponential stability of the resulting closed loop pde in the spatial l norm but also ensures the mixed h 2 h performance index defined in the spatial l norm if the sufficient conditions presented in terms of standard linear matrix inequalities lmis are fulfilled the satisfactory and better performance of the suggested sampled data feedback controller is demonstrated by numerical simulation results of an illustrative example

diffusion in a symmetric bistable potential a variational approach an approximation procedure for the solution of stochastic nonlinear equations which was derived from a variational principle in a previous paper is applied to the problem of a particle that diffuses in a symmetric bistable potential starting from the point of unstable equilibrium the second moment left langle widetilde x 2 t right rangle and variance left langle widetilde x 4 t right rangle left langle widetilde x 2 t right rangle 2 for the particle s position widetilde x t are calculated as functions of the timet good agreement is found with results recently obtained by baibuzet al from an approximate evaluation of a path integral expression for the probability density

dynamic analysis of a hybrid bioeconomic plankton system with double time delays and stochastic fluctuations abstract in this paper we establish a double delayed hybrid bioeconomic plankton system with stochastic fluctuations and commercial harvesting on zooplankton where maturation delay for toxin producing phytoplankton and gestation delay for zooplankton are considered stochastic fluctuations are incorporated into the proposed system in form of gaussian white noises to depict stochastic environmental factors in plankton system for deterministic system without double time delays existence of singularity induced bifurcation is studied due to variation of economic interest of commercial harvesting and state feedback controllers are designed to eliminate singularity induced bifurcation for deterministic system with double time delays positivity and uniform persistence of solutions are studied and some sufficient conditions associated with asymptotic stability of interior equilibrium are investigated for stochastic system without double time delays stochastic stability and existence of stochastic hopf bifurcation are discussed based on singular boundary theory of diffusion process and invariant measure theory for stochastic system with double time delays existence and uniqueness of global positive solution are investigated and asymptotic behaviors of the interior equilibrium are studied by constructing appropriate lyapunov functions numerical simulations are carried out to validate theoretical analysis

extension of statistical replacement to systems with time correlated fluctuations nonlinear systems with correlated stochastic parameters are approximated by simpler systems this method is an extension of an earlier version of statistical replacement and statistical linearization the extended method is applicable to systems with correlated fluctuations we show how this general method reduces to the earlier methods in special cases

sampled data observers for semilinear damped wave equations under spatially sampled state measurements abstract sampled data observers controllers under the sampled in space and time measurements were suggested in the past for parabolic systems in the present paper for the first time a sampled data observer is constructed for a hyperbolic system governed by 1d semilinear wave equation with either viscous or boundary damping the measurements are sampled in space and time sufficient conditions for the exponential stability of the estimation error are derived by using the time delay approach to sampled data control and appropriate lyapunov krasovskii functionals the dual sampled data controller problems are formulated numerical examples including observer design for unstable damped sine gordon equation illustrate the efficiency of the method

optimal gaussian solutions of nonlinear stochastic partial differential equations we present a linearization procedure of a stochastic partial differential equation for a vector field x i t x t 0 x r d i l n t x i t x b i x t x d x i t x i f i t x here is the laplace beltrami operator inr d and f i t x is a gaussian random field with f i t x f j t x ij t t x x the procedure is a natural extension of the equivalent linearization for stochastic ordinary differential equations the linearized solution is optimal in the sense that the distance between true and approximate solutions is minimal when it is measured by the kullback leibler entropy the procedure is applied to the scalar valued ginzburg landau model in r1 withb 1 z z vz 3 stationary values of mean variance and correlation length are calculated they almost agree with exact ones if 1 24 2 1 4 d 1 1 3 c when c there appear quasistationary states fluctuating around one of the bottoms of the potentialu z b 1 z dz the second moment at the quasistationary states almost agrees with the exact one transient phenomena are also discussed half width at half maximum of a structure function decays liket 1 2 for small t the diffusion term 2 x accelerates the relaxation from the neighborhood of an unstable initial statex 0 x 0

construction of solutions with exactly k blow up points for the schrodinger equation with critical nonlinearity we consider the nonlinear schrodinger equation i partial u mathord left vphantom i partial u partial t right kern nulldelimiterspace partial t delta u left u right 4 mathord left vphantom 4 n right kern nulldelimiterspace n uandu left 0 right varphi left right 1 whereu 0 t n for any given pointsx1 x2 x k in n we construct a solution of eq 1 u t which blows up in a finite timet at exactlyx1 x2 x k in addition we describe the precise behavior of the solutionu t whent t at the blow up points x1 x2 x k and in n x1 x2 x k

output feedback boundary control of a heat pde sandwiched between two odes we present designs for exponential stabilization of an ordinary differential equation ode heat partial differential equation pde ode coupled system where the control actuation only acts in one ode the combination of pde backstepping and ode backstepping is employed in a state feedback control law and in an observer that estimates pde and two ode states using only one pde boundary measurement based on the state feedback control law and the observer the output feedback control law is then proposed the exponential stability of the closed loop system and the boundedness and exponential convergence of the control law are proved via lyapunov analysis finally numerical simulations validate the effectiveness of this method for the sandwiched system

fuzzy boundary control design for a class of nonlinear parabolic distributed parameter systems this paper deals with the problem of fuzzy boundary control design for a class of nonlinear distributed parameter systems which are described by semilinear parabolic partial differential equations pdes both distributed measurement form and collocated boundary measurement form are considered a takagi sugeno t s fuzzy pde model is first applied to accurately represent the semilinear parabolic pde system based on the t s fuzzy pde model two types of fuzzy boundary controllers which are easily implemented since only boundary actuators are used are proposed to ensure the exponential stability of the resulting closed loop system sufficient conditions of exponential stabilization are established by employing the lyapunov direct method and the vector valued wirtinger s inequality and presented in terms of standard linear matrix inequalities finally the advantages and effectiveness of the proposed control methodology are demonstrated by the simulation results of two examples

exact propagator and eigenfunctions for multistable models with arbitrarily prescribed n lowest eigenvalues the author presents a method to construct potentials for schrodinger equations with some prescribed features for which all eigenfunctions and the time dependent propagator can be explicitly calculated the prescribed features can be formulated by choosing arbitrarily the n lowest eigenvalues alternatively one can prescribe some qualitative behaviour for the potential like the number and relative depths of wells and barriers the results can also be applied to the construction of fokker planck models with prescribed properties and explicitly calculable transition probability density the method is based on ideas of supersymmetric quantum mechanics and the theory of solitons that can be traced back to the work of darboux 1894 and crum 1955

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

numerical analysis of the smoluchowski equation using the split operator method abstract we apply the split operator method of feit fleck and steiger to the numerical solution of the smoluchowski equation we study the time evolution of the first few moments and of the equilibrium time correlation function for bistable potentials the time evolution of the probability density obtained by us is compared with the one predicted by the scaling theory of suzuki we also analyze the dependence of the total equilibrium relaxation time on the noise intensity d for small and large values of d the relaxation from an initial condition away from the unstable point is also discussed we find that the one well population is well described by a single exponential law with decay times which grow extremely large as the value of d is decreased

a numerical approach to blow up issues for dispersive perturbations of burgers equation abstract we provide a detailed numerical study of various issues pertaining to the dynamics of the burgers equation perturbed by a weak dispersive term blow up in finite time versus global existence nature of the blow up existence for long times and the decomposition of the initial data into solitary waves plus radiation we numerically construct solitary waves for fractional korteweg de vries equations

asymptotic stability of solitary waves we show that the family of solitary waves 1 solitons of the korteweg de vries equation partial t u u partial x u partial x 3 u 0 is asymptotically stable our methods also apply for the solitary waves of a class of generalized korteweg de vries equations partial t u partial x f u partial x 3 u 0 in particular we study the case wheref u up 1 p 1 p 1 2 3 and 3 0 withf c4 the same asymptotic stability result for kdv is also proved for the casep 2 the modified korteweg de vries equation we also prove asymptotic stability for the family of solitary waves for all but a finite number of values ofp between 3 and 4 the solitary waves are known to undergo a transition from stability to instability as the parameterp increases beyond the critical valuep 4 the solution is decomposed into a modulating solitary wave with time varying speedc t and phase t bound state part and an infinite dimensional perturbation radiating part the perturbation is shown to decay exponentially in time in a local sense relative to a frame moving with the solitary wave asp 4 the local decay or radiation rate decreases due to the presence of aresonance pole associated with the linearized evolution equation for solitary wave perturbations

stability of distributed parameter systems with finite dimensional controller compensators using singular perturbations abstract using a singular perturbation formulation of the linear time invariant distributed parameter system we develop a method to design finite dimensional feedback compensators of any fixed order which will stabilize the infinite dimensional distributed parameter system the synthesis conditions are given entirely in terms of a finite dimensional reduced order model the stability results depend on an infinite dimensional version of the klimushchev krasovskii lemma presented here

spatially piecewise fuzzy control design for sampled data exponential stabilization of semilinear parabolic pde systems this paper employs a takagi sugeno t s fuzzy partial differential equation pde model to solve the problem of sampled data exponential stabilization in the sense of spatial l infty norm vert cdot vert infty for a class of nonlinear parabolic distributed parameter systems dpss where only a few actuators and sensors are discretely distributed in space initially a t s fuzzy pde model is assumed to be derived by the sector nonlinearity method to accurately describe complex spatiotemporal dynamics of the nonlinear dpss subsequently a static sampled data fuzzy local state feedback controller is constructed based on the t s fuzzy pde model by constructing an appropriate lyapunov krasovskii functional candidate and employing vector valued wirtinger s inequalities a variation of vector valued poincare wirtinger inequality in one dimensional spatial domain as well as a vector valued agmon s inequality it is shown that the suggested sampled data fuzzy controller exponentially stabilizes the nonlinear dpss in the sense of vert cdot vert infty if sufficient conditions presented in term of standard linear matrix inequalities lmis are fulfilled moreover an lmi relaxation technique is utilized to enhance exponential stabilization ability of the suggested sampled data fuzzy controller finally the satisfactory and better performance of the suggested sampled data fuzzy controller are demonstrated by numerical simulation results of two examples

l2 concentration of blow up solutions for the nonlinear schrodinger equation with critical power nonlinearity abstract we consider the solution of the nonlinear schrodinger equation i u t u f u and u 0 where u is defined on 0 t rn and f is a nonlinear complex valued function we consider the case of the critical power that is f z z 4 n z as z in a sense which will be made precise in addition we suppose that u t blows up in the norm h1 at time t we prove that u t has no limit in l2 as t t in the particular case where has a spherical symmetry we further show a phenomenon of l2 concentration at the origin

dynamics of gompertzian tumour growth under environmental fluctuations abstract we investigate the effect of correlated additive and multiplicative gaussian white noise on the gompertzian growth of tumours our results are obtained by solving numerically the time dependent fokker planck equation fpe associated with the stochastic dynamics in our numerical approach we have adopted b spline functions as a truncated basis to expand the approximated eigenfunctions the eigenfunctions and eigenvalues obtained using this method are used to derive approximate solutions of the dynamics under study we perform simulations to analyze various aspects of the probability distribution of the tumour cell populations in the transient and steady state regimes more precisely we are concerned mainly with the behaviour of the relaxation time to the steady state distribution as a function of i of the correlation strength between the additive noise and the multiplicative noise and ii as a function of the multiplicative noise intensity d and additive noise intensity it is observed that both the correlation strength and the intensities of additive and multiplicative noise affect the relaxation time

finite dimensional control of parabolic pde systems using approximate inertial manifolds abstract this paper introduces a methodology for the synthesis of nonlinear finite dimensional output feedback controllers for systems of quasi linear parabolic partial differential equations pdes for which the eigenspectrum of the spatial differential operator can be partitioned into a finite dimensional slow one and an infinite dimensional stable fast complement combination of galerkin s method with a novel procedure for the construction of approximate inertial manifolds for the pde system is employed for the derivation of ordinary differential equation ode systems whose dimension is equal to the number of slow modes that yield solutions which are close up to a desired accuracy to the ones of the pde system for almost all times these ode systems are used as the basis for the synthesis of nonlinear output feedback controllers that guarantee stability and enforce the output of the closed loop system to follow up to a desired accuracy a prespecified response for almost all times

on the robustness of sampled data control to unmodeled high frequency dynamics robustness of sampled data control designs to unmodeled high frequency dynamics is studied using singular perturbation theory it is argued that when the plant is preceded by a zero order hold a direct transmission term of the reduced order model which results from neglecting high frequency dynamics should be modeled as a delay element in order to ensure robustness

a variational approach to stochastic nonlinear problems a variational principle is formulated which enables the mean value and higher moments of the solution of a stochastic nonlinear differential equation to be expressed as stationary values of certain quantities approximations are generated by using suitable trial functions in this variational principle and some of these are investigated numerically for the case of a bernoulli oscillator driven by white noise comparison with exact data available for this system shows that the variational approach to such problems can be quite effective

statistical replacement for systems with delta correlated fluctuations nonlinear systems with stochastic parameters are approximated by simpler systems using a method that we call statistical replacement this method is an extension of the previously developed agree which was restricted to systems with additive fluctuations statistical replacement incorporates the distinctions between globally stable thermodynamically closed systems and thermodynamically open systems that can be unstable

quasilinear approximations for the propagator of the fokker planck equation abstract four novel classes of the approximate fokker planck propagators are developed by means of a decoupling operator treatment for a small time increment t these are correct at least through second order t the propagators have an ornstein uhlenbeck form thus leading to novel discrete path integrals with a linear reference system their ambiguities are characterized by an arbitrary function of the time increment and by a matrix and a vector field which may be both time and state dependent the efficiency of the treatment used is very sensitive to the choice of these functions several forms of which are proposed the power of the treatment is illustrated by a physically meaningful fokker planck equation relevant to laser physics being used as the global approximations the approximate propagators obtained are shown to provide a correct description of a bistable system in the entire time domain a scaling theory of suzuki including its unified version emerges from them in a very natural way another attractive feature of our general treatment when compared to the others known in the literature is that it allows for equations with singular diffusion matrices two of which a kramers equation and a colored noise problem are considered

piecewise optimal linearization method for nonlinear stochastic differential equations a method for estimating the dynamical statistical properties of the solutions of nonlinear langevin type stochastic differential equations is presented the non linear equation is linearized within a small interval of the independent variable and statistical properties are expressed analytically within the interval the linearization procedure is optimal in the sense of the chebyshev inequality long term behavior of the solution process is obtained by appropriately matching the approximate solutions at the boundaries between intervals the method is applied to a model nonlinear equation for which the exact time dependent moments can be obtained by numerical methods the calculations demonstrate that the method represents a significant improvement over the method of statistical linearization in time regimes far from equilibrium

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

approximate gaussian representation of evolution equations i single degree of freedom nonlinear equations we have developed a generalization of the method of statistical linearization to enable us to describe transient and other nonstationary phenomena obeying stochastic nonlinear differential equations this approximation technique provides an optimal gaussian representation with time dependent parameters the algorithm specifies a set of ordinary differential equations for the gaussian parameters in terms of the time dependent average nonlinearities we apply the general formalism developed herein for single degree of freedom dissipative systems to a particular example

suppression of thermally activated escape by heating the problem of thermally activated escape over a potential barrier is solved by means of path integrals for one dimensional reaction dynamics with very general time dependences for a suitably chosen but still quite simple static potential landscape the net escape rate may be substantially reduced by temporally increasing the temperature above its unperturbed constant level

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

existence of blow up solutions in the energy space for the critical generalized kdv equation from these conservation laws h appears as an energy space so that it is a natural space in which to study the solutions note that p 2 is a special case for equation 2 indeed from the integrability theory see lax 14 we have for suitable u0 u0 and its derivatives with fast decay at infinity an infinite number of conservation laws the general question is to understand the dynamics induced by such equations local existence in time of solutions of 2 in the energy space is now well understood see kato 10 ginibre and tsutsumi 8 for the h theory s 32 kenig ponce and vega 11 for the l theory in the case of equation 1 and sharp h theory for 2 and bourgain 3 and 4 for the periodic case

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

on the generalized benjamin ono equation we study well posedness of the initial value problem for the generalized benjamin ono equation tu ukdxu 0 k e in sobolev spaces hs r for small data and higher nonlinearities k 2 new local and global including scattering results are established our method of proof is quite general it combines several estimates concerning the associated linear problem with the contraction principle hence it applies to other dispersive models in particular it allows us to extend the results for the generalized benjamin ono to nonlinear schrodinger equations or systems of the form atu i2u p u oxu ui axi 0

diffusion in a bistable potential a comparative study of different methods of solution the problem of diffusion in a bistable potential is studied by considering the associated nonlinear langevin equation and its equivalent fokker planck equation two numerically exact methods of solution namely the monte carlo solution of the nonlinear langevin equation and the solution of the fokker planck equation via the finite difference technique are considered the latter method has the advantage that it directly gives the evolution of the probability distribution function approximate analyses of the fluctuations using the system size expansion the gaussian decoupling procedure and the scaling approach are also carried out these investigations are performed on a representative problem for two specific cases 1 evolution from intrinsically unstable states and 2 evolution from extensive regime the fluctuations obtained using these approximate methods are compared with those obtained via the numerically exact methods the study brings out the advantages and limitations of each of the methods considered

diffusion in a bistable potential the functional integral approach we study with the help of the onsager machlup functional integral approach the distributionp of a single stochastic variable the evolution of which is described by a fokker planck equation with a first moment deriving from a bistable potential we set up the approximation scheme appropriate in this approach to the limit of constant and small diffusion coefficient two regimes are to be distinguished very long times kramers regime are treated within the frame of a free instanton molecule gas approximation and at intermediate times suzuki regime a standard semiclassical calculation is legitimate we thus rederive exactly the results obtained from the mode expansion and wkb method

diffusion in a potential field path integral approach abstract within the framework of the path integral formalism we develop a simple method to solve the fokker planck diffusion problems in a potential field including the decay of an unstable state unlike previous approaches it yields a unified treatment of all time regimes and is legitimate for any value of the diffusion coefficient an attractive feature of our method is that it provides numerical results in regions where there are no analytical solutions in particular being valid for an arbitrary shape of the potential it is very useful for studying a diffusion problem near and in the critical point where the standard treatments break down moreover our method after a simple transformation permits to obtain also the first passage times of the process

stability of blow up profile and lower bounds for blow up rate for the critical generalized kdv equation the generalized korteweg de vries equations are a class of hamiltonian systems in infinite dimension derived from the kdv equation where the quadratic term is replaced by a higher order power term these equations have two conservation laws in the energy space h 1 l 2 norm and energy we consider in this paper the critical generalized kdv equation which corresponds to the smallest power of the nonlinearity such that the two conservation laws do not imply a bound in h 1 uniform in time for all h 1 solutions and thus global existence from 15 there do exist for this equation solutions u t such that u t h1 as t t where t we call them blow up solutions the question is to describe in a qualitative way how blow up occurs for solutions with l 2 mass close to the minimal mass allowing blow up and with decay in l 2 at the right we prove after rescaling and translation which leave invariant the l 2 norm that the solution converges to a universal profile locally in space at the blow up time t from the nature of this profile we improve the standard lower bound on the blow up rate for finite time blow up solutions

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 n soliton like solutions of the subcritical and critical generalized korteweg de vries equations we consider the generalized korteweg de vries equations ut uxx upx 0t x r inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i in the subcritical and critical cases p 2 3 4 or 5 let r j t x qc j x c j t x j where j 1 n be n soliton solutions of this equation with corresponding speeds 0 c 1 c 2 c n in this paper we construct a solution u t of the generalized korteweg de vries equation such that inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i this solution behaves asymptotically as t as the sum of n solitons without loss of mass by dispersion this is an exceptional behavior indeed being given the parameters c j 1 j n x j 1 j n we prove uniqueness of such a solution in the integrable cases p 2 and 3 such solutions are explicitly known and their properties were extensively studied in the literature they are called n soliton solutions therefore the existence result is new only for the nonintegrable cases the uniqueness result is new for all cases

unstable state dynamics treatment of colored noise the diffusion of a particle set near an unstable point in a bistable potential is considered the scaling theory of fluctuations proposed originally for onedimensional systems driven by gaussian white noise is extended to arbitrary dimensions the merits and drawbacks of the scaling theory are discussed by taking a model problem in one dimension it is shown in passing that the saddle point approximation enables one to get analytic expressions for various moments of the stochastic process the two different methods to include asymptotic fluctuations which are absent in the usual scaling solution are shown to be equivalent an alternate way of including asymptotic fluctuations is attempted by solving the associated fokker planck equation using the fer formula the reason for the failure of this method is traced after this it is argued that the unified scaling theory should be applicable for treatment of colored noise as well for the scaling assumption is independent of the statistical property of the driving noise explicit monte carlo simulation of a model onedimensional system driven by exponentially correlated gaussian noise is performed and compared with the scaling solution to bolster this point the agreement is very good

a spatial domain decomposition approach to distributed h observer design of a linear unstable parabolic distributed parameter system with spatially discrete sensors abstractthis paper discusses the design problem of distributed h luenberger type partial differential equation pde observer for state estimation of a linear unstable parabolic distributed parameter system dps with external disturbance and measurement disturbance both pointwise measurement in space and local piecewise uniform measurement in space are considered that is sensors are only active at some specified points or applied at part thereof of the spatial domain the spatial domain is decomposed into multiple subdomains according to the location of the sensors such that only one sensor is located at each subdomain by using lyapunov technique wirtinger s inequality at each subdomain and integration by parts a lyapunov based design of luenberger type pde observer is developed such that the resulting estimation error system is exponentially stable with an h performance constraint and presented in terms of standard linear matrix inequalities lmis for the case of local piecewise uniform meas

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

nonlocal models for nonlinear dispersive waves abstract considered herein are model equations for the unidirectional propagation of small amplitude nonlinear dispersive long waves such as those governed by the classical korteweg de vries equation of special interest are physical situations in which the linear dispersion relation is not appropriately approximated by a polynomial so that the operator modelling dispersive effect is nonlocal particular cases in view here are the benjamin ono equation and the intermediate long wave equation which arise in internal wave theory and smith s equation which governs certain types of continental shelf waves the initial value problem for these equations is shown to be globally well posed in the classical sense including continuous dependence upon the initial data and in certain cases upon the modelling of nonlinear and dispersive effects whilst the results are stated for the specific equations listed above the techniques utilized are seen to have a considerable range of generality as regards application to nonlinear dispersive evolution equations particularly worthy of note is our theorem implying that solutions of the intermediate long wave equation converge strongly to solutions of the korteweg de vries equation or to solutions of the benjamin ono equation in appropriate asymptotic limits

the duffing oscillator in the low friction limit theory and analog simulation by a joint use of theory and analog simulation the low friction regime of the duffing oscillator is explored in the weak temperature case it is shown that the low friction regime in turn must be divided in two well distinct subregimes in the former one characterized by the friction ranging from 2 0 0 is the harmonic frequency up to a lower bound t the method of statistical linearization applies and a continued fraction procedure cfp generated by the zwanzig mori projection techniques is shown to provide correct information for both the renormalized frequency of the oscillator and the corresponding line shape the latter subregime characterized by the friction ranging from 0 to t is fraught with the complete breakdown of the statistical linearization method the cfp is shown to provide an incorrect picture of the line shape while suggesting a novel mean field approximation which is then proven analytically via an alternative method of calculation this method consists of expressing the system in a form reminiscent of the model of kubo s stochastic oscillator by using this alternative approach we are in a position to account for the residual linewidth which is shown by the analog experiment to survive for 0

numerical estimation of the path integral and the diffusion in an anharmonic oscillator abstract the solution of the fokker planck equation given in the form of the path integral is estimated by the contribution from paths around the most probable path a numerical procedure for this estimation is given and applied to the regression of an anharmonic oscillator under a random force from unstable positions to a stable position

solution of fokker planck equation using trotter s formula solution of fokker planck equation using trotter s formula is discussed the method is illustrated on the linear fokker planck equation and the ornstein uhlenbeck solution is obtained for the case of a general nonlinear fokker planck equation the method yields an integral representation amenable to approximations in the lowest order approximation suzuki s scaling result emerges physical interpretation and limitations of the approximations are also discussed

on the linearization of nonlinear langevin type stochastic differential equations we show that a logical extension of the piecewise optimal linearization procedure leads to the gaussian decoupling scheme where no iteration is required the scheme is equivalent to solving a few coupled equations the method is applied to models which represent a a single steady state b passage from an initial unstable state to a final preferred stable state by virtue of a finite displacement from the unstable state and c a bivariate case of passage from an unstable state to a final stable state the results are shown to be in very good agreement with the monte carlo calculations carried out for these cases the method should be of much value in multidimensional cases

on universality of blow up profile for l 2 critical nonlinear schrodinger equation we consider finite time blow up solutions to the critical nonlinear schrodinger equation iut u u 4 nu with initial condition u0 h1 existence of such solutions is known but the complete blow up dynamic is not understood so far for a specific set of initial data finite time blow up with a universal sharp upper bound on the blow up rate has been proved in 22 23

the application of statistical linearization to the problem of decay from an unstable state abstract it is shown that the method of statistical linearization for the approximate treatment of stochastic nonlinear systems driven by gaussian white noise may be applied successfully to systems which evolve from a state of unstable equilibrium thereby developing non gaussian statistics the essential idea is the replacement of the full ensemble of realizations by subensembles according to the values taken by a small number of linear functionals of the noise the method is illustrated by applying it to two problems i strongly damped linear motion of a particle in a symmetric bistable potential ii the single mode laser in both cases it is found that low order moments can be calculated quite easily to an accuracy of a few per cent and that corrections with error estimates can be found by using the approximation to provide a control variate for numerical simulation

robustness of linear feedback control systems to unmodelled high frequency dynamics a basic issue in the control of linear time invariant plants is the effect of neglected high frequency dynamics on the performance in particular on the closed loop stability of the control system in this paper the robustness of various output feedback control designs based on a reduced order model with neglected high frequency dynamics is investigated using singular perturbation techniques a general robust design rule is to avoid static output feedback for systems with unmodelled high frequency dynamics from a frequency domain standpoint the robust design rule is to avoid closing high frequency plant loops by using strictly proper controllers or controllers with a low pass filtering property

dynamics of generalist predator in a stochastic environment modified ratio dependent holling tanner model with prey refuge is considered delays are taken in logistic growth of prey as feedback mechanism and gestation period of predator model system exhibits hopf bifurcation when the delay parameters cross their critical values stochastic analysis of the model system is carried out by incorporating gaussian white noise using fourier transform technique fluctuation and stability of the stochastic model is studied in this paper an attempt has been made to understand the dynamics of a prey predator system with multiple time delays where the predator population is regarded as a generalist type in this regard we consider a modified holling tanner prey predator system where a constant time delay is incorporated in the logistic growth of the prey to represent a delayed density dependent feedback mechanism and the second time delay is considered to account for the length of the gestation period of the predator predator s interference in prey predator relationship provides better descriptions of predator s feeding over a range of prey predator abundances so the predator s functional response is considered to be type ii ratio dependent and foraging efficiency of predator largely varies with the refuge strategy of prey population in accordance with previous studies it is observed that delay destabilizes the system in general and stability loss occurs via hopf bifurcation in particular we show that there exists critical values of the delay parameters below which the coexistence equilibrium is stable and above which it is unstable hopf bifurcation occurs when the delay parameters cross their critical values also environmental stochasticity in the form of gaussian white noise plays a significant role to describe the system and its values numerical computation is also performed to validate and visualize different theoretical results presented the analysis and results in this work are interesting both in mathematical and biological point of views

construction of a minimal mass blow up solution of the modified benjamin ono equation we construct a minimal mass blow up solution of the modified benjamin ono equation mbo begin aligned u t u 3 d 1 u x 0 end aligned mbo which is a standard mass critical dispersive model let q in h frac 1 2 q 0 be the unique ground state solution of d 1 q q q 3 constructed using variational arguments by weinstein commun part differ equations 12 1133 1173 1987a j differ equations 69 192 203 1987b and albert et al proc r soc lond a 453 1233 1260 1997 and whose uniqueness was recently proved by frank and lenzmann acta math 210 261 318 2013 we show the existence of a solution s of mbo satisfying vert s vert l 2 vert q vert l 2 and begin aligned s t frac 1 lambda frac 1 2 t q left frac cdot x t lambda t right rightarrow 0 quad text in h frac 1 2 mathbb r text as t downarrow 0 end aligned where begin aligned lambda t sim t quad x t sim ln t quad hbox and quad vert s t vert dot h frac 1 2 sim t frac 1 2 vert q vert dot h frac 1 2 quad hbox as t downarrow 0 end aligned this existence result is analogous to the one obtained by martel et al j eur math soc 17 1855 1925 2015 for the mass critical generalized korteweg de vries equation however in contrast with the gkdv equation for which the blow up problem is now well understood in a neighborhood of the ground state s is the first example of blow up solution for mbo the proof involves the construction of a blow up profile energy estimates as well as refined localization arguments developed in the context of benjamin ono type equations by kenig et al ann inst h poincare anal non lin 28 853 887 2011 due to the lack of information on the mbo flow around the ground state the energy estimates have to be considerably sharpened in the present paper

decay to the nonequilibrium steady state of the thermal diffusion in a tilted periodic potential abstract we investigate the phenomenon of asymptotic decay towards the nonequilibrium steady state of the thermal diffusion in the presence of a tilted periodic potential the parameter dependence of the decay rate is revealed by investigating the fokker planck fp equation in the low temperature case under the spatially periodic boundary condition pbc we apply the wkb method to the associated schrodinger equation while eigenvalues of the non hermitian fp operator are complex in general in a small tilting case the imaginary parts of the eigenvalues are almost vanishing then the schrodinger equation is solved with pbc the decay rate is analyzed in the context of quantum tunneling through a triple well effective periodic potential in a large tilting case the imaginary parts of the eigenvalues of the fp operator are crucial we apply the complex valued wkb method to the schrodinger equation with the absorbing boundary condition finding that the decay rate saturates and depends only on the temperature the potential periodicity and the viscous constant the intermediate tilting case is also explored the analytic results agree well with the numerical data for a wide range of tilting

blow up for the critical generalized korteweg de vries equation i dynamics near the soliton we consider the quintic generalized korteweg de vries equation gkdv u t u xx u 5 x 0 which is a canonical mass critical problem for initial data in h1 close to the soliton in earlier works on this problem finite or infinite time blow up was proved for non positive energy solutions and the solitary wave was shown to be the universal blow up profile see 16 26 and 20 for well localized initial data finite time blow up with an upper bound on blow up rate was obtained in 18