steady states of the one dimensional cahn hilliard equation the steady states of the cahn hilliard equation are studied as a function of interval length l and average mass m we count the number of nontrivial monotone increasing steady state solutions and demonstrate that if m lies within the spinodal region then for a e there is an even number of such solutions and for a e there is an odd number of such solutions if m lies within the metastable region then for a e l 0 there is an even number of solutions furthermore we prove that for all values of m there are no secondary bifurcations

stabilization of nonaffine systems a constructive method for polynomial systems this note focuses on the stabilization of nonaffine systems described by continuous nonlinear ordinary differential equations first conditions of stabilization using control lyapunov function give theoretical but nonconstructive and restrictive results second some particular nonaffine systems are considered this is polynomial system in the control variable of order two and three a method is also given for high order systems the main result is a method of construction of feedbacks for this class of polynomial systems in the end the polynomial example of the levitation system is stabilized using an extension of this method to discontinuous feedback

solitons in a nonlinear model medium the periodic stationary solutions of a model nonlinear evolution equation simulating the propagation of short wave perturbations in a relaxing medium are studied solutions expressed by a multiple valued function are shown to exist a method for determining the nonlinear interaction between solitary waves is suggested an example of a collision of solitons is given

stable adaptive control for a class of nonlinear systems using a modified lyapunov function investigates the adaptive control design for a class of nonlinear systems using lyapunov s stability theory the proposed method is developed based on a novel lyapunov function which removes the possible controller singularity problem in some of the existing adaptive control schemes using feedback linearization techniques the resulting closed loop system is proven to be globally stable and the output tracking error converges to an adjustable neighborhood of zero

adaptive output feedback control of nonlinear systems using neural networks a direct adaptive output feedback control design procedure is developed for highly uncertain nonlinear systems that does not rely on state estimation the approach is also applicable to systems of unknown but bounded dimension in particular we consider single input single output nonlinear systems whose output has known but otherwise arbitrary relative degree this includes systems with both parameter uncertainty and unmodeled dynamics the result is achieved by extending the universal function approximation property of linearly parameterized neural networks to model unknown system dynamics from input output data the network weight adaptation rule is derived from lyapunov stability analysis and guarantees that the adapted weight errors and the tracking error are bounded numerical simulations of an output feedback controlled van der pol oscillator coupled with a linear oscillator is used to illustrate the practical potential of the theoretical results

universal approximation using radial basis function networks there have been several recent studies concerning feedforward networks and the problem of approximating arbitrary functionals of a finite number of real variables some of these studies deal with cases in which the hidden layer nonlinearity is not a sigmoid this was motivated by successful applications of feedforward networks with nonsigmoidal hidden layer units this paper reports on a related study of radial basis function rbf networks and it is proved that rbf networks having one hidden layer are capable of universal approximation here the emphasis is on the case of typical rbf networks and the results show that a certain class of rbf networks with the same smoothing factor in each kernel node is broad enough for universal approximation

optimal decay rate of solutions for cahn hilliard equation with inertial term in multi dimensions abstract the cauchy problem of the cahn hilliard equation with inertial term is considered based on green s function method together with energy estimates we get the global in time existence and optimal decay rate of solutions furthermore the viscous case is investigated in the last section

linear stability of steady states for thin film and cahn hilliard type equations we study the linear stability of smooth steady states of the evolution equation h t f h xxx x g h h x x ah under both periodic and neumann boundary conditions if a 0 we assume f 1 in particular we consider positive periodic steady states of thin film equations where a 0 and f g might have degeneracies such as f 0 0 as well as singularities like g 0

energy levels of steady states for thin film type equations abstract we study the phase space of the evolution equationht f h hxxx x g h hx x by means of a dissipated energy here h x t 0 and at h 0 the coefficient functions f 0 and g can either degenerate to 0 or blow up to or tend to a nonzero constant we first show that all positive periodic steady states are saddles in the energy landscape with respect to zero mean perturbations if g f 0 or if the perturbations are allowed to have period longer than that of the steady state for power law coefficients f y yn and g y b ym for some b 0 we analytically determine the relative energy levels of distinct steady states for example with m n 1 2 and for suitable choices of the period and mean value we find three fundamentally different steady states the first is a constant steady state that is stable and is a local minimum of the energy the second is a positive periodic steady state that is linearly unstable and has higher energy than the constant steady state it is a saddle point for the energy the third is a periodic collection of droplet compactly supported steady states having lower energy than either the positive steady state or the constant one since the energy must decrease along every orbit these results significantly constrain the dynamics of the evolution equation

unexpectedly linear behavior for the cahn hilliard equation this paper gives theoretical results on spinodal decomposition for the cahn hillard equation we prove a mechanism which explains why most solutions for the cahn hilliard equation starting near a homogeneous equilibrium within the spinodal interval exhibit phase separation with a characteristic wavelength when exiting a ball of radius r namely most solutions are driven into a region of phase space in which linear behavior dominates for much longer than expected the cahn hilliard equation depends on a small parameter epsilon modeling the atomic scale interaction length we quantify the behavior of solutions as epsilon rightarrow 0 specifically we show that most solutions starting close to the homogeneous equilibrium remain close to the corresponding solution of the linearized equation with relative distance o epsilon 2 n 2 up to a ball of radius r in the h 2 omega norm where r is proportional to epsilon 1 rho n 4 as epsilon to 0 here n in 1 2 3 denotes the

adaptive neural control of mimo nonlinear state time varying delay systems with unknown dead zones and gain signs in this paper adaptive neural control is proposed for a class of uncertain multi input multi output mimo nonlinear state time varying delay systems in a triangular control structure with unknown nonlinear dead zones and gain signs the design is based on the principle of sliding mode control and the use of nussbaum type functions in solving the problem of the completely unknown control directions the unknown time varying delays are compensated for using appropriate lyapunov krasovskii functionals in the design the approach removes the assumption of linear functions outside the deadband as an added contribution by utilizing the integral lyapunov function and introducing an adaptive compensation term for the upper bound of the residual and optimal approximation error as well as the dead zone disturbance the closed loop control system is proved to be semi globally uniformly ultimately bounded simulation results demonstrate the effectiveness of the approach

wave breaking in a class of non local conservation laws abstract for models describing water waves constantin and escher 3 s works have long been considered as the cornerstone method for proving wave breaking phenomena their rigorous analytic proof shows that if the lowest slope of flows can be controlled by its highest slope initially then the wave breaking occur for the whitham type equation since this breakthrough there have been numerous refined wave breaking results established by generalizing the kernel which describes the dispersion relation of water waves in this work from a rich class of non local conservation laws a riccati type system that governs the flow s gradient is extracted and investigated the system s leading coefficient functions are allowed to change their values and signs over time as opposed to the ones in many of other previous works are fixed constants the blow up analysis is illustrated via the whitham type equation with nonlinear drift

nonlinear aspects of the cahn hilliard equation abstract this paper treats phase separation within the context of the phenomenological cahn hilliard equation c t m c f c k 2 c where c is the concentration m c is the mobility and f c is homogeneou s free energy which is assumed here to be a fourth degree polynomial natural boundary conditions are introduced the full set of equilibrium solutions is specified a comparison theorem for stability criteria which was postulated by langer is proved here within the framework of the natural boundary conditions energy methods are used to define and estimate the limit of monotonic global stability it is pointed out that within the parameter region where the uniform homogeneous state is the only equilibrium solution there may still exist some internal excitable region in which the homogeneous solution possesses growing fluctuations furthermore a periodic instability is shown to exist in the metastable region in addition to the well known nucleation instability

stable adaptive tracking of uncertain systems using nonlinearly parametrized on line approximators the design of stable adaptive neural controllers for uncertain nonlinear dynamical systems with unknown nonlinearities is considered the lyapunov synthesis approach is used to develop state feedback adaptive control schemes based on a general class of nonlinearly parametrized on line approximation models the key assumptions are that the system uncertainty satisfies a strict feedback condition and that the network reconstruction error and higher order terms of the on line approximator with respect to the network weights satisfy certain bounding conditions an adaptive bounding design is used to show that the overall neural control system guarantees semi global uniform ultimate boundedness within a neighbourhood of zero tracking error the theoretical results are illustrated through a simulation example

thresholds in three dimensional restricted euler poisson equations abstract this work provides a description of the critical threshold phenomenon in multi dimensional restricted euler poisson rep equations introduced in h liu e tadmor spectral dynamics of the velocity gradient field in restricted fluid flows comm math phys 228 2002 435 466 for three dimensional rep equations we identified both upper thresholds for the finite time blow up of solutions and subthresholds for the global existence of solutions with the thresholds depending on the relative size of the eigenvalues of the initial velocity gradient matrix and the initial density for the attractive forcing case these one sided threshold conditions of the initial configurations are optimal and the corresponding results also hold for arbitrary n dimensions n 3

adaptive dynamic inversion via time scale separation this paper presents a full state feedback adaptive dynamic inversion method for uncertain systems that depend nonlinearly upon the control input using a specialized set of basis functions that respect the monotonic property of the system nonlinearities with respect to control input a state predictor is defined for derivation of the adaptive laws the adaptive dynamic inversion controller is defined as a solution of a fast dynamical equation which achieves time scale separation between the state predictor and the controller dynamics lyapunov based adaptive laws ensure that the predictor tracks the state of the nonlinear system with bounded errors as a result the system state tracks the desired reference model with bounded errors benefits of the proposed design method are demonstrated using van der pol dynamics with nonlinear control input

spectral dynamics of the velocity gradient field in restricted flows we study the velocity gradients of the fundamental eulerian equation t u u u f which shows up in different contexts dictated by the different modeling of f s to this end we utilize a basic description for the spectral dynamics of u expressed in terms of the possibly complex eigenvalues u which are governed by the ricatti like equation t u 2 l fr we focus our investigation on four prototype models associated with different forcing f ranging from simple linear damping and viscous dusty medium models to the main thrust of the paper the restricted models of euler navier stokes equations and euler poisson equations in particular we address the question of the time regularity for these models that is whether they admit a finite time breakdown a global smooth solution or an intermediate scenario of critical threshold phenomena where global regularity depends on initial configurations using the spectral dynamics as our essential tool in these investigations we obtain a simple form of a critical threshold for the linear damping model and we identify the 2d vanishing viscosity limit for the viscous irrotational dusty medium model moreover for the n dimensional restricted euler equations we obtain n 2 1 global invariants interesting for their own sake which enable us to precisely characterize the local topology at breakdown time extending previous studies in the n 3 dimensional case finally as a fourth model we introduce the n dimensional restricted euler poisson rep system identifying a set of n 2 global invariants which in turn yield i sufficient conditions for finite time breakdown and ii characterization of a large class of 2 dimensional

nonlinear inversion based output tracking an inversion procedure is introduced for nonlinear systems which constructs a bounded input trajectory in the preimage of a desired output trajectory in the case of minimum phase systems the trajectory produced agrees with that generated by hirschorn s inverse dynamic system however the preimage trajectory is noncausal rather than unstable in the nonminimum phase case in addition the analysis leads to a simple geometric connection between the unstable manifold of the system zero dynamics and noncausality in the nonminimum phase case with the addition of stabilizing feedback to the preimage trajectory asymptotically exact output tracking is achieved tracking is demonstrated with a numerical example and compared to the well known byrnes isidori regulator rather than solving a partial differential equation to construct a regulator the inverse is calculated using a picard like interaction when preactuation is not possible noncausal inverse trajectories can be truncated resulting in the tracking error transients found in other control schemes

robust stability of interval time delay systems with delay dependence abstract this paper proposes a sufficient robust stability condition for interval time delay systems with delay dependence the properties of the comparison theorem and matrix measure are employed to investigate the problem the stability criteria are delay dependent and less conservative than delay independent stability criteria 5 6 12 16 and delay dependent stability criteria 1 14 15 17 when delay is small however the results of this paper indeed give us one more choice for the stability examination of the interval time delay systems simulation examples are given to demonstrate the application of our result

stable adaptive fuzzy sliding mode control of interconnected systems the problem of stable adaptive fuzzy control for a class of interconnected systems with unknown control gains is studied in this paper by using fuzzy modeling method a design scheme of an adaptive fuzzy controller is proposed the design is capable of incorporating linguistic and numerical information into controllers by theoretical analysis the closed loop adaptive fuzzy control system is proved to be globally stable in the sense that all signals involved are bounded with tracking errors converging to a neighborhood of zero

adaptive control of a class of nonlinear systems with fuzzy logic an adaptive tracking control architecture is proposed for a class of continuous time nonlinear dynamic systems for which an explicit linear parameterization of the uncertainty in the dynamics is either unknown or impossible the architecture employs fuzzy systems which are expressed as a series expansion of basis functions to adaptively compensate for the plant nonlinearities global asymptotic stability of the algorithm is established in the lyapunov sense with tracking errors converging to a neighborhood of zero simulation results for an unstable nonlinear plant are included to demonstrate that incorporating the linguistic fuzzy information from human experts results in superior tracking performance

singularities in a modified kuramoto sivashinsky equation describing interface motion for phase transition abstract phase transitions can be modeled by the motion of an interface between two locally stable phases a modified kuramoto sivashinsky equation h t 2 h 4 h 1 h 2 2 h 2 h xx h yy h xy 2 describes near planar interfaces which are marginally long wave unstable we study the question of finite time singularity formation in this equation in one and two space dimensions on a periodic domain such singularity formation does not occur in the kuramoto sivashinsky equation 0 for all 1 0 we provide sufficient conditions on the initial data and size of the domain to guarantee a finite time blow up in which a second derivative of h becomes unbounded using a bifurcation theory analysis we show a parallel between the stability of steady periodic solutions and the question of finite time blow up in one dimension finally we consider the local structure of the blow up in the one dimensional case via similarity solutions and numerical simulations that employ a dynamically adaptive self similar grid the simulations resolve the singularity to over 25 decades in h xx l and indicate that the singularities are all locally described by a unique self similar profile in h xx we discuss the relevance of these observations to the full intrinsic equations of motion and the associated physics

random attractors for locally monotone stochastic partial differential equations abstract we prove the existence of random dynamical systems and random attractors for a large class of locally monotone stochastic partial differential equations perturbed by additive levy noise the main result is applicable to various types of spde such as stochastic burgers type equations stochastic 2d navier stokes equations the stochastic 3d leray model stochastic power law fluids the stochastic ladyzhenskaya model stochastic cahn hilliard type equations stochastic kuramoto sivashinsky type equations stochastic generalized porous media equations and stochastic generalized p laplace equations

a novel error observer based adaptive output feedback approach for control of uncertain systems we develop an adaptive output feedback control methodology for nonaffine in control of uncertain systems having full relative degree given a smooth reference trajectory the objective is to design a controller that forces the system measurement to track it with bounded errors a neural network with linear parameters is introduced as an adaptive signal a simple linear observer is proposed to generate an error signal for the adaptive laws ultimate boundedness is shown through lyapunov s direct method simulations of a nonlinear second order system illustrate the theoretical results

a fourth order parabolic equation modeling epitaxial thin film growth abstract we study the continuum model for epitaxial thin film growth from phys d 132 1999 520 542 which is known to simulate experimentally observed dynamics very well we show existence uniqueness and regularity of solutions in an appropriate function space and we characterize the existence of nontrivial equilibria in terms of the size of the underlying domain in an investigation of asymptotical behavior we give a weak assumption under which the limit set of the dynamical system consists only of steady states in the one dimensional setting we can characterize the set of steady states and determine its unique asymptotically stable element the article closes with some illustrative numerical examples

pointwise estimates of solutions for the cahn hilliard equation with inertial term in multi dimensions abstract in this paper we study the pointwise estimates of solutions for the cahn hilliard equation with inertial term in multi dimensions by the green function method furthermore we obtain the l p convergence rate of the solution

thresholds for shock formation in traffic flow models with nonlocal concave convex flux abstract we identify sub thresholds for finite time shock formation in a class of non local conservation law with concavity changing flux from a class of non local conservation laws the riccati type ode system that governs a solution s gradient is obtained the changes in concavity of the flux function correspond to the sign changes in the leading coefficient functions of the ode system we identify the blow up condition of this structurally generalized riccati type ode the method is illustrated via the traffic flow models with nonlocal concave convex flux the techniques and ideas developed in this paper is applicable to a large class of non local conservation laws

large amplitude solutions to the sivashinsky and riley davis equations for directional solidification abstract directional solidification of a dilute binary alloy is often characterized by the appearance of deep root finger type interfaces as a model of this phenomenon we investigate the large amplitude behavior of long wave evolution equations previously derived for directional solidification in this paper we conduct a detailed asymptotic and numerical study of the periodic solutions of the sivashinsky and riley davis equations the sivashinsky equation which describes the limit of small segregation coefficient is shown to form deep root solutions which can be described in terms of elliptic functions however this equation is known to be ill posed in the sense that the solution branches are unstable and large classes of initial conditions lead to finite time blow up by considering asymptotically large surface energy riley and davis derived an evolution equation which may be considered a regularizing perturbation of the sivashinsky equation this equation has steady large amplitude solutions which are linearly stable in the absence of solutal segregation we may describe these solutions asymptotically in this limit amplitudes and wavelengths may become arbitrarily large a stability analysis suggests that with periodic boundary conditions the solution may coarsen until a single finger fills the domain if segregation effects are included both the amplitude and wavelength are bounded independent of domain size this wavelength bound gives the system a finger width selection mechanism associated with a tip splitting bifurcation that has also been observed in experiments

strong solutions for the generalized perona malik equation for image restoration abstract in this paper we establish the existence and uniqueness of strong solutions for the generalized perona malik equation of the fourth order for image restoration in dimension one

feedback linearization using neural networks abstract for a class of single input single output continuous time nonlinear systems a multilayer neural network based controller that feedback linearizes the system is presented control action is used to achieve tracking performance for a state feedback linearizable but unknown nonlinear system the control structure consists of a feedback linearization portion provided by two neural networks plus a robustifying portion that keeps the control magniture bounded a stability proof is given in the sense of lyapunov it is shown that all the signals in the closed loop system are uniformly ultimately bounded no off line learning phase is needed initialization of the network weights is straightforward

on the regularity of a generalized diffusion problem arising in population dynamics set in a cylindrical domain abstract in this paper we consider a generalized diffusion problem arising in population dynamics to this end we study a fourth order operational equation of elliptic type with various boundary conditions we show existence uniqueness and regularity of a classical solution on a cylindrical domain under some necessary and sufficient conditions on the data this elliptic problem is solved in l p a b x p 1 where a b r and x is a umd banach space our techniques use essentially the functional calculus and the semigroup theory

performance recovery of a class of uncertain non affine systems with unmodelled dynamics an indirect dynamic inversion method abstractthis paper presents an output feedback indirect dynamic inversion idi approach for a class of uncertain nonaffine systems with input unmodelled dynamics compared with previous approaches to achieve performance recovery the proposed method aims at dealing with a broader class of nonaffine in control systems with triangular structure an idi state feedback law is designed first in which less knowledge of the model plant is needed compared to earlier approximate dynamic inversion methods thus yielding more robust performance after that an extended high gain observer is designed to accomplish the task with output feedback finally we prove that the designed idi controller is equivalent to an adaptive proportional integral pi controller with respect to both time response equivalence and robustness equivalence the conclusion implies that for the studied strict feedback non affine systems with unmodelled dynamics there always exits a pi controller to stabilise the systems the effectiveness

symmetries and solutions to the thin film equations we study symmetries and solutions of the generalized fourth order nonlinear partial differential equations which arise from studies of thin liquid films it is shown that the equations admit extended scaling and rotation symmetries and a class of generalized conditional symmetries for certain coefficient functions exact solutions associated to the symmetries are obtained

a new global carleman estimate for cahn hilliard type equation and its applications abstract we establish a new carleman inequality for linear cahn hilliard type equation when the norms of right hand sides are in sobolev spaces of negative orders and apply it to obtain the unique continuation property ucp for cahn hilliard type equation with the coefficients of terms of lower order that are not regular and null controllability for semilinear cahn hilliard type equation whose semilinear term contains derivatives of first and second order of solutions

a universal adaptive stabilizer for a class of nonlinear systems abstract an adaptive stabilizer is described which is universal for a class of single input nonlinear systems modelled as a controlled differential inclusion on r n the stabilizer has a discontinous feedback structure and incorporates a gain of nussbaum type the analysis draws on an extension to differential inclusions of lasalle s principle for ordinary differential equations

deadzone compensation in motion control systems using neural networks a compensation scheme is presented for general nonlinear actuator deadzones of unknown width the compensator uses two neural networks nns one to estimate the unknown deadzone and another to provide adaptive compensation in the feedforward path the compensator nn has a special augmented form containing extra neurons whose activation functions provide a jump function basis set for approximating piecewise continuous functions rigorous proofs of closed loop stability for the deadzone compensator are provided and yield tuning algorithms for the weights of the two nns the technique provides a general procedure for using nns to determine the preinverse of an unknown right invertible function

global solutions of nonconcave hyperbolic conservation laws with relaxation arising from traffic flow abstract we establish global solutions of nonconcave hyperbolic equations with relaxation arising from traffic flow one of the characteristic fields of the system is neither linearly degenerate nor genuinely nonlinear furthermore there is no dissipative mechanism in the relaxation system characteristics travel no faster than traffic the global existence and uniqueness of the solution to the cauchy problem are established by means of a finite difference approximation to deal with the nonconcavity we use a modified argument of oleinik amer math soc translations 26 1963 95 it is also shown that the zero relaxation limit of the solutions exists and is the unique entropy solution of the equilibrium equation

the keller segel model with logistic sensitivity function and small diffusivity the keller segel model is the classical model for chemotaxis of cell populations it consists of a drift diffusion equation for the cell density coupled to an equation for the chemoattractant here a variant of this model is studied in one dimensional position space where the chemotactic drift is turned off for a limiting cell density by a logistic term and where the chemoattractant density solves an elliptic equation modeling a quasi stationary balance of reaction and diffusion with production of the chemoattractant by the cells the case of small cell diffusivity is studied by asymptotic and numerical methods on a time scale characteristic for the convective effects convergence of solutions to weak entropy solutions of the limiting nonlinear hyperbolic conservation law is proven numerical and analytic evidence indicates that solutions of this problem converge to irregular patterns of cell aggregates separated by entropic shocks from vacuum regions as time tends to infinity close to each of these p

minimum phase properties for input nonaffine nonlinear systems for input nonaffine nonlinear control systems the minimum phase property of the system in general depends on the control law switching or discontinuous controllers may offer advantages in this context in particular there may not exist a continuous control law which would keep the output identically equal to zero and for which the zero output constrained dynamics are locally stable whereas a discontinuous controller which achieves this exists for single input single output input nonaffine nonlinear systems we give sufficient conditions for existence and present a method for the design of discontinuous switching controllers which yield locally stable zero dynamics

technical communique a direct adaptive controller for dynamic systems with a class of nonlinear parameterizations in this note the adaptive control problem is considered for a class of nonlinearly parametrized systems by introducing a novel kind of lyapunov functions a direct adaptive controller is developed for achieving asymptotic tracking control the transient performance of the resulting closed loop system can be guaranteed by suitably choosing the lyapunov function to construct the controller the effectiveness of the proposed scheme is illustrated with two examples