stochastic output feedback control convex lifting approach abstract this paper presents a method to design output feedback control for discrete time linear systems affected by bounded additive state output disturbances and subject to chance constraints on the state and hard constraints on the control input this method relies on a so called convex lifting which is a nonnegative convex piecewise affine function equal to 0 over a given stochastic positively invariant set and strictly positive outside this set accordingly it is shown that this function is strictly decreasing along the closed loop dynamics outside this invariant set and convergent to 0 as time tends to infinity consequently the state is convergent to the given invariant set while the method only requires solving a linear program at each sampling instant

some remarks on the blow up rate for the 3d magnetic zakharov system we continue our study 8 on the cauchy problem for the three dimensional magnetic zakharov system zsm iet e ne i e b e 0 nt v vt n e 2 b i e e b 0 with initial data zsm 1 e 0 x e0 x n 0 x n0 x v 0 x v0 x let e n v c 0 t h1 r3 l2 r3 l2 r3 be a blow up solution to the cauchy problem zsm zsm 1 and let t be its blow up time then e n satisfies the space time integral estimate e 1 0t r3 n t x qdx 1q r3 e t x 2qdx 1q dt where 1e e 0 14 and q 32 1 e 32 2 the estimate e 1 implies that for a 1 supt 0 t t t ae n t lq r3 e t l2q r3 2 in particular if e n v is a radially symmetric solution to zsm zsm 1 then supt 0 t t t ae n t lq d e t l2q d 2 where a 0 13 and d x r3 x 1

stability and instability of periodic travelling wave solutions for the critical korteweg de vries and nonlinear schrodinger equations abstract in this paper we establish new results about the existence stability and instability of periodic travelling wave solutions related to the critical korteweg de vries equation u t 5 u 4 u x u x x x 0 and the critical nonlinear schrodinger equation i v t v x x v 4 v 0 the periodic travelling wave solutions obtained in our study tend to the classical solitary wave solutions in the infinite wavelength scenario the stability approach is based on the theory developed by angulo natali in j angulo f natali positivity properties of the fourier transform and the stability of periodic travelling wave solutions siam j math anal 40 2008 1123 1151 for positive periodic travelling wave solutions associated to dispersive evolution equations of korteweg de vries type the instability approach is based on an extension to the periodic setting of arguments found in bona souganidis strauss j l bona p e souganidis w a strauss stability and instability of solitary waves of korteweg de vries type proc roy soc london ser a 411 1987 395 412 regarding the critical schrodinger equation stability instability theories similar to the critical korteweg de vries equation are obtained by using the classical grillakis shatah strauss theory in m grillakis j shatah w strauss stability theory of solitary waves in the presence of symmetry ii j funct anal 94 1990 308 348 m grillakis j shatah w strauss stability theory of solitary waves in the presence of symmetry i j funct anal 74 1987 160 197 the arguments presented in this investigation have prospects for the study of the stability of periodic travelling wave solutions of other nonlinear evolution equations

complexity for extended dynamical systems we consider dynamical systems for which the spatial extension plays an important role for these systems the notions of attractor entropy and topological entropy per unit time and volume have been introduced previously in this paper we use the notion of kolmogorov complexity to introduce for extended dynamical systems a notion of complexity per unit time and volume which plays the same role as the metric entropy for classical dynamical systems we introduce this notion as an almost sure limit on orbits of the system moreover we prove a kind of variational principle for this complexity

localization and coherence in nonintegrable systems we study the irreversible dynamics of nonlinear nonintegrable hamiltonian oscillator chains approaching their statistical asymptotic states in systems constrained by more than one conserved quantity the partitioning of the conserved quantities leads naturally to localized and coherent structures if the phase space is compact the final equilibrium state is governed by entropy maximization and the coherent structures are stable lumps in systems where the phase space is not compact the coherent structures can be collapsed represented in phase space by a heteroclinic connection of some unstable saddle to infinity

stochastic mpc framework for controlling the average constraint violation this technical note considers linear discrete time systems with additive bounded disturbances subject to hard control input bounds and a stochastic constraint on the amount of state constraint violation averaged over time the amount of violations is quantified by a loss function and the averaging can be weighted corresponding to exponential forgetting of past violations the freedom in the choice of the loss function makes this formulation highly flexible for instance probabilistic constraints or integrated chance constraints can be enforced by an appropriate choice of the loss function for the type of constraint considered we develop a recursively feasible receding horizon control scheme exploiting the averaged over time nature by explicitly taking into account the amount of past constraint violations when determining the current control input this leads to a significant reduction in conservatism as a simple extension of the proposed approach we show how time varying state constraints can be handled within our framework the computational complexity online as well as offline is comparable to existing model predictive control schemes the effectiveness of the proposed methodology is demonstrated by means of a numerical example from building climate control

periodic nonlinear schrodinger equation and invariant measures in this paper we continue some investigations on the periodic nlseiu u iu xx u u p 2 p 6 started in lrs we prove that the equation is globally wellposed for a set of data of full normalized gibbs measrue e beta h phi hd phi x h phi tfrac 1 2 int left phi right 2 tfrac 1 p int left phi right p after suitablel 2 truncation the set and the measure are invariant under the flow the proof of a similar result for the kdv and modified kdv equations is outlined the main ingredients used are some estimates from b1 on periodic nls and kdv type equations

statistical mechanics of nonlinear wave equations 4 cubic schrodinger the cubic schrodinger equation is considered on the circle both in the de focussing and the focussing case the existence of the flow is proved together with the invariance of the appropriate gibbsian measure namely the petit canonical measure in the defocussing case and the micro canonical measure in the focussing case

invariant measures for nls in infinite volume we study gibbs measures and invariant dynamics for 1d defocusing nls iut uxx u u u p 2 0 when the period l for all p weak solutions are typically holder continuous on bounded space time regions if p 4 uniqueness and continuous dependence on the initial data may be shown

concentration of the invariant measures for the periodic zakharov kdv nls and gross piatevskii equations in 1d and 2d abstract this paper concerns gibbs measures for some nonlinear pde over the d torus t d the hamiltonian h t d u 2 t d u p has canonical equations with solutions in n u l 2 t d u 2 n this n is a parameter in quantum field theory analogous to the number of particles in a classical system for d 1 and 2 p 6 n supports the gibbs measure d u z 1 e h u x t d u x which is normalized and formally invariant under the flow generated by the pde the paper proves that n l 2 is a metric probability space of finite diameter that satisfies the logarithmic sobolev inequalities for the periodic kdv the focussing cubic nonlinear schrodinger equation and the periodic zakharov system for suitable subset of n a logarithmic sobolev inequality also holds in the critical case p 6 for d 2 the gross piatevskii equation has h t 2 u 2 t 2 v u 2 u 2 for a suitable bounded interaction potential v and the gibbs measure lies on a metric probability space h s which satisfies lsi in the above cases d is the limit in l 2 transportation distance of finite dimensional n n given by fourier sums

brief paper stochastic tube mpc with state estimation an output feedback model predictive control mpc strategy for linear systems with additive stochastic disturbances and probabilistic constraints is proposed given the probability distributions of the disturbance input the measurement noise and the initial state estimation error the distributions of future realizations of the constrained variables are predicted using the dynamics of the plant and a linear state estimator from these distributions a set of deterministic constraints is computed for the predictions of a nominal model the constraints are incorporated in a receding horizon optimization of an expected quadratic cost which is formulated as a quadratic program the constraints are constructed so as to provide a guarantee of recursive feasibility and the closed loop system is stable in a mean square sense all uncertainties in this paper are taken to be bounded in most control applications this gives a more realistic representation of process and measurement noise than the more traditional gaussian assumption

survey constrained model predictive control stability and optimality model predictive control is a form of control in which the current control action is obtained by solving at each sampling instant a finite horizon open loop optimal control problem using the current state of the plant as the initial state the optimization yields an optimal control sequence and the first control in this sequence is applied to the plant an important advantage of this type of control is its ability to cope with hard constraints on controls and states it has therefore been widely applied in petro chemical and related industries where satisfaction of constraints is particularly important because efficiency demands operating points on or close to the boundary of the set of admissible states and controls in this review we focus on model predictive control of constrained systems both linear and nonlinear and discuss only briefly model predictive control of unconstrained nonlinear and or time varying systems we concentrate our attention on research dealing with stability and optimality in these areas the subject has developed in our opinion to a stage where it has achieved sufficient maturity to warrant the active interest of researchers in nonlinear control we distill from an extensive literature essential principles that ensure stability and use these to present a concise characterization of most of the model predictive controllers that have been proposed in the literature in some cases the finite horizon optimal control problem solved on line is exactly equivalent to the same problem with an infinite horizon in other cases it is equivalent to a modified infinite horizon optimal control problem in both situations known advantages of infinite horizon optimal control accrue

invariant measures for some partial differential equations we construct invariant measures for hamiltonian systems such as the nonlinear schrrdinger equation or the wave equation in order to prove poisson s recurrence the particular case of schemes finite dimensional spaces is also treated in order to explain the recurrence phenomenon which is observed during numerical simulations

invariant measure for the schrodinger equation on the real line abstract in this paper we build a gibbs measure for the cubic defocusing schrodinger equation on the real line with a decreasing interaction potential in the sense that the non linearity u 2 u is multiplied by a function which we assume integrable and smooth enough we prove that this equation is globally well posed in the support of this measure and that the measure is invariant under the flow of the equation what is more the support of the measure the set of initial data is disjoint from l 2

a complexity approach to the soliton resolution conjecture the soliton resolution conjecture is one of the most interesting open problems in the theory of nonlinear dispersive equations roughly speaking it asserts that a solution with generic initial condition converges to a finite number of solitons plus a radiative term in this paper we use the complexity of a finite object a notion introduced in algorithmic information theory to show that the soliton resolution conjecture is equivalent to the analogous of the second law of thermodynamics for the complexity of a solution of a dispersive equation

chaotic and homoclinic behavior for numerical discretizations of the nonlinear schro dinger equation abstract certain conservative discretizations of the nls can produce irregular behavior we consider the diagonal discretization as a conservative perturbation of the integrable discretization and study the homoclinic crossings in its nonlinear spectrum we find that irregularity sets in when two homoclinic structures are present and in this case many and continual homoclinic crossings occur throughout the irregular time series we indicate a melnikov analysis to study the consequences of this homoclinic behavior

an additional gibbs state for the cubic schrodinger equation on the circle an invariant gibbs state for the nonlinear schrodinger equation on the circle was constructed by bourgain 2 3 and mckean 6 out of the basic hamiltonian using a trigonometric cutoff the cubic nonlinear schrodinger equation is a completely integrable system having an infinite number of additional integrals of motion in this paper we construct the second invariant gibbs state from one of these additional integrals for the cubic nls on the circle this additional gibbs state is singular with respect to the gibbs state previously constructed from the basic hamiltonian our approach employs the ablowitz ladik system a completely integrable discretization of the cubic schrodinger equation 2001 john wiley sons inc

linear systems with state and control constraints the theory and application of maximal output admissible sets the initial state of an unforced linear system is output admissible with respect to a constraint set y if the resulting output function satisfies the pointwise in time condition y t in y t or 0 the set of all possible such initial conditions is the maximal output admissible set o sub infinity the properties of o sub infinity and its characterization are investigated in the discrete time case it is generally possible to represent o sub infinity or a close approximation of it by a finite number of functional inequalities practical algorithms for generating the functions are described in the continuous time case simple representations of the maximal output admissible set are not available however it is shown that the discrete time results may be used to obtain approximate representations

unique ergodicity for a class of stochastic hyperbolic equations with additive space time white noise in this paper we consider a certain class of second order nonlinear pdes with damping and space time white noise forcing posed on the d dimensional torus this class includes the wave equation for d 1 and the beam equation for d le 3 we show that the gibbs measure is the unique invariant measure for this system since the flow does not satisfy the strong feller property we introduce a new technique for showing unique ergodicity this approach may be also useful in situations in which finite time blowup is possible

a microscopic derivation of time dependent correlation functions of the 1d cubic nonlinear schrodinger equation abstract we give a microscopic derivation of time dependent correlation functions of the 1d cubic nonlinear schrodinger equation nls from many body quantum theory the starting point of our proof is 11 on the time independent problem and 16 on the corresponding problem on a finite lattice an important new obstacle in our analysis is the need to work with a cutoff in the number of particles which breaks the gaussian structure of the free quantum field and prevents the use of the wick theorem we overcome it by means of complex analytic methods our methods apply to the nonlocal nls with bounded convolution potential in the periodic setting we also consider the local nls arising from short range interactions in the many body setting to that end we need the dispersion of the nls in the form of periodic strichartz estimates in x s b spaces

nonquadratic stochastic model predictive control a tractable approach this paper deals with the finite horizon stochastic optimal control problem with the expectation of the p norm as the objective function and jointly gaussian although not necessarily independent additive disturbance process we develop an approximation strategy that solves the problem in a certain class of nonlinear feedback policies while ensuring satisfaction of hard input constraints a bound on suboptimality of the proposed strategy in this class of nonlinear feedback policies is given for the special case of p 1 we also develop a recursively feasible receding horizon policy with respect to state chance constraints and or hard control input constraints in the presence of bounded disturbances the performance of the proposed policies is examined in two numerical examples

distributed stochastic mpc of linear systems with additive uncertainty and coupled probabilistic constraints this technical note develops a new form of distributed stochastic model predictive control dsmpc algorithm for a group of linear stochastic subsystems subject to additive uncertainty and coupled probabilistic constraints we provide an appropriate way to design the dsmpc algorithm by extending a centralized smpc csmpc scheme to achieve the satisfaction of coupled probabilistic constraints in a distributed manner only one subsystem is permitted to optimize at each time step in addition by making explicit use of the probabilistic distribution of the uncertainties probabilistic constraints are converted into a set of deterministic constraints for the predictions of nominal models the distributed controller can achieve recursive feasibility and ensure closed loop stability for any choice of update sequence numerical examples illustrate the efficacy of the algorithm

statistical mechanics of the nonlinear schrodinger equation we investigate the statistical mechanics of a complex fieldo whose dynamics is governed by the nonlinear schrodinger equation such fields describe in suitable idealizations langmuir waves in a plasma a propagating laser field in a nonlinear medium and other phenomena their hamiltonian h phi int omega frac 1 2 nabla phi 2 1 p phi p dx is unbounded below and the system will under certain conditions develop self focusing singularities in a finite time we show that when is the circle and thel 2 norm of the field which is conserved by the dynamics is bounded byn the gibbs measure obtained is absolutely continuous with respect to wiener measure and normalizable if and only ifp andn are such that classical solutions exist for all time no collapse of the solitons this measure is essentially the same as that of a one dimensional version of the more realisitc zakharov model of coupled langmuir and ion acoustic waves in a plasma we also obtain some properties of the gibbs state by both analytic and numerical methods asn and the temperature are varied

bounded subquotients of pseudodifferential operator modules recently there have been several papers on the action of the virasoro lie algebra on the projective decompositions of the modules of pseudodifferential operators on the circle we use their results to prove that a wide class of the uniserial completely indecomposable bounded modules of the virasoro lie algebra may be realized as subquotients of such modules of pseudodifferential operators this gives easy proofs of the existence of many previously known uniserial modules and moreover yields some hitherto undiscovered

continuity of the flow of kdv with regard to the wasserstein metrics and application to an invariant measure abstract in this paper we prove the continuity of the flow of kdv on spaces of probability measures with respect to a combination of wasserstein distances on h s s 0 and l 2 we are motivated by the existence of an invariant measure belonging to the spaces onto which these distances are defined

statistical mechanics of the periodic benjamin ono equation the periodic benjamin ono equation is an autonomous hamiltonian system with a gibbs measure on l2 t this paper shows that the gibbs measures on bounded balls of l2 satisfy some logarithmic sobolev inequalities the space of n soliton solutions of the periodic benjamin ono equation as discovered by case is a hamiltonian system with an invariant gibbs measure as n these gibbs measures exhibit a concentration of measure phenomenon case introduced soliton solutions that are parameterized by atomic measures in the complex plane the limiting distributions of these measures give the density of a compressible gas that satisfies the isentropic euler equations the periodic benjamin ono equation is an autonomous hamiltonian system with a gibbs measure on l2 t this paper shows that the gibbs measures on bounded balls of l2 satisfy some logarithmic sobolev inequalities the space of n soliton solutions of the periodic benjamin ono equation as discovered by case is a hamiltonian system with an invariant gibbs measure as n these gibbs measures exhibit a concentration of measure phenomenon case introduced soliton solutions that are parameterized by atomic measures in the complex plane the limiting distributions of these measures give the density of a compressible gas that satisfies the isentropic euler equations

an approach to output feedback mpc of stochastic linear discrete time systems in this paper we propose an output feedback model predictive control mpc algorithm for linear discrete time systems affected by a possibly unbounded additive noise and subject to probabilistic constraints in case the noise distribution is unknown the probabilistic constraints on the input and state variables are reformulated by means of the chebyshev cantelli inequality the recursive feasibility is guaranteed the convergence of the state to a suitable neighbor of the origin is proved under mild assumptions and the implementation issues are thoroughly addressed two examples are discussed in detail with the aim of providing an insight into the performance achievable by the proposed control scheme

the scenario approach for stochastic model predictive control with bounds on closed loop constraint violations many practical applications in control require that constraints on the inputs and states of the system are respected while some performance criterion is optimized in the presence of model uncertainties or disturbances it is often sufficient to satisfy the state constraints for at least a prescribed share of the time such as in building climate control or load mitigation for wind turbines for such systems this paper presents a new method of scenario based model predictive control scmpc the basic idea is to optimize the control inputs over a finite horizon subject to robust constraint satisfaction under a finite number of random scenarios of the uncertainty and or disturbances previous scmpc approaches have suffered from a substantial gap between the rate of constraint violations specified in the optimal control problem and that actually observed in closed loop operation of the controlled system this paper identifies the two theoretical explanations for this gap first accounting for the special structure of the optimal control problem leads to a substantial reduction of the problem dimension second the probabilistic constraints have to be interpreted as average in time rather than pointwise in time based on these insights a novel scmpc method can be devised for general linear systems with additive and multiplicative disturbances for which the number of scenarios is significantly reduced the presented method retains the essential advantages of the general scmpc approach namely a low computational complexity and the ability to handle arbitrary probability distributions moreover the computational complexity can be adjusted by a sample and remove strategy

on the volume limit of the focusing cubic schrodinger equation we revisit the question of an invariant measure for the focusing cubic schrodinger equation on the line for the periodic problem the appropriate ensemble was introduced by lebowitz rose and speer 3 and proved to be invariant under the flow by mckean 5 these parties and others have also discussed the thermodynamic limit though without consensus simulations carried out in 3 indicated the possibility of a phase transition similar experiments in 1 appeared to contradict that interpretation later a proof was put forward in 6 that the full thermodynamic limit did not exist suggesting a possible explanation for the disparate conclusions drawn from the numerics unfortunately the latter contains an error the main result here is that in the infinite volume the ensemble collapses onto the unit mass on the trivial path along the way sharp estimates for the partition function are established 2002 wiley periodicals inc

annihilators of tensor density modules abstract we describe the two sided ideals in the universal enveloping algebras of the lie algebras of vector fields on the line and the circle which annihilate the tensor density modules both of these lie algebras contain the projective subalgebra a copy of sl 2 the restrictions of the tensor density modules to this subalgebra are duals of verma modules of sl 2 for vec r and principal series modules of sl 2 for vec s 1 thus our results are related to the well known theorem of duflo describing the annihilating ideals of verma modules of reductive lie algebras we find that in general the annihilator of a tensor density module of vec r or vec s 1 is generated by the duflo generator of its annihilator over sl 2 the casimir operator minus a scalar together with one other generator a cubic element of u vec r not contained in u sl 2

stochastic tubes in model predictive control with probabilistic constraints stochastic model predictive control mpc strategies can provide guarantees of stability and constraint satisfaction but their online computation can be formidable this difficulty is avoided in the current technical note through the use of tubes of fixed cross section and variable scaling a model describing the evolution of predicted tube scalings facilitates the computation of stochastic tubes furthermore this procedure can be performed offline the resulting mpc scheme has a low online computational load even for long prediction horizons thus allowing for performance improvements the efficacy of the approach is illustrated by numerical examples

probabilistic tubes in linear stochastic model predictive control this paper considers constrained control of linear systems with additive and multiplicative stochastic uncertainty and linear input state constraints both hard and soft constraints are considered and bounds are imposed on the probability of soft constraint violation assuming the plant parameters to be finitely supported a method of constraint handling is proposed in which a sequence of tubes corresponding to a sequence of confidence levels on the predicted future plant state is constructed online around nominal state trajectories a set of linear constraints is derived by imposing bounds on the probability of constraint violation at each point on an infinite prediction horizon through constraints on one step ahead predictions a guarantee of the recursive feasibility of the online optimization ensures that the closed loop system trajectories satisfy both the hard and probabilistic soft constraints the approach is illustrated by a numerical example

brief paper explicit use of probabilistic distributions in linear predictive control the guarantee of feasibility given feasibility at initial time is an issue that has been overlooked by many of the recent papers on stochastic model predictive control effective solutions have recently been proposed but these carry considerable online computational load and a degree of conservativism for the case that the elements of the random additive disturbance vector are independent the current paper ensures that probabilistic constraints are met and that a quadratic stability condition is satisfied a numerical example illustrates the efficacy of the proposed algorithm which achieves tight satisfaction of constraints and thereby attains near optimal performance

brief a probabilistically constrained model predictive controller we propose a novel control algorithm probabilistically constrained predictive control to deal with the uncertainties of system disturbances the output is to be controlled in the constrained range with a desired probability under the assumption of a linear system the formulated joint probabilistically constrained problem is convex thus it can be solved with a nonlinear programming solver the probabilities and gradients of the constraints composed of disturbance sequences with multivariate normal distribution are computed using an efficient simulation approach the results of a test problem show the effectiveness of the proposed algorithm

convergence of optimal prediction for nonlinear hamiltonian systems optimal prediction is a computational method for systems that cannot be properly resolved in which the unresolved variables are viewed as random this paper presents a first analysis of optimal prediction as a numerical method we prove the convergence of the scheme for a class of equations of schrodinger type and derive error bounds for the mean error between the optimal prediction solution and the set of exact solutions with random initial data it is shown that optimal prediction is the scheme that minimizes the mean truncation error

probabilistic methods for discrete nonlinear schrodinger equations we show that the thermodynamics of the focusing cubic discrete nonlinear schrodinger equation are exactly solvable in dimension 3 and higher a number of explicit formulas are derived

statistical mechanics of the 2 dimensional focusing nonlinear schrodinger equation we study a natural construction of an invariant measure for the 2 dimensional periodic focusing nonlinear schrodinger equation with the critical cubic nonlinearity we find that a phase transition occurs as the coupling constant defining the strength of the nonlinearity is increased but that the natural construction successful for the 1 dimensional case and for the 2 dimensional defocusing case cannot produce an invariant measure our methods rely on an analysis of a statistical mechanical model closely related to the spherical model of berlin and kac

stability of solitary waves in finite ablowitz ladik lattices we discuss the effects of different types of boundary conditions b c on the linear stability of a solitary wave in a finite length dynamical lattice described by the ablowitz ladik al model types of b c considered are fixed dirichlet no flux neumann and free as well as periodic b c the behaviour of eigenvalues around a stationary nonlinear wave consistent with several types of b c is studied analytically and numerically the translational eigenvalues are found to move away from the origin it is shown that for b c of the fixed type these eigenvalues bifurcate into a neutrally stable oscillatory kind while the no flux and free b c lead to an exponentially weak instability the rotational eigenmodes those related to the gauge invariance of the lattice strictly remain at the origin as gauge invariance is not broken by the linear homogeneous b c these numerical results are compared to the analytical and semi analytical predictions obtained within the approximation of the images generated by the solitary wave beyond the lattice edges a potential effect of continuous spectrum band edge bifurcations on stability is also evaluated with the conclusion that that these bifurcations cannot destabilize the soliton for periodic b c the translational invariance is preserved and so is the lax pair structure sustaining the integrable nature of the problem the al model was chosen so that the effects of boundary conditions are monitored in the absence of non integrable discreteness finally the possibility of the generation of multipulse configurations due to the interplay of finiteness effects with the exponential tail tail interaction of the pulses is examined

random data cauchy theory for supercritical wave equations ii a global existence result we prove that the subquartic wave equation on the three dimensional ball with dirichlet boundary conditions admits global strong solutions for a large set of random supercritical initial data in bigcap s 1 2 h s theta times h s 1 theta we obtain this result as a consequence of a general random data cauchy theory for supercritical wave equations developed in our previous work 6 and invariant measure considerations inspired by earlier works by bourgain 2 3 on the non linear schrodinger equation which allow us to obtain also precise large time dynamical informations on our solutions

a mean field statistical theory for the nonlinear schrodinger equation abstract a statistical model of self organization in a generic class of one dimensional nonlinear schrodinger nls equations on a bounded interval is developed the main prediction of this model is that the statistically preferred state for such equations consists of a deterministic coherent structure coupled with fine scale random fluctuations or radiation the model is derived from equilibrium statistical mechanics by using a mean field approximation of the conserved hamiltonian and particle number l2 norm squared for finite dimensional spectral truncations of the nls dynamics the continuum limits of these approximated statistical equilibrium ensembles on finite dimensional phase spaces are analyzed holding the energy and particle number at fixed finite values the analysis shows that the coherent structure minimizes total energy for a given value of particle number and hence is a solution to the nls ground state equation and that the remaining energy resides in gaussian fluctuations equipartitioned over wave numbers some results of direct numerical integration of the nls equation are included to validate empirically these properties of the most probable states for the statistical model moreover a theoretical justification of the mean field approximation is given in which the approximate ensembles are shown to concentrate on the associated microcanonical ensemble in the continuum limit

random data cauchy theory for supercritical wave equations i local theory we study the local existence of strong solutions for the cubic nonlinear wave equation with data in hs m s 1 2 where m is a three dimensional compact riemannian manifold this problem is supercritical and can be shown to be strongly ill posed in the hadamard sense however after a suitable randomization we are able to construct local strong solution for a large set of initial data in hs m where s 1 4 in the case of a boundary less manifold and s 8 21 in the case of a manifold with boundary

coupled map lattices some topological and ergodic properties abstract lattice dynamical systems ldss form the class of extended systems that is the intermediate one between partial differential equations pdes and cellular automata the most popular class of ldss is formed by coupled map lattices cmls while being introduced rather recently ldss allowed already to clarify and to define exactly some notions that form the basis of the modern phenomenological theory of spatio temporal dynamics and to obtain some new and even rigorous results on space time chaos intermittency and pattern formation we discuss the rigorous results that were obtained in this area but eventually give also some applications

invariant measures and the soliton resolution conjecture the soliton resolution conjecture for the focusing nonlinear schrodinger equation nls is the vaguely worded claim that a global solution of the nls for generic initial data will eventually resolve into a radiation component that disperses like a linear solution plus a localized component that behaves like a soliton or multisoliton solution considered to be one of the fundamental open problems in the area of nonlinear dispersive equations this conjecture has eluded a proof or even a precise formulation to date this paper proves a statistical version of this conjecture at mass subcritical nonlinearity in the following sense the uniform probability distribution on the set of all functions with a given mass and energy if such a thing existed would be a natural invariant measure for the nls flow and would reflect the long term behavior for generic initial data with that mass and energy unfortunately such a probability measure does not exist we circumvent this problem by constructing a sequence of discrete measures that in principle approximate this fictitious probability distribution as the grid size goes to 0 we then show that a continuum limit of this sequence of probability measures does exist in a certain sense and in agreement with the soliton resolution conjecture the limit measure concentrates on the unique ground state soliton combining this with results from ergodic theory we present a tentative formulation and proof of the soliton resolution conjecture in the discrete setting the above results following in the footsteps of a program of studying the long term behavior of nonlinear dispersive equations through their natural invariant measures initiated by lebowitz rose and speer and carried forward by bourgain mckean tzvetkov oh and others are proved using a combination of techniques from large deviations pdes harmonic analysis and bare hands probability theory it is valid in any dimension 2014 wiley periodicals inc

breakdown of weak turbulence and nonlinear wave condensation abstract the formation of a large scale coherent structure a condensate as a result of the long time evolution of the initial value problem of a classical partial differential nonlinear wave equation is considered we consider the nonintegrable and unforced defocusing nonlinear schrodinger nls equation as a representative model in spite of the formal reversibility of the nls equation the nonlinear wave exhibits an irreversible evolution towards a thermodynamic equilibrium state the equilibrium state is characterized by a homogeneous solution condensate with small scale fluctuations superposed uncondensed particles which store the information necessary for time reversal we analyze the evolution of the cumulants of the random wave as originally formulated by d j benney and p g saffman d j benney p g saffman proc roy soc london a 289 1966 301 and a c newell a c newell rev geophys 6 1968 1 this allows us to provide a self consistent weak turbulence theory of the condensation process in which the nonequilibrium formation of the condensate is a natural consequence of the spontaneous regeneration of a non vanishing first order cumulant in the hierarchy of the cumulants equations more precisely we show that in the presence of a small condensate amplitude all relevant statistical information is contained in the off diagonal second order cumulant as described by the usual weak turbulence theory conversely in the presence of a high amplitude condensate the diagonal second order cumulants no longer vanish in the long time limit which signals a breakdown of the weak turbulence theory however we show that an asymptotic closure of the hierarchy of the cumulants equations is still possible provided one considers the bogoliubov s basis rather than the standard fourier s free particle basis the nonequilibrium dynamics turns out to be governed by the bogoliubov s off diagonal second order cumulant while the corresponding diagonal cumulants as well as the higher order cumulants are shown to vanish asymptotically the numerical discretization of the nls equation implicitly introduces an ultraviolet frequency cut off the simulations are in quantitative agreement with the weak turbulence theory without adjustable parameters despite the fact that the theory is expected to breakdown nearby the transition to condensation the fraction of condensed particles vs energy is characterized by two distinct regimes for small energies h h c the bogoliubov s regime is established whereas for h h c the small amplitude condensate regime is described by the weak turbulence theory in both regimes we derive coupled kinetic equations that describe the coupled evolution of the condensate amplitude and the incoherent field component the influence of finite size effects and of the dimensionality of the system are also considered it is shown that beyond the thermodynamic limit wave condensation is reestablished in two spatial dimensions in complete analogy with uniform and ideal 2d bose gases

almost sure local well posedness and scattering for the 4d cubic nonlinear schrodinger equation abstract we consider the cauchy problem for the defocusing cubic nonlinear schrodinger equation in four space dimensions and establish almost sure local well posedness and conditional almost sure scattering for random initial data in h x s r 4 with 1 3 s 1 the main ingredient in the proofs is the introduction of a functional framework for the study of the associated forced cubic nonlinear schrodinger equation which is inspired by certain function spaces used in the study of the schrodinger maps problem and is based on strichartz spaces as well as variants of local smoothing inhomogeneous local smoothing and maximal function spaces additionally we prove an almost sure scattering result for randomized radially symmetric initial data in h x s r 4 with 1 2 s 1

a priori estimates and weak solutions for the derivative nonlinear schrodinger equation on torus below h 1 2 abstract we propose a priori estimates for a weak solution to the derivative nonlinear schrodinger equation dnls on torus with small l 2 norm datum in low regularity sobolev spaces these estimates allow us to show the existence of solutions in h s t with some s 1 2 in a relatively weak sense furthermore we make some remarks on the error estimates arising from the finite dimensional approximation solutions of dnls using the fourier lesbesgue type as auxiliary spaces despite the fact that nahmod oh rey bellet and staffilani 12 have already seen them

morse and melnikov functions for nls pde s the theory of the focusing nls equation under periodic boundary conditions together with the floquet spectral theory of its associated zakharov shabat linear operator hat l is developed in sufficient detail for later use in studies of perturbations of the nls equation counting lemmas for the non selfadjoint operator hat l are established which control its spectrum and show that all of its eccentricities are finite in number and must reside within a finite discd in the complex eigenvalue plane the radius of the discd is controlled by theh 1 norm of the potential overrightarrow q for this integrable nls hamiltonian system unstable tori are identified and backlund transformations are then used to construct global representations of their stable and unstable manifolds whiskered tori for the nls pde the floquet discriminant delta lambda overrightarrow q is used to introduce a natural sequence of nls constants of motion f j overrightarrow q equiv delta lambda lambda j c overrightarrow q overrightarrow q where denotes thej th critical point of the floquet discriminant a taylor series expansion of the constants f j overrightarrow q with explicit representations of the first and second variations is then used to study neighborhoods of the whiskered tori in particular critical tori with hyperbolic structure are identified through the first and second variations of f j overrightarrow q which themselves are expressed in terms of quadratic products of eigenfunctions of hat l the second variation permits identification within the disc d of important bifurcations in the spectral configurations of the operator hat l the constant f j overrightarrow q as the height of the floquet discriminant over the critical point admits a natural interpretation as a morse function for nls isospectral level sets this morse interpretation is studied in some detail it is valid globally for the infinite tail f j overrightarrow q j n which is associated with critical points outside the discd within this disc the interpretation is only valid locally with the same obstruction to its global validity as to a global ordering of the spectrum nevertheless this local morse theory together with the backlund representations of the whiskered tori produces extremely clear pictures of the stratification of nls invariant sets near these whiskered tori pictures which are useful in the study of perturbations of nls finally a natural connection is noted between the constants f j overrightarrow q of the integrable theory and melnikov functions for the theory of perturbations of the nls equation this connection generates a simple but general representations of the melnikov functions

a statistical approach to the asymptotic behavior of a class of generalized nonlinear schrodinger equations a statistical relaxation phenomenon is studied for a general class of dispersive wave equations of nonlinear schrodinger type which govern non integrable non singular dynamics in a bounded domain the solutions of these equations have been shown numerically to tend in the long time limit toward a gibbsian statistical equilibrium state consisting of a ground state solitary wave on the large scales and gaussian fluctuations on the small scales the main result of the paper is a large deviation principle that expresses this concentration phenomenon precisely in the relevant continuum limit the large deviation principle pertains to a process governed by a gibbs ensemble that is canonical in energy and microcanonical in particle number some supporting monte carlo simulations of these ensembles are also included to show the dependence of the concentration phenomenon on the properties of the dispersive wave equation especially the high frequency growth of the dispersion relation the large deviation principle for the process governed by the gibbs ensemble is based on a large deviation principle for gaussian processes for which two independent proofs are given

nonequilibrium statistical behavior of nonlinear schrodinger equations this paper explores the equilibrium and near equilibrium statistical behavior of nonlinear schrodinger equations through numerical simulations the investigation centers on a class of finite dimensional hamiltonian systems obtained by discretizing nls equations defined on a bounded interval with saturated focusing nonlinearities these nonlinear systems govern a wave dynamics that is nonintegrable and nonsingular and for which typical solutions are composed of large scale coherent structures interacting within a background of small scale fluctuations the equilibrium statistics of such a system is most naturally defined by a gibbs distribution that is canonical in energy and microcanonical in particle number by sampling these equilibrium distributions using a metropolis type algorithm the typical excitations of the system are determined over the full range of system parameters nonequilibrium relaxations from initial perturbations of these equilibrium distributions are then computed by means of linear response theory the computations reveal a surprising dependence of the qualitative properties of relaxation toward equilibrium on the temperature and the nonlinearity on the one hand a simple exponential decay to equilibrium is observed for sufficiently high temperature or sufficiently strong nonlinearity for low temperature and moderate nonlinearity on the other hand the system rapidly adjusts to a metaequilibrium state that equilibrates extremely slowly this two stage thermalization may be related to quasiperiodicity in the underlying dynamics indeed the metaequilibria are found to occur in a range of system parameters for which approximately quasiperiodic solutions are prevalent in equilibrium

blow up of solutions to the nonlinear schrodinger equations on manifolds in this paper we partially settle down the long standing open problem of the finite time blow up property about the nonlinear schrodinger equations on some riemannian manifolds such as the standard 2 sphere s2 and the hyperbolic 2 space h2 1 using the similar idea we establish such blow up results on higher dimensional standard sphere and hyperbolic n space extensions to n dimensional riemannian warped product manifolds with n 2 are also given

efficient robust predictive control predictive constrained control of time varying and or uncertain linear systems has been effected through the use of ellipsoidal invariant sets kothare et al 1996 linear matrix inequalities lmis have been used to design a state dependent state feedback law that maintains the state vector inside invariant feasible sets for the purposes of prediction however at each time instant the state feedback law is assumed constant in addition due to the large number of lmis involved online computation becomes intractable for anything other than small dimensional systems here we propose an approach that deploys a fixed state feedback law but introduces extra degrees of freedom through the use of perturbations on the fixed state feedback law the problem is so formulated that all demanding computations can be performed offline leaving only a simple optimization problem to be solved online over and above the very significant reduction in computational cost the extra degrees of freedom allow for better performance and wider applicability

invariant gibbs measure evolution for the radial nonlinear wave equation on the 3d ball abstract we establish new global well posedness results along gibbs measure evolution for the nonlinear wave equation posed on the unit ball in r 3 via two distinct approaches the first approach invokes the method established in the works bourgain 1994 1996 3 4 5 based on a contraction mapping principle and applies to a certain range of nonlinearities the second approach allows to cover the full range of nonlinearities admissible to treatment by gibbs measure working instead with a delicate analysis of convergence properties of solutions the method of the second approach is quite general and we shall give applications to the nonlinear schrodinger equation on the unit ball in subsequent works bourgain and bulut 2013 10 11

invariant measures for the periodic derivative nonlinear schrodinger equation we construct invariant measures associated to the integrals of motion of the periodic derivative nonlinear schrodinger equation dnls for small data in l 2 and we show these measures to be absolutely continuous with respect to the gaussian measure the key ingredient of the proof is the analysis of the gauge group of transformations associated to dnls as an intermediate step for our main result we prove quasi invariance with respect to the gauge maps of the gaussian measure on l 2 with covariance mathbb i delta k 1 for any k geqslant 2

cooperative distributed stochastic mpc for systems with state estimation and coupled probabilistic constraints a cooperative distributed stochastic model predictive control cdsmpc algorithm is given for multiple dynamically decoupled subsystems with additive stochastic disturbances and coupled probabilistic constraints for which states are not measurable cooperation between subsystems is promoted by a scheme in which a local subsystem designs hypothetical plans for others in some cooperating set and considers the weighted costs of these subsystems in its objective to achieve satisfaction of coupled probabilistic constraints in a distributed way only one subsystem is permitted to optimize at each time step in addition by using a lifting technique and the probabilistic information on additive disturbances measurement noise and the state estimation error a set of deterministic constraints is constructed for the predictions of nominal models recursive feasibility with respect to both local and coupled probabilistic constraints is guaranteed and stability for any choice of update sequence and any structure of cooperation is ensured numerical examples illustrate the efficacy of the proposed algorithm

brownian motion with restoring drift the petit and micro canonical ensembles letf q be odd and positive near then the non linear wave equation 2q t2 2q x2 f q 0 considered on the circle 0 x l can be written in hamiltonian formq h p p h q with p q cdot and h tfrac 1 2 mathop smallint limits 0 l q 2 mathop smallint limits 0 l f q tfrac 1 2 mathop smallint limits 0 l p 2 the corresponding flow preserves the suitably interpreted petit ensemble e h d qd p and forl q settles down to the stationary diffusion with infinitesimal operator 1 2 2 q2 m q q m being the logarithmic derivative of the ground state of d2 dq2 f q this diffusion is the brownian motion with restoring drift see mckean vaninsky 1993 1 for reasons suggested by the paper of lebowitz rose speer 1988 on nls it is interesting to study the micro canonical ensemble obtained by restricting to the sphere int limits 0 l q 2 n and makingl with fixedd n l now forf q q2 the same type of diffusion appears but with drift arising from the modified potentialf q cq2 c being chosen so that the mean ofq2 is the assigned numberd the proof employs doblin s method of loops 1937 and steepest descent the same is true forf q m2q2 only now the proof is elementary the outcome is also the same iff q q2 0 providedd is smaller than the petit canonical mean ofq2 ford larger than this mean the matter is more subtle and the outcome is unknown

invariant measures for the 2d defocusing nonlinear schrodinger equation consider the2d defocusing cubic nlsiu t u u u 2 0 with hamiltonian smallint left left nabla phi right 2 tfrac 1 2 left phi right 4 right it is shown that the gibbs measure constructed from the wick ordered hamiltonian i e replacing 4 by 4 is an invariant measure for the appropriately modified equationiu t u u u 2 2 u 2 dx u 0 there is a well defined flow on thesupport of the measure in fact it is shown that for almost all data the solutionu u 0 satisfiesu t e it c hs for somes 0 first a result local in time is established and next measure invariance considerations are used to extend the local result to a global one cf b2

ideal magnetofluid turbulence in two dimensions a continuum model of coherent structures in two dimensional magnetohydro dynamic turbulence is developed these structures are macroscopic states which persist among the turbulent microscopic fluctuations typically as magnetic islands with flow they are modeled as statistical equilibrium states for the non dissipative dynamics which conserves energy and families of cross helicity and flux integrals the model predicts that from a given initial state an ideal magnetofluid will evolve into a final state having steady mean magnetic and velocity fields and gaussian local fluctuations in these fields excellent qualitative and quantitative agreement is found with the known results of direct numerical simulations a rigorous justification of the theory is also provided in the sense that the continuum model is derived from a lattice model in a fixed volume small spacing limit this construction uses the discrete fourier transform to link the discretization ofx space with the truncation ofk space under the ergodic hypothesis and a separation of scales hypothesis the lattice model is defined by a mean field approximation to the gibbs measure on the discretized phase space a concentration property shows that this measure is equivalent to the microcanonical measure in the continuum limit

classification of the indecomposable bounded admissible modules over the virasoro lie algebra with weightspaces of dimension not exceeding two in view of 1 2 any bounded admissible modulea over the virasoro lie algebra open image in new window is a finite length extension of irreducible modules with one dimensional weightspaces to each extension of finite lengthn are associatedn 1 invariants a1 1 n we prove that we have i j 0 1 6 n 1b for all i j with 1 i j n in the casen 2 this result allows us to construct all the indecomposable bounded admissible open image in new window modules where the dimensions of the weightspaces are less than or equal to two in particular we obtain all the extensions of two irreducible bounded open image in new window modules