two novel methods and multi mode periodic solutions for the fermi pasta ulam model we introduce two novel methods for studying periodic solutions of the fpu model both numerically and rigorously one is a variational approach based on the dual formulation of the problem and the other involves computer assisted proofs these methods are used e g to construct a new type of solutions whose energy is spread among several modes associated with closely spaced resonances

asymptotic behavior of solutions of functional difference equations for linear functional difference equations we obtain some results on the asymptotic behavior of solutions which correspond to a perron type theorem for linear ordinary difference equations we also apply our results to volterra difference equations with infinite delay

lqg control with an h sup infinity performance bound a riccati equation approach an lqg linear quadratic gaussian control design problem involving a constraint on h sup infinity disturbance attenuation is considered the h sup infinity performance constraint is embedded within the optimization process by replacing the covariance lyapunov equation by a riccati equation whose solution leads to an upper bound on l sub 2 performance in contrast to the pair of separated riccati equations of standard lqg theory the h sup infinity constrained gains are given by a coupled system of three modified riccati equations the coupling illustrates the breakdown of the separation principle for the h sup infinity constrained problem both full and reduced order design problems are considered with an h sup infinity attenuation constraint involving both state and control variables an algorithm is developed for the full order design problem and illustrative numerical results are given

lp boundedness properties for volterra difference equations this work deals with l p boundedness of solutions of a class of volterra functional difference equation the effect of different kinds of the nonhomogeneous term lipschitz local lipschitz completely continuous are taken into consideration

bushes of vibrational modes for fermi pasta ulam chains abstract some exact solutions and multimode invariant submanifolds were found for the fermi pasta ulam fpu model by poggi and ruffo physica d 103 1997 251 in the present paper we demonstrate how results of such a type can be obtained for an arbitrary n particle chain with periodic boundary conditions with the aid of our group theoretical approach physica d 117 1998 43 based on the concept of bushes of normal modes in mechanical systems with discrete symmetry the integro differential equation describing the fpu dynamics in the modal space is derived the loss of stability of the bushes of modes for the fpu model in particular for the limiting case n for the dynamical regime with displacement pattern having period twice the lattice spacing mode is studied our results for the fpu chain are compared with those by poggi and ruffo for the fpu chain

asymptotic behavior of volterra difference equations abstract in this paper we investigate boundedness and stability properties of some classes of discrete volterra equations the main tool in this work is the use of a representation formula which allows us to express the solution of discrete volterra equations in terms of the resolvent matrix of the corresponding system of volterra difference equations

an alternative approach to norm bound computation for inverses of linear operators in hilbert spaces abstract in the present paper we propose a computer assisted procedure to prove the invertibility of a linear operator in a hilbert space and to compute a verified norm bound of its inverse a number of the authors have previously proposed two verification approaches that are based on projection and constructive a priori error estimates the approach of the present paper is expected to bridge the gap between the two previous procedures in actual numerical verifications several verification examples that confirm the actual effectiveness of the proposed procedure are reported

symmetric invariant manifolds in the fermi pasta ulam lattice abstract the fermi pasta ulam fpu lattice with periodic boundary conditions and n particles admits a large group of discrete symmetries the fixed point sets of these symmetries naturally form invariant symplectic manifolds that are investigated in this short note for each k dividing n we find k degree of freedom invariant manifolds they represent short wavelength solutions composed of k fourier modes and can be interpreted as embedded lattices with periodic boundary conditions and only k particles inside these invariant manifolds other invariant structures and exact solutions are found which represent for instance periodic and quasi periodic solutions and standing and travelling waves some of these results have been found previously by other authors via a study of mode coupling coefficients and recently also by investigating bushes of normal modes the method of this paper is similar to the latter method and much more systematic than the former we arrive at previously unknown results without any difficult computations it is shown moreover that similar invariant manifolds exist also in the klein gordon kg lattice and in the thermodynamic and continuum limits

asymptotic resonance interaction of modes and subharmonic bifurcation we study the existence of small amplitude oscillations near elliptic equilibria of autonomous systems which mix different normal modes the reference problem is the fermi pasta ulam model a chain of nonlinear oscillators with nearest neighborhood interaction we develop a new bifurcation approach that locates secondary bifurcations from the unimodal primary branches two sufficient conditions for bifurcation are given one involves only the arithmetic properties of the eigenvalues of the linearized system asymptotic resonance while the other takes into account the nonlinear character of the interaction between normal modes nonlinear coupling both conditions are checked for the fermi pasta ulam problem

structured stability radii and exponential stability tests for volterra difference systems uniform exponential ue stability of linear difference equations with infinite delay is studied using the notions of a stability radius and a phase space the state space x is an abstract banach space we work with non fading phase spaces c 0 z x and z x and with exponentially fading phase spaces of the p and c 0 types for equations of the convolution type several criteria of ue stability are obtained in terms of the z transform email protected z of the convolution kernel k in terms of the input state operator and of the resolvent fundamental matrix these criteria do not impose additional positivity or compactness assumptions on coefficients k j time varying non convolution difference equations are studied via structured ue stability radii r t of convolution equations these radii correspond to a feedback scheme with delayed output and time varying disturbances we also consider stability radii r c associated with a time invariant disturbance operator unstructured stability radii and stability radii corresponding to delayed feedback for all these types of stability radii two sided estimates are obtained the estimates from above are given in terms of the z transform email protected z the estimate from below via the norm of the input output operator these estimates turn into explicit formulas if the state space x is hilbert or if disturbances are time invariant the results on stability radii are applied to obtain various exponential stability tests for non convolution equations several examples are provided

almost periodic and almost automorphic solutions of linear differential difference equations without favard s separation condition ii abstract the well known favard s theorem states that the linear differential equation 1 x a t x f t with bohr almost periodic coefficients admits at least one bohr almost periodic solution if it has a bounded solution the main assumption in this theorem is the separation among bounded solutions of homogeneous equations 2 x b t x where b h a b b t lim n a t t n if there are bounded solutions which are non separated sometimes almost periodic solutions do not exist r johnson r ortega and m tarallo v zhikov and b levitan in this paper we prove that linear differential equation 1 with levitan almost periodic coefficients has a unique levitan almost periodic solution if it has at least one bounded solution and the bounded solutions of the homogeneous equation 3 x a t x are homoclinic to zero i e lim t t 0 for all bounded solutions of 3 if the coefficients of 1 are bohr almost periodic and all bounded solutions of all limiting equations 2 are homoclinic to zero then eq 1 admits a unique almost automorphic solution the analogue of this result for difference equations is also given we study the problem of existence of bohr levitan almost periodic solutions of eq 1 in the framework of general non autonomous dynamical systems cocycles

existence theorem for solitary waves on lattices in this article we give an existence theorem for localized travelling wave solutions on one dimensional lattices with hamiltonian h sum limits n in mathbb z left tfrac 1 2 p n 2 v q n 1 q n right wherev is the potential energy due to nearest neighbour interactions until now apart from rare integrable lattices like the toda latticev ab 1 e b b 1 the only evidence for existence of such solutions has been numerical our result in particular recovers existence of solitary waves in the toda lattice establishes for the first time existence of solitary waves in the nonintegrable cubic and quartic latticesv 1 2 2 1 3a 3 v 1 2 2 1 4b 4 thereby confirming the numerical findings in 1 and shedding new light on the recurrence phenomena in these systems observed first by fermi pasta and ulam 2 and shows that contrary to widespread belief the presence of exact solitary waves is not a peculiarity of integrable systems but generic in this class of nonlinear lattices the approach presented here is new and quite general and should also be applicable to other forms of lattice equations the travelling waves are sought as minimisers of a naturally associated variational problem obtained via hamilton s principle and existence of minimisers is then established using modern methods in the calculus of variations the concentration compactness principle of p l lions 3

convergent solutions of linear functional difference equations in phase space by using weighted summable dichotomies and schauder s fixed point theorem we prove the existence of convergent solutions of linear functional difference equations we apply our result to volterra difference equations with infinite delay

entire solutions for a semilinear fourth order elliptic problem with exponential nonlinearity we investigate entire radial solutions of the semilinear biharmonic equation 2 u exp u in r n n 5 0 being a parameter we show that singular radial solutions of the corresponding dirichlet problem in the unit ball cannot be extended as solutions of the equation to the whole of r n in particular they cannot be expanded as power series in the natural variable s log x next we prove the existence of infinitely many entire regular radial solutions they all diverge to as x and we specify their asymptotic behaviour as in the case with power type nonlinearities f gazzola h ch grunau radial entire solutions for supercritical biharmonic equations math ann 334 2006 905 936 the entire singular solution x 4l og x plays the role of a separatrix in the bifurcation picture finally a technique for the computer assisted study of a broad class of equations is developed it is applied to obtain a computer assisted proof of the underlying dynamical behaviour for the bifurcation diagram of a corresponding autonomous system of odes in the case n 5

l sub 2 gain analysis of nonlinear systems and nonlinear state feedback h sub infinity control previously obtained results on l2 gain analysis of smooth nonlinear systems are unified and extended using an approach based on hamilton jacobi equations and inequalities and their relation to invariant manifolds of an associated hamiltonian vector field on the basis of these results a nonlinear analog is obtained of the simplest part of a state space approach to linear h sub infinity control namely the state feedback h sub infinity optimal control problem furthermore the relation with h sub infinity control of the linearized system is dealt with

computer assisted methods for the study of stationary solutions in dissipative systems applied to the kuramoto sivashinski equation we develop some computer assisted techniques for the analysis of stationary solutions of dissipative partial differential equations of their stability and of their bifurcation diagrams as a case study these methods are applied to the kuramoto sivashinski equation this equation has been investigated extensively and its bifurcation diagram is well known from a numerical point of view here we rigorously describe the full graph of solutions branching off the trivial branch complete with all secondary bifurcations for parameter values between 0 and 80 we also determine the dimension of the unstable manifold for the flow at some stationary solution in each branch

multiple periodic solutions for lattice dynamical systems with superquadratic potentials abstract in this paper we consider one dimensional lattices consisting of infinitely many particles with nearest neighbor interaction the autonomous dynamical system is described by the following infinite system of second order differential equations q i i 1 q i 1 q i i q i q i 1 i z where i denotes the interaction potential between two neighboring particles and q i t is the state of the i th particle supposing i is superquadratic at infinity for all t 0 we obtain a nonzero t periodic solution of finite energy which may be nonconstant in some range of period if in addition i x is even in x we also obtain infinitely many geometrically distinct solutions for any period t 0 in particular a prescribed number of geometrically distinct nonconstant periodic solutions is obtained for some range of period since the functional associated to the above system is invariant under the actions of the non compact group z and the continuous compact group s 1 under our assumptions in order to prove our results we need to extend the abstract critical point theorem about strongly indefinite functional developed by bartsch and ding math nachr 279 2006 1267 1288 to a more general class of symmetry

nonlinear integral type sliding surface for both matched and unmatched uncertain systems this note presents a new nonlinear integral type sliding surface which incorporates a virtual nonlinear nominal control to achieve prescribed specifications first the plant with matched uncertainties is considered the resultant closed loop system during ideal sliding mode behaves exactly like the nominal plant under the nonlinear nominal control which completely nullifies the matched uncertainties and consequently satisfies the prescribed specifications second the stability analysis of the proposed sliding mode for the systems with unmatched uncertainties is performed to exploit the stability conditions numerical results demonstrate the validity of the proposed concept

all fixed order h sub spl infin controllers observer based structure and covariance bounds this note obtains a parameterization of the set of all stabilizing controllers of order less than or equal to the plant which yields for the closed loop transfer matrix a specified h sub spl infin norm bound the algebraic results of covariance control are applied to the h sub spl infin control problem to yield a parameterization in terms of the lyapunov matrix which carries many system properties such as h sub 2 performance covariance bounds system entropy at infinity etc all low order h sub spl infin controllers are shown to have observer based structure for reduced order models of the plant and are characterized by two riccati equations with a coupling condition

all controllers for the general h control problem lmi existence conditions and state space formulas this paper presents all controllers for the general h control problem with no assumptions on the plant matrices necessary and sufficient conditions for the existence of an h controller of any order are given in terms of three linear matrix inequalities lmis our existence conditions are equivalent to scherer s results but with a more elementary derivation furthermore we provide the set of all h controllers explicitly parametrized in the state space using the positive definite solutions to the lmis even under standard assumptions full rank etc our controller parametrization has an advantage over the q parametrization the freedom q a real rational stable transfer matrix with the h norm bounded above by a specified number is replaced by a constant matrix l of fixed dimension with a norm bound and the solutions x y to the lmis the inequality formulation converts the existence conditions to a convex feasibility problem and also the free matrix l and the pair x y define a finite dimensional design space as opposed to the infinite dimensional space associated with the q parametrization

analysis and design of integral sliding manifolds for systems with unmatched perturbations the robustness properties of integral sliding mode controllers are studied this note shows how to select the projection matrix in such a way that the euclidean norm of the resulting perturbation is minimal it is also shown that when the minimum is attained the resulting perturbation is not amplified this selection is particularly useful if integral sliding mode control is to be combined with other methods to further robustify against unmatched perturbations h sub spl infin is taken as a special case simulations support the general analysis and show the effectiveness of this particular combination

ℒ2 gain tuning of variable structure siso systems of relative degree n abstractthe present paper extends the l2 gain analysis towards variable structure systems vss with sliding mode sm dynamics the developed analysis is applied to feedback linearisable vss driven by standard sm algorithms of the first order constructive parameter tuning of the resulting closed loop system to possess the l2 gain less than a desired attenuation level is additionally deduced the proposed tuning is shown to be capable not only of rejecting matched uniformly bounded disturbances but also of attenuating unbounded ones including mismatched disturbances numerical simulations of an inverted pendulum driven by a first order sm controller are involved to support the theory in the presence of the position measurement inaccuracy being a source of a mismatched disturbance

asymptotic properties of solutions to nonautonomous volterra difference systems with infinite delay abstract we show that under certain conditions there is a one to one correspondence between weighted bounded solutions of a linear nonautonomous volterra difference system with infinite delay and its perturbed system under additional assumptions this correspondence is elevated to asymptotic equivalence on the other side we prove the existence of weighted convergent solutions

state space formulae for all stabilizing controllers that satisfy and h norm bound and relations to risk sensitivity abstract given a linear system all stabilizing controllers such that a specified closed loop transfer function has h norm less than a given scalar are parametrized this characterization involves the solution to two algebraic riccati equations each with the same order as the system and further gives gives feasible controllers also with this order the relationship to the risk sensitive lqg stochastic control problem is established giving an equivalence between robust and stochastic control

a new parametrization of h suboptimal controllers elementary state space concepts are used to derive a transparent solution to the h control problem on an infinite horizon the main contribution of this solution is a novel representation of suboptimal controllers in terms of a pair of parametric riccati equations with a coupling constraint unlike the classical parametrization in terms of linear fractional transformation this state space representation has a homogeneous riccati formulation which should help to make the most of the suboptimal controller diversity potential applications include the design of reduced order controllers and more generally the selection of suboptimal controllers to meet or optimize additional constraints

symmetry and resonance in periodic fpu chains the symmetry and resonance properties of the fermi pasta ulam chain with periodic boundary conditions are exploited to construct a near identity transformation bringing this hamiltonian system into a particularly simple form this birkhoff gustavson normal form retains the symmetries of the original system and we show that in most cases this allows us to view the periodic fpu hamiltonian as a perturbation of a nondegenerate liouville integrable hamiltonian according to the kam theorem this proves the existence of many invariant tori on which motion is quasiperiodic experiments confirm this qualitative behaviour we note that one can not expect this in lower order resonant hamiltonian systems so the periodic fpu chain is an exception and its special features are caused by a combination of special resonances and symmetries

solitary waves with prescribed speed on infinite lattices using a variant of the mountain pass theorem we prove the existence of solitary waves with prescribed speed on infinite lattices of particles with nearest neighbor interaction the problem is to solve a second order forward backward differential difference equation c 1997 academic press

a new branch of mountain pass solutions for the choreographical 3 body problem we prove the existence of a new branch of solutions of mountain pass type for the periodic 3 body problem with choreographical constraint at first we describe the variational structure of the action functional associated to the choreographical three body problem in mathbb r 3 in the second part using a bisection algorithm we provide a numerical non rigorous solution of mountain pass type for this problem in a rotating frame with angular velocity 1 5 the last step consists in the rigorous computer assisted proof of the existence of a full branch of solutions for the problem starting from the mountain pass solution detected numerically

state space solutions to standard h sub 2 and h sub infinity control problems simple state space formulas are derived for all controllers solving the following standard h sub infinity problem for a given number gamma 0 find all controllers such that the h sub infinity norm of the closed loop transfer function is strictly less than gamma it is known that a controller exists if and only if the unique stabilizing solutions to two algebraic riccati equations are positive definite and the spectral radius of their product is less than gamma sup 2 under these conditions a parameterization of all controllers solving the problem is given as a linear fractional transformation lft on a contractive stable free parameter the state dimension of the coefficient matrix for the lft constructed using the two riccati solutions equals that of the plant and has a separation structure reminiscent of classical lqg i e h sub 2 theory this paper is intended to be of tutorial value so a standard h sub 2 solution is developed in parallel

an asymptotic theory for retarded functional difference equations using discrete dichotomies and krasnoselsky s theorem we obtain existence and asymptotic behavior of convergent solutions for retarded functional difference equations we also will get some global properties for the set of convergent solutions applications on volterra difference equations with infinite delay are shown

about the behavior of solutions for volterra difference equations with infinite delay we study the existence of discrete almost automorphic solutions and asymptotic behavior for non linear volterra difference equations of convolution type with infinite delay where the nonlinear perturbation is considered not necessarily globally lipschitz the results are a consequence of application of different fixed point theorems namely the contraction principle the leray schauder the krasnoselkii and matkowski s fixed point technique

exact tracking using backstepping control design and high order sliding modes in this technical note a controller ensuring exponential exact tracking in the presence of matched and unmatched disturbances for the nonlinear systems in the block controllable form is proposed the controller is designed using the backstepping procedure and combines the feedback linearization technique with the high order sliding modes the matched and unmatched disturbances are compensated by the injection of a continuous term generated by the robust exact high order sliding modes differentiator the obtained control law is differentiable and can be applied directly to the system simulations verify the performance of the proposed controller

dynamic switching surfaces for output sliding mode control the robustness properties of sliding mode and h controllers are exploited to produce a dynamic output feedback controller that is insensitive to matched perturbations and attenuates the unmatched ones the assumptions on the plant differ from the standard assumptions of the riccati state space approach to h control the sliding mode controller drives the state into a reduced order manifold for which the equivalent system does satisfy the standard assumptions and hence the standard theory can be applied the resulting riccati equations are of reduced order

the zone boundary mode in periodic nonlinear electrical lattices abstract we study the existence and linear stability of the zone boundary mode in a nonlinear electrical lattice consisting of n inductors and n voltage dependent capacitors with periodic boundary conditions the inductances are allowed to alternate while the capacitors are identical and each have a quadratic dependence on voltage by block diagonalizing a 2 n 2 n floquet problem we reduce the question of the stability of the mode to a single hill s equation that is analyzed using methods of perturbation theory and averaging we show that periodicity of the lattice inductances degrades stability and also show that the instability threshold is proportional to n 2 numerical computations validate the perturbative results