shape and morse theory of attractors we study the shape in the sense of borsuk of attractors of continuous semi dynamical systems on general metric spaces we show in particular that the natural inclusion of the global attractor into the state space is a shape equivalence this and other results of the paper are used to develop an elementary morse theory of an attractor 2000 john wiley sons inc

vector norms as lyapunov functions for linear systems a unified theory of quadratic and piecewise linear lyapunov functions for continuous and discrete time linear systems is presented the key to this work is the description of these lyapunov functions by vector norms the main results are sufficient and necessary conditions for a vector norm to be a lyapunov function as well as a method based on these conditions of constructing such lyapunov functions

generalized h2 control of switched discrete time systems with unknown delays the generalized h 2 control problem for a class of linear discrete time switched systems with constant unknown delays is investigated new linear matrix inequalities lmis characterization based on switched lyapunov krasovskii functional are established to guarantee the asymptotic stability and generalized h 2 performance design of generalized h 2 state feedback and dynamic output feedback controllers are developed a numerical example is worked out in detail to illustrate the theoretical developments

further results on infinity norms as lyapunov functions for linear systems this note continues on the results proposed in two previous papers for computing vector 1 or infin norm based lyapunov functions for linear discrete time systems by presenting a finitely terminating algorithm for the construction of lyapunov functions of that class the algorithm utilizes the solution of a finite sequence of feasibility linear programs lps with very few constraints or equivalently very simple algebraic tests in contrast to solving the original problem where a bilinear matrix equation with rank and norm constraint needs to be solved

robust event triggered mpc with guaranteed asymptotic bound and average sampling rate we propose a robust event triggered model predictive control mpc scheme for linear time invariant discrete time systems subject to bounded additive stochastic disturbances and hard constraints on the input and state for given probability distributions of the disturbances acting on the system we design event conditions such that the average frequency of communication between the controller and the actuator in the closed loop system attains a given value we employ tube mpc methods to guarantee robust constraint satisfaction and a robust asymptotic bound on the system state moreover we show that instead of a given periodically updated tube mpc scheme an appropriate event triggered mpc scheme can be applied with the same guarantees on constraints and region of attraction but with a reduced number of average communications

composite systems with uncertain couplings of fixed structure scaled riccati equations and the problem of quadratic stability we consider large scale systems consisting of a finite number of separate uncertain subsystems which interact via uncertain couplings of arbitrarily prescribed structure the couplings are viewed as structured perturbations of the block diagonal system representing the collection of the separate nominal subsystems the nominal system we define spectral value sets and stability radii for these time invariant structured perturbations and derive formulas for their computation scaled riccati equations are introduced to obtain explicit formulas for the stability radii with respect to time varying and possibly nonlinear perturbations of the given structure from these we derive necessary and sufficient conditions under which the stability radii with respect to time invariant and time varying perturbations are equal these results are obtained by constructing joint quadratic liapunov functions of optimal robustness with their help we prove necessary and sufficient conditions for quadratic stability and sufficient conditions for the validity of a generalized aizerman conjecture

inertial manifolds for the hyperviscous navier stokes equations abstract we prove the existence of inertial manifolds for the incompressible hyperviscous navier stokes equations on the two or three dimensional torus u t u u u p f t x r t d div u 0 where d 2 or 3 and 3 2 since the spectral gap condition is not necessarily satisfied for the aforementioned problem in three dimensions we employ the spatial averaging method introduced by mallet paret and sell in 26

inertial manifolds for nonlinear evolutionary equations abstract in this paper we introduce the concept of an inertial manifold for nonlinear evolutionary equations in particular for ordinary and partial differential equations these manifolds which are finite dimensional invariant lipschitz manifolds seem to be an appropriate tool for the study of questions related to the long time behavior of solutions of the evolutionary equations the inertial manifolds contain the global attractor they attract exponentially all solutions and they are stable with respect to perturbations furthermore in the infinite dimensional case they allow for the reduction of the dynamics to a finite dimensional ordinary differential equation

a robust stabilizing law for switched linear systems in this paper we address some robustness issues for a class of switched linear systems with non linear perturbations we illustrate through an example that the well known state space partition based switching law may lead to chattering in the event that the systems undergo perturbations to overcome this we propose a state feedback switching law with a non zero level set we prove that this switching law can prevent the system from chattering in addition this switching law makes the perturbed systems bounded with bounded perturbations and practically convergent with convergent perturbations we also propose an observer based switching law to steer the switched system stable with nice robustness performances

some backward uniqueness results soit 7b u un operateur non borne lineaire ou non lineaire sur un espace de hilbert h tel que 7b u 0 0 soit une fonction solution de l equation d evolution d dt 7b u 0 et telle que t 0 pour un t 0 on etudie la propriete d unicite retrograde sur 0 t pour des inegalites d t dt a t t h n t t d a 1 2 t ou a t t 0 est une famille d operateurs lineaires non bornes autoadjoint sur h et u l 2 0 t

on infinity norms as lyapunov functions for linear systems the note presents a construction of the lyapunov function defined by l sub spl infin sup n vector norm for both continuous and discrete time linear systems in contrast to the previous results no restrictions on the positions or multiplicity of system matrix eigenvalues are required

formulation and validation of boundary conditions at a branched junction for nonlinear waves abstract a numerical model is presented for the calculation of nonlinear waves through the branched junctions in a gas transmission pipe system in the one dimensional time domain analysis of the wave dynamics the behaviour of the nonlinear waves is determined by quasi steady boundary conditions at junctions in a pipe system the quasi steady treatment through the junction yields a system of nonlinear balance equations the iterative solution procedure of the system of nonlinear equations causes the problems of divergence and multiple solutions in order to overcome the difficulties a new set of quasi linear balance equations is formulated without a sacrifice of accuracy the present model therefore requires a non iterative solution procedure that results in a unique solution and a smaller computational effort thompson s boundary condition is used at each pipe a new time derivative form of balance equations based thompson boundary condition gives a set of linear algebraic equations for the balance equations when implementing the new time derivative form of the balance equations corrected terms are added numerically the present numerical model is implemented and tested for the calculation of behaviour of nonlinear waves in a branched pipe and the calculation of sound attenuation of a helmholtz resonator and the prediction of radiated orifice noise of an internal combustion engine calculation results show good agreements with previous iterative calculations and measurement data

persistence of overflowing manifolds for semiflow this paper which is a sequel to a previous one 4 by the same authors is devoted to the persistence of overflowing manifolds and inflowing manifolds for a semiflow in a banach space we consider a c1 semiflow defined on a banach space x that is it is continuous on 0 x and for each t 0 t t x x is c1 and t t t x t t s x for all t s 0 and x x a typical example is the solution operator for a differential equation in 4 we proved that a compact normally hyperbolic invariant manifold m persists under small c1 perturbations in the semiflow we also showed that in a neighborhood of m there exist a center stable manifold and a center unstable manifold that intersect in the manifold m in 4 the compactness and invariance of the manifold m were important assumptions in the present paper we study the more general case where the manifold m is overflowing negatively invariant and the semiflow crosses the boundary transversally or inflowing positively invariant and the semiflow crosses the boundary transversally we do not assume that m is compact or finite dimensional also m is not necessarily an imbedded manifold but an immersed manifold as an example a local unstable manifold of an equilibrium point is an overflowing manifold in brief our main results on the overflowing manifolds may be summarized as follows the precise statements are given in section 2 we assume that the immersed manifoldm does not twist very much locally m is covered by the image under t t of a subset a positive distance away from boundary dt t contracts along the normal direction and does so more strongly than it does along the tangential direction and dt t has a certain uniform continuity in a neighborhood of m if the c1 perturbation t t of t t is sufficiently close to t t then t t has a unique c1 immersed overflowing manifold m nearm furthermore if t t isck and a spectral gap condition holds then m isck similar results for inflowing manifolds are also obtained and given in section 7

application of maccormack schemes to i c engine exhaust noise prediction non linear flow calculations are a usual tool for describing the flow inside intake and exhaust systems of internal combustion engines therefore they may also be used to estimate the noise radiated by these systems most of the work reported in this field is related to the use of the method of characteristics and some references to the use of the lax wendroff finite differences scheme are also found in this paper maccormack predictor corrector schemes are shown to be less dispersive and slightly faster than the lax wendroff scheme and are used to solve the one dimensional flow equations in the case of the exhaust system of a single cylinder engine a modification of the treatment of the source terms is proposed which results in a considerable reduction in the computation time without any loss of information an iterative procedure has been devised that allows the use with this finite differences scheme of boundary conditions developed to be used with the method of characteristics comparison with experimental results is given for the case of an exhaust duct without muffler and with an expansion chamber showing fair agreement in both cases

a lagrange duality characterisation for stability under arbitrary switching in switched positive linear systems abstractthe present communication is concerned with uniform exponential stability under arbitrary switching in discrete time switched positive linear systems lagrange duality is used in order to obtain a new characterisation for uniform exponential stability which is in terms of sets of inequalities involving each of the matrices that represent the modes of the system these sets of inequalities are shown to generalise the classical linear lyapunov inequality that characterises in positive matrices the property of being schur each solution to these sets of inequalities is shown to provide a representation in terms of a number of linear functionals for a common lyapunov function for the switched positive linear system a result is further presented which conveys to a conservative upper bound on the minimum required number of linear functionals in the above mentioned representation and also to a method for computing them our proof for the aforementioned characterisation is based on another equ

robust minkowski lyapunov functions abstract a necessary and sufficient condition for solvability of the lyapunov inequality for polytopic linear inclusions over the space of minkowski functions of nonempty compact and convex sets that contain the origin as an interior point is derived it is demonstrated that the introduced framework results in a nonconservative stability analysis of polytopic linear inclusions

lyapunov function construction by linear programming in this paper the lyapunov function given by the infinity norm is considered a new proof of the theorem of molchanov and pyatintskii 1986 is given the presented proof enables us to give an explicit calculation of the matrix which was only demonstrated to exist in the original paper combining the obtained result with the idea of scaling faces of a polytope gives the method of fitting the shape of level surfaces of the lyapunov function given by the infinity norm to the specified demands

graph lyapunov function for switching stabilization and distributed computation abstract this paper studies stabilization of discrete time switched linear systems slss using the notion of graph control lyapunov functions gclfs a gclf is a set of lyapunov functions defined on a weighted digraph where each lyapunov function is represented by a node in the digraph and there is a lyapunov inequality associated with each subgraph consisting of a node and its out neighbors the weight of a directed edge indicates the decay or growth rate of the lyapunov functions it is proved that an sls is switching stabilizable if and only if there exists a gclf the main benefits of gclfs are reduced computational cost and conservatism for stabilizability tests besides we show that the proposed gclf framework unifies several control lyapunov functions and the related stabilization theorems moreover we propose a distributed algorithm to evaluate the stabilizability with reduced computational costs by taking benefits of the graph structure of gclfs several examples are given to demonstrate the efficiency of the algorithm

constrained regulation of linear systems abstract in this paper the linear constrained regulation problem without an assigned set of initial states is investigated this problem consists of the determination of a linear state feedback control law and of a set of admissible initial states d so that every trajectory of the resulting closed loop system which emanates from the set converges asymptotically to the origin without violation of the control constraints by applying well known results on the positive invariance of polyhedral sets an eigenstructure assignment approach to this problem is established if the number of unstable open loop eigenvalues does not exceed the number of control variables then this approach enables one to derive a linear state feedback stabilizing control law which makes the set of states where the control constraints are respected positively invariant this positively invariant set is the maximal admissible set of initial states for the control law under consideration for the case where the number of control variables is greater than the number of the unstable open loop eigenvalues a modified algorithm is proposed

positively invariant polyhedral sets of discrete time linear systems the problem of the existence of positively invariant polyhedral sets for linear discrete time dynamical systems is studied in the first part of the paper necessary and sufficient conditions for a

prediction of the acoustical performances of muffling pipe systems by the method of characteristics abstract a numerical model is presented for the calculation of sound attenuation of flow duct systems in the frequency domain the method of characteristics is used to solve the non linear hyperbolic problem of plane wave motion in ducts the basic conservation equations are not linearized to achieve a comparison between linear model and gas dynamic model predictions boundary conditions providing acoustic excitation of the duct system are explained and details are given about the calculation of the transfer functions of muffling pipe systems including some numerical algorithms to achieve satisfactory predictions by the method of characteristics end corrections have been introduced to take account of phase shifts in the wave reflection at open terminations as well as at abrupt area changes a comparison between experimental attenuation curves measured in a semi anechoic room and calculated ones is displayed for some simple geometries including helmholtz resonators and expansion chambers and shows a fairly good agreement

inertial forms of navier stokes equations on the sphere abstract the navier stokes equations for incompressible flows past a two dimensional sphere are considered in this article the existence of an inertial form of the equations is established furthermore for the first time for fluid equations we derive an upper bound on the dimension of the differential system inertial manifold which fully reproduces the infinite dimensional dynamics this bound is expressed in terms of grashof numbers

a numerical technique for the stability analysis of linear switched systems in this paper the ray gridding approach a new numerical technique for the stability analysis of linear switched systems is presented it is based on uniform partitions of the state space in terms of ray directions which allow refinable families of polytopes of adjustable complexity to be examined for invariance in this framework the existence of a polyhedral lyapunov function that is common to a family of asymptotically stable subsystems can be checked efficiently via simple iterative algorithms the technique can be used to prove the stability of switched linear systems classes of linear time varying systems and linear differential lnclusions we also present preliminary results on another related problem namely the construction of multiple polyhedral lyapunov functions for the specification of stabilizing switching sequences for a switched system constructed from a family of stable linear subsystems

variational approach to closure of nonlinear dynamical systems autonomous case a general approach for the derivation of nonlinear parameterizations of neglected scales is presented for nonlinear systems subject to an autonomous forcing in that respect dynamically based formulas are derived subject to a free scalar parameter to be determined per mode to parameterize for each high mode this free parameter is obtained by minimizing a cost functional a parameterization defect depending on solutions from direct numerical simulation dns but over short training periods of length comparable to a characteristic recurrence or decorrelation time of the dynamics an important class of dynamically based formulas for our parameterizations to optimize are obtained as parametric variations of manifolds approximating the invariant ones to better appreciate the origins of the modified manifolds thus obtained the standard approximation theory of invariant manifolds is revisited in part i of this article a special emphasis is put on backward forward bf systems naturally associated with the original system whose asymptotic integration provides the leading order approximation of invariant manifolds part ii presents then i the modifications of these approximating manifolds based also on integration of the same bf systems but this time over a finite time tau and ii the variational approach aimed at making an efficient selection of tau per mode to parameterize the parametric class of leading interaction approximation lia of the high modes obtained this way is completed by another parametric class built from the quasi stationary approximation qsa close to the first criticality the qsa is an approximation to the lia but it differs as one moves away from criticality rigorous results are derived that show that given a cutoff dimension the best manifolds that can be obtained through our variational approach are manifolds which are in general no longer invariant the minimizers are objects called the optimal parameterizing manifolds pms that are intimately tied to the conditional expectation of the original system i e the best vector field of the reduced state space resulting from averaging of the unresolved variables with respect to a probability measure conditioned on the resolved variables applications to the closure of low order models of atmospheric primitive equations and rayleigh benard convection are then discussed the approach is finally illustrated in the context of the kuramoto sivashinsky turbulence as providing efficient closures without slaving for a cutoff scale k mathfrak c placed within the inertial range and the reduced state space is just spanned by the unstable modes without inclusion of any stable modes whatsoever the underlying optimal pms obtained by our variational approach are far from slaving and allow for remedying the excessive backscatter transfer of energy to the low modes encountered by the lia or the qsa parameterizations in their standard forms when they are used at this cutoff wavelength

stabilization of second order lti switched systems this paper studies and solves the problem of asymptotic stabilization of switched systems consisting of unstable secondorder linear time invariant lti subsystems necessary and sufficient conditi

inertial manifolds associated to partly dissipative reaction diffusion systems abstract in a previous work we established the existence of finite dimensional attractors for partly dissipative reaction diffusion systems of two types systems with a polynomial growth nonlinearity and systems admitting an invariant region the aim of this article is to show that these problems possess inertial manifolds in space dimension n 2 these inertial manifolds contain the universal attractor attract exponentially all the solutions as time goes to infinity and due to the weak dissipation are here infinite dimensional our results are applied to several classical systems borrowed from mathematical biology physics and chemistry

criteria of asymptotic stability of differential and difference inclusions encountered in control theory abstract for a class of differential inclusions to which many of the practically important control systems can be reduced necessary and sufficient conditions for asymptotic stability of the zero solution are established by the method of lyapunov functions similar results are obtained for the corresponding class of difference inclusions

inertial manifolds for the 3d modified leray α model abstract in this paper we investigate the critical modified leray model in t 3 and prove the existence of an n dimensional inertial manifold for this problem

comparison principle positive invariance and constrained regulation of nonlinear systems abstract in this paper the comparison principle is applied to the design of feedback controllers for nonlinear discrete time systems subject to physical constraints first the connection between the comparison principle and the existence of positively invariant sets of a special form is established and polyhedral positively invariant sets for nonlinear systems are obtained then these results are applied to the determination of state feedback control laws that transfer to the origin all initial states of a prespecified region of the state space without violation of physical constraints on the control vector

nonquadratic lyapunov functions for robust control abstract this paper deals with the problem of controlling a linear continuous time system with structured time varying parameter uncertainties and input disturbances with a lyapunov function approach in contrast with most of the previous results in the literature we do not confine our attention to the class of quadratic lyapunov functions conversely the basic motivation of this paper is to determine whether there exist other functions that can be conveniently used as candidate lyapunov functions this question has a positive answer the proposed class is that of polyhedral norms or more generally of polyhedral minkowski functionals we show that the class of these functions is universal in the sense that if the problem of ultimately bounding the state in an assigned convex set via state feedback control can be solved via a lyapunov function and a continuous state feedback compensator then it can be solved via a polyhedral lyapunov function and a possibly different continuous control moreover we show that the control can be piecewise linear a numerical technique for constructing the controller is presented for the case in which the uncertainty constraint sets are polyhedral

common lyapunov functions and gradient algorithms this note is concerned with the problem of finding a quadratic common lyapunov function for a large family of stable linear systems we present gradient iteration algorithms which give deterministic convergence for finite system families and probabilistic convergence for infinite families

stability and robustness of planar switching linear systems this paper presents a decision algorithm for the analysis of the stability of a class of planar switched linear systems modeled by hybrid automata the dynamics in each location of the hybrid automaton is assumed to be linear and asymptotically stable the guards on the transitions are hyperplanes in the state space we show that for every pair of an ingoing and an outgoing transition related to a location the exact gain in the norm of the vector induced by the dynamics in that location can be computed these exact gains are used in defining a gain automaton which forms the basis of an algorithmic criterion to determine if a planar hybrid automaton is stable or not

inertial manifolds and finite dimensional reduction for dissipative pdes this paper is devoted to the problem of finite dimensional reduction for parabolic partial differential equations we give a detailed exposition of the classical theory of inertial manifolds as well as various attempts to generalize it based on the so called mane projection theorems the recent counter examples showing that the underlying dynamics may in a sense be infinite dimensional if the spectral gap condition is violated as well as a discussion of the most important open problems are also included

generating functions of switched linear systems analysis computation and stability applications in this paper a unified framework is proposed to study the exponential stability of discrete time switched linear systems and more generally the exponential growth rates of their trajectories under three types of switching rules arbitrary switching optimal switching and random switching it is shown that the maximum exponential growth rates of system trajectories over all initial states under these three switching rules are completely characterized by the radii of convergence of three suitably defined families of functions called the strong the weak and the mean generating functions respectively in particular necessary and sufficient conditions for the exponential stability of the switched linear systems are derived based on these radii of convergence various properties of the generating functions are established and their relations are discussed algorithms for computing the generating functions and their radii of convergence are also developed and illustrated through examples

switched discrete time systems with time varying delays a generalized h2 approach in this paper we investigate the delay dependent performance analysis and control synthesis of a class of linear discrete time hybrid systems with time varying delays we employ hybrid lyapunov krasovskii functionals to establish the desired results under arbitrary switching a generalized h 2 approach is adopted and new parameterized linear matrix inequalities lmis characterization are developed to guarantee the delay dependent asymptotic stability design of generalized h 2 state feedback and dynamic output feedback controllers are subsequently constructed a numerical example is solved in detail to illustrate the theoretical developments

almost bi lipschitz embeddings and almost homogeneous sets this paper is concerned with embeddings of homogeneous spaces into euclidean spaces we show that any homogeneous metric space can be em bedded into a hilbert space using an almost bi lipschitz mapping bi lipschitz to within logarithmic corrections the image of this set is no longer homo geneous but almost homogeneous we therefore study the problem of em bedding an almost homogeneous subset x of a hilbert space h into a finite dimensional euclidean space in fact we show that if x is a compact subset of a banach space and x x is almost homogeneous then for n sufficiently large a prevalent set of linear maps from x into r n are almost bi lipschitz between x and its image we are then able to use the kuratowski embedding of x d into l x to prove a similar result for compact metric spaces

design of stabilizing switching control laws for discrete and continuous time linear systems using piecewise linear lyapunov functions in this paper the stability of switched linear systems is investigated using piecewise linear lyapunov functions in particular we identify classes of switching sequences that result in stable trajectories given a switched linear system we present a systematic methodology for computing switching laws that guarantee stability based on the matrices of the system in the proposed approach we assume that each individual subsystem is stable and admits a piecewise linear lyapunov function based on these lyapunov functions we compose global lyapunov functions that guarantee stability of the switched linear system a large class of stabilizing switching sequences for switched linear systems is characterized by computing conic partitions of the state space the approach is applied to both discrete time and continuous time switched linear systems

further comments on vector norms as lyapunov functions for linear systems this paper deals with the conditions for a vector norm to be a lyapunov function for a linear time invariant system an error in the proof of the related theorem given in hmamed 1994 is pointed out and a counterexample proving that this theorem is generally false is provided it is demonstrated that the considered problem is related to the one of extending a linear operator without increasing its norm

affine characterizations of minimal and mode dependent dwell times for uncertain linear switched systems an alternative approach for minimum and mode dependent dwell time characterization for switched systems is derived while minimum dwell time results require the subsystems to be asymptotically stable mode dependent dwell time results can consider unstable subsystems and dwell times within a possibly unbounded range of values the proposed approach is related to lyapunov looped functionals a new type of functionals leading to stability conditions affine in the system matrices unlike standard results for minimum dwell time these conditions are expressed as infinite dimensional lmis which can be solved using recent polynomial optimization techniques such as sum of squares the specific structure of the conditions is finally utilized in order to derive dwell time stability results for uncertain switched systems several examples illustrate the efficiency of the approach

survey paper set invariance in control the properties of positively invariant sets are involved in many different problems in control theory such as constrained control robustness analysis synthesis and optimization in this paper we provide an overview of the literature concerning positively invariant sets and their application to the analysis and synthesis of control systems

convex liftings based robust control design this paper presents a new approach for control design of constrained linear systems affected by bounded additive disturbances and polytopic uncertainties this method hinges on so called convex liftings which emulate control lyapunov function by providing a constructive framework for optimization based control implementation it will be shown that this method can guarantee the recursive feasibility and robust stability finally a numerical example will be presented to illustrate this method