on a common quadratic lyapunov function for widely distant systems a sufficient condition concerning the construction of a common quadratic lyapunov function for widely distant linear systems is presented the condition is set on a mutual relation matrix whose size is just equal to the number of the coefficient matrices

hamiltonian dynamics of vortex and magnetic lines in hydrodynamic type systems vortex line and magnetic line representations are introduced for a description of flows in ideal hydrodynamics and magnetohydrodynamics mhd respectively for incompressible fluids it is shown with the help of this transformation that the equations of motion for vorticity omega and magnetic field follow from a variational principle by means of this representation it is possible to integrate the hydrodynamic type system with the hamiltonian h integral omega dr and some other systems it is also demonstrated that these representations allow one to remove from the noncanonical poisson brackets defined in the space of divergence free vector fields the degeneracy connected with the vorticity frozenness for the euler equation and with magnetic field frozenness for ideal mhd for mhd a new weber type transformation is found it is shown how this transformation can be obtained from the two fluid model when electrons and ions can be considered as two independent fluids the weber type transformation for ideal mhd gives the whole lagrangian vector invariant when this invariant is absent this transformation coincides with the clebsch representation analog introduced by v e zakharov and e a kuznetsov dokl ajad nauk 194 1288 1970 sov phys dokl 15 913 1971

a kalman decomposition for possibly controllable uncertain linear systems this paper considers the structure of uncertain linear systems building on concepts of robust unobservability and possible controllability the paper presents a new geometric characterization of the possibly controllable states when combined with previous geometric results on robust unobservability the results of this paper lead to a general kalman type decomposition for uncertain linear systems which can be applied to the problem of obtaining reduced order uncertain system models

minimality and local state decompositions of a nonlinear state space realization using energy functions in this paper a set of sufficient conditions is developed in terms of controllability and observability functions under which a given state space realization of a formal power series is minimal specifically it is shown that positivity of these functions in addition to a stability requirement and a few technical conditions implies minimality using the nonlinear analogue of the kalman decomposition connections are then established between minimality singular value functions balanced realizations and various notions of reachability and observability for nonlinear systems

a general theorem on local controllability we prove a general sufficient condition for local controllability of a nonlinear system at an equilibrium point earlier results of brunovsky hermes jurdjevic crouch and byrnes sussmann and grossmann are shown to be particular cases of this result also a number of new sufficient conditions are obtained all these results follow from one simple general principle namely that local controllability follows whenever brackets with certain symmetries can be neutralized in a suitable way by writing them as linear combinations of brackets of a lower degree both the class of symmetries and the definition of degree can be chosen to suit the problem

a converse lyapunov theorem for nonlinear switched systems abstract in this paper we present a converse lyapunov theorem for uniform asymptotic stability of switched nonlinear systems its proof is a simple consequence of some results on converse lyapunov theorems for systems with bounded disturbances obtained by lin et al siam j control optim 34 1996 124 160 once an association of the switched system with a nonlinear system with disturbances is established

balanced realization and model order reduction for nonlinear systems based on singular value analysis this paper discusses balanced realization and model order reduction for both continuous time and discrete time general nonlinear systems based on singular value analysis of the corresponding hankel operators singular value analysis clarifies the gain structure of a given nonlinear operator here it is proved that singular value analysis of smooth hankel operators defined on hilbert spaces can be characterized by simple equations in terms of their states a balanced realization and model order reduction procedure is derived based on it and several important properties such as stability balanced form hankel norm controllability and observability of the original system are preserved the work improves the earlier results of k fujimoto and j m a scherpen ieee trans automat control 50 2005 pp 2 18 and then continues with new balancing and model reduction results

non local symmetries and conservation laws for one dimensional gas dynamics equations the theory on the generation of conservation laws symmetry conditions and construction of conservation laws for differential equations preprint for systems of partial differential equations by the use of lie backlund symmetries is generalised to include the use of non local symmetries it is shown how the action of a symmetry on the appropriate conservation laws can be utilised to extend the symmetry generator with respect to a covering system systems of differential equations that describe one dimensional gas flow are used to demonstrate the direct calculation of new non local symmetries as well as the generation of new conservation laws also a solution that is invariant under a non local symmetry is found

scaling symmetries conservation laws and action principles in one dimensional gas dynamics scaling symmetries of the planar one dimensional gas dynamic equations with adiabatic index are used to obtain lagrangian and eulerian conservation laws associated with the symmetries the known eulerian symmetry operators for the scaling symmetries are converted to the lagrangian form in which the eulerian spatial position of the fluid element is given in terms of the lagrangian fluid labels conditions for a linear combination of the three scaling symmetries to be a divergence or variational symmetry of the action are established the corresponding lagrangian and eulerian form of the conservation laws are determined by application of noether s theorem a nonlocal conservation law associated with the scaling symmetries is obtained by applying a nonlocal symmetry operator to the scaling symmetry conserved vector an action principle incorporating known conservation laws using lagrangian constraints is developed noether s theorem for the constrained action principle gives the same formulas for the conserved vector as the classical noether theorem except that the lie symmetry vector field now includes the effects of nonlocal potentials noether s theorem for the constrained action principle is used to obtain nonlocal conservation laws the scaling symmetry conservation laws only apply for special forms of the entropy of the gas

multi symplectic lagrangian one dimensional gas dynamics the equations of lagrangian ideal one dimensional compressible gas dynamics are written in a multi symplectic form using the lagrangian mass coordinate m and time t as independent variables and in which the eulerian position of the fluid element x x m t is one of the dependent variables this approach differs from the eulerian multi symplectic approach using clebsch variables lagrangian constraints are used to specify equations for xm xt and st consistent with the lagrangian map where s is the entropy of the gas we require st 0 corresponding to advection of the entropy s with the flow we show that the lagrangian hamiltonian equations are related to the de donder weyl multi momentum formulation the pullback conservation laws and the symplecticity conservation laws are discussed the pullback conservation laws correspond to invariance of the action with respect to translations in time energy conservation and translations in m in noether s theorem the conservation law due to m translatio

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

model reduction by moment matching for linear and nonlinear systems the model reduction problem for single input single output linear and nonlinear systems is addressed using the notion of moment a re visitation of the linear theory allows to obtain novel results for linear systems and to develop a nonlinear enhancement of the notion of moment this in turn is used to pose and solve the model reduction problem by moment matching for nonlinear systems to develop a notion of frequency response for nonlinear systems and to solve model reduction problems in the presence of constraints on the reduced model connections between the proposed results projection methods the covariance extension problem and interpolation theory are presented finally the theory is illustrated by means of simple worked out examples and case studies

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

an extended procedure for finding exact solutions of partial differential equations arising from potential symmetries applications to gas dynamics lie point symmetries and nonlocal symmetries of partial differential equation pde systems are widely used for construction of exact invariant solutions in this paper we describe an extended algorithmic procedure that for a given nonlocal potential symmetry can yield additional exact solutions which cannot be found using the usual algorithm in particular such additional solutions are exact solutions of the given pde system but are not invariant solutions of the corresponding potential system as an example we consider a tree of nonlocally related pde systems for lagrange planar gas dynamics equations and classify its nonlocal symmetries for an ideal polytropic gas for two different nonlocal symmetries of the lagrange system we demonstrate that the extended method yields wider classes of exact solutions than the usual method

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

nonlinear input normal realizations based on the differential eigenstructure of hankel operators this paper investigates the differential eigenstructure of hankel operators for nonlinear systems first it is proven that the variational system and the hamiltonian extension with extended input and output spaces can be interpreted as the ga spl circ teaux differential and its adjoint of a dynamical input output system respectively second the ga spl circ teaux differential is utilized to clarify the main result the differential eigenstructure of the nonlinear hankel operator which is closely related to the hankel norm of the original system third a new characterization of the nonlinear extension of hankel singular values are given based on the differential eigenstructure finally a balancing procedure to obtain a new input normal output diagonal realization is derived the results in this paper thus provide new insights to the realization and balancing theory for nonlinear systems

observability functions for linear and nonlinear systems a generalization of the zero state observability function is considered for nonlinear systems the linear time invariant case is considered as an application in model reduction problems

brief paper a notion of possible controllability for uncertain linear systems with structured uncertainty this paper introduces a notion of possible controllability for a class of uncertain linear systems with structured uncertainty described by averaged integral quadratic constraints this notion relates to the question of when a state is controllable for some possible value of the uncertainty the notion of possible controllability is motivated by a desire to extend the theory of minimal realization for linear time invariant systems to the case of uncertain systems with structured uncertainty

multisymplectic conservation laws for differential and differential difference equations many well known partial differential equations can be written as multisymplectic systems such systems have a structural conservation law from which scalar conservation laws can be derived these c

vorticity and symplecticity in lagrangian fluid dynamics the relationship between potential vorticity pv and the symplectic form is explored for the shallow water equations governing lagrangian particle paths starting with the symplectic form the pv is found by the pull back operation to the reference space at first sight the encoding of pv in the symplectic form appears to be independent of the particle relabelling symmetry the analysis is carried a step further in two ways using the conservation of symplecticity as a starting point the fluxes of symplecticity arise as differential forms and a complete pull back of the flux forms leads to a geometric description of pv conservation secondly symmetry methods are used to give a rigorous connection between particle relabelling symplecticity and pv conservation generalizations of these issues to semi geostrophic flow and three dimensional lagrangian fluid flows and the connection with ertel s theorem are also discussed

hamiltonian realizations of nonlinear adjoint operators this paper addresses the issue of state space realizations for nonlinear adjoint operators in particular the relationships between nonlinear hilbert adjoint operators hamiltonian extensions and port controlled hamiltonian systems are established then characterizations of the adjoints of controllability observability and hankel operators are derived from this analysis the state space realizations of such adjoint operators provide new insights on singular value analysis and duality issues in nonlinear control systems theory finally a duality between the controllability and observability energy functions is proved

on the nonuniqueness of singular value functions and balanced nonlinear realizations the notion of balanced realizations for nonlinear state space model reduction was first introduced in 1993 analogous to the linear case the so called singular value functions of a system describe the relative importance of each state component from an input output point of view in this paper it is shown that the usual procedure for nonlinear balancing has some interesting ambiguities that do not occur in the linear case specifically it appears that the singular value functions as currently defined are dependent on a particular factorization of the observability function it is shown by example that in a fixed coordinate frame this factorization is not unique and thus other distinct definitions for the singular value functions and balanced realizations are possible one method relating singular value functions from different factorizations is presented

stability of switched systems a lie algebraic condition we present a sucient condition for asymptotic stability of a switched linear system in terms of the lie algebra generated by the individual matrices namely if this lie algebra is solvable then the switched system is exponentially stable for arbitrary switching in fact we show that any family of linear systems satisfying this condition possesses a quadratic common lyapunov function we also discuss the implications of this result for switched nonlinear systems c 1999 elsevier science b v all rights reserved

framework for nonlocally related partial differential equation systems and nonlocal symmetries extension simplification and examples any partial differential equation pde system can be effectively analyzed through consideration of its tree of nonlocally related systems if a given pde system has n local conservation laws then each conservation law yields potential equations and a corresponding nonlocally related potential system moreover from these n conservation laws one can directly construct 2n 1 independent nonlocally related systems by considering these potential systems individually n singlets in pairs n n 1 2couplets taken all together one n plet in turn any one of these 2n 1 systems could lead to the discovery of new nonlocal symmetries and or nonlocal conservation laws of the given pde system moreover such nonlocal conservation laws could yield further nonlocally related systems a theorem is proved that simplifies this framework to find such extended trees by eliminating redundant systems the planar gas dynamics equations and nonlinear telegraph equations are used as illustrative examples many new local

hankel singular value functions from schmidt pairs for nonlinear input output systems this paper presents three results in singular value analysis of hankel operators for nonlinear input output systems first the notion of a schmidt pair is defined for a nonlinear hankel operator this makes it possible to define a hankel singular value function from a purely input output point of view and without introducing a state space setting however if a state space realization is known to exist then a set of sufficient conditions is given for the existence of a schmidt pair and the state space provides a convenient representation of the corresponding singular value function finally it is shown that in a specific coordinate frame it is possible to relate this new singular value function definition to the original state space notion used to describe nonlinear balanced realizations

framework for potential systems and nonlocal symmetries algorithmic approach an algorithmic framework is presented to find an extended tree of nonlocally related systems for a given system of differential equations des each system in an extended tree is equivalent in the sense that the solution set for any system in a tree can be found from the solution set for any other system in the tree useful conservation laws play an essential role in the construction of an extended tree a useful conservation law yields potential variables and equivalent nonlocally related potential systems and subsystems for any given system nonlocal symmetries for a given system of des can arise from any system in its extended tree we construct extended trees for the systems of planar gas dynamics and nonlinear telegraph equations and in both cases obtain new nonlocal symmetries more importantly due to the equivalence of solution sets any coordinate independent method of analysis qualitative numerical perturbation etc can be applied to any system within the tree and may yield simpler computations and or results that cannot be obtained when the method is directly applied to the given system

on a new class of completely integrable nonlinear wave equations ii multi hamiltonian structure the multi hamiltonian structure of a class of nonlinear wave equations governing the propagation of finite amplitude waves is discussed infinitely many conservation laws had earlier been obtained for these equations starting from a primary hamiltonian formulation of these equations the necessary and sufficient conditions for the existence of bi hamiltonian structure are obtained and it is shown that the second hamiltonian operator can be constructed solely through a knowledge of the first hamiltonian function the recursion operator which first appears at the level of bi hamiltonian structure gives rise to an infinite sequence of conserved hamiltonians it is found that in general there exist two different infinite sequences of conserved quantities for these equations the recursion relation defining higher hamiltonian structures enables one to obtain the necessary and sufficient conditions for the existence of the k 1 st hamiltonian operator which depends on the kth hamiltonian function the infinite sequence of conserved hamiltonians are common to all the higher hamiltonian structures the equations of gas dynamics are discussed as an illustration of this formalism and it is shown that in general they admit tri hamiltonian structure with two distinct infinite sets of conserved quantities the isothermal case of 1 is an exceptional one that requires separate treatment this corresponds to a specialization of the equations governing the expansion of plasma into vacuum which will be shown to be equivalent to poisson s equation in nonlinear acoustics

multiplicity of phase transitions and mean field criticality on highly non amenable graphs we consider independent percolation ising and potts models and the contact process on infinite locally finite connected graphs

symmetry groups and similarity solutions of mhd equations a similarity analysis of the nonlinear nonstationary real mhd equations is presented for several cases of real mhd the complete lie algebra of infinitesimal symmetries is calculated using subalgebras of this lie algebra reduced systems for similarity solutions can be obtained some similarity solutions of the ideal mhd equations are discussed

model reduction by differential balancing based on nonlinear hankel operators in this paper we construct balancing theory for nonlinear systems in the contraction framework first we define two novel controllability and observability functions via prolonged systems we analyze their properties in relation to controllability and observability and use them for so called differential balancing and its application to model order reduction one of the main contribution of this paper is showing that differential balancing has close relationships with the frechet derivative of the nonlinear hankel operator inspired by 3 we provide a generalization in order to have a computationally more feasible method moreover error bounds for model reduction by generalized balancing are provided

balanced realizations near stable invariant manifolds it is shown that the notion of balancing a state space realization about a stable isolated equilibrium point can be generalized to systems possessing stable invariant regular submanifolds of arbitrary dimension for example a stable limit cycle ycle

minimal state space realization in linear system theory an overview we give a survey of the results in connection with the minimal state space realization problem for linear time invariant systems we start with a brief historical overview and a short introduction to linear system theory next we present some of the basic algorithms for the reduction of nonminimal state space realizations and for the minimal state space realization of infinite or finite sequences of markov parameters of linear time invariant systems finally we discuss some extensions of this problem to other classes of systems and point out some related problems

a converse lyapunov theorem for a class of dynamical systems which undergo switching the authors investigate the stability of a system in which the dynamics at any instant in time will follow one of a fixed set of vector fields they allow switching between members of the family of vector fields to be completely random the focus of this paper is to prove a converse lyapunov theorem for this class of systems

nonsquare spectral factorization for nonlinear control systems this paper considers nonsquare spectral factorization of nonlinear input affine state space systems in continuous time more specifically we obtain a parametrization of nonsquare spectral factors in terms of invariant lagrangian submanifolds and associated solutions of hamilton jacobi inequalities this inequality is a nonlinear analogue of the bounded real lemma and the control algebraic riccati inequality by way of an application we discuss an alternative characterization of minimum and maximum phase spectral factors and introduce the notion of a rigid nonlinear system

computation of nonlinear balanced realization and model reduction based on taylor series expansion in this paper a computational algorithm for nonlinear balanced realization and model reduction based on taylor series expansion is proposed this algorithm requires recursive computations with respect to the order of the taylor series in which we need to solve linear equations with unknown parameters in each step furthermore the proposed method is applied to a double pendulum system some numerical simulations demonstrate the effectiveness of the proposed algorithm

poisson brackets and clebsch representations for magnetohydrodynamics multifluid plasmas and elasticity abstract poisson brackets are constructed by the same mathematical procedure for three physical theories ideal magnetohydrodynamics multifluid plasmas and elasticity each of these brackets is given a simple lie algebraic interpretation moreover each bracket is induced to physical space by use of a canonical poisson bracket in the space of clebsch potentials which are constructed for each physical theory by the standard procedure of constrained lagrangians

fliess operators on lp spaces convergence and continuity fliess operators as input output mappings are particularly useful in a number of fundamental problems concerning nonlinear realization theory in the classical analysis of these operators certain growth conditions on the coe3cients in their series representations insure uniform and absolute convergence provided every input is uniformly bounded by some 4xed upperbound in some emerging applications however it is more natural to consider other classes of inputs in this paper lp function spaces are considered in particular it is shown that the classic growth conditions also provide su3cient conditions for convergence and continuity when the admissible inputs are from a ball in lp t0 t 0 t where t is bounded and p 1 in addition stronger global growth conditions are given that apply even for the case where t is unbounded when the coe3cients of a fliess operator have a state space representation it is shown that the state space model will always locally realize the correspondinginput output map on lp t0 t 0 t for su3ciently small t 0 if certain well posedness conditions are satis4ed then the state space model will globally realized the input output mapping for unbounded t when the coe3cients satisfy the global growth condition c 2002 elsevier science b v all rights reserved

pbh tests for nonlinear systems abstract recently concepts of nonlinear eigenvalues and eigenvectors are introduced in this paper we establish connections between the nonlinear eigenvalues and nonlinear accessibility observability in particular we provide a generalization of popov belevitch hautus pbh controllability observability test to nonlinear accessibility observability our pbh accessibility observability test is equivalent to the accessibility observability rank condition under a stranded assumption

branching random walks and contact processes on homogeneous trees branching random walks and contact processes on the homogeneous tree in which each site has d 1 neighbors have three possible types of behavior for d 2 local survival local extinction with global survival and global extinction for branching random walks we show that if there is local extinction then the probability that an individual ever has a descendent at a site n units away from that individual s location is at most d n 2 while if there is global extinction this probability is at most d n next we consider the structure of the set of invariant measures with finite intensity for the system and see how this structure depends on whether or not there is local and or global survival these results suggest some problems and conjectures for contact processes on trees we prove some and leave others open in particular we prove that for some values of the infection parameter there are nontrivial invariant measures which have a density tending to zero in all directions and hence are different from those constructed by durrett and schinazi in a recent paper