system analysis via integral quadratic constraints this paper introduces a unified approach to robustness analysis with respect to nonlinearities time variations and uncertain parameters from an original idea by yakubovich 1967 the approach has been developed under a combination of influences from the western and russian traditions of control theory it is shown how a complex system can be described using integral quadratic constraints iqc for its elementary components a stability theorem for systems described by iqcs is presented that covers classical passivity dissipativity arguments but simplifies the use of multipliers and the treatment of causality a systematic computational approach is described and relations to other methods of stability analysis are discussed last but not least the paper contains a summarizing list of iqcs for important types of system components

second order perturbation theory for embedded eigenvalues we study second order perturbation theory for embedded eigenvalues of an abstract class of self adjoint operators using an extension of the mourre theory under assumptions on the regularity of bound states with respect to a conjugate operator we prove upper semicontinuity of the point spectrum and establish the fermi golden rule criterion our results apply to massless pauli fierz hamiltonians for arbitrary coupling

classical dirichlet forms on topological vector spaces the construction of the associated diffusion process given a minimal classical dirichlet form onl 2 e we construct the associated diffusion process heree is a locally convex topological vector space and is a not necessarily quasi invariant probability measure one the construction is carried out under certain assumptions one and which can be easily verified in many examples in particular we explicitly apply our results to time zero and space time quantum fields with or with out cut off

interface fluctuations for the d 1 stochastic ginzburg landau equation with nonsymmetric reaction term we consider a ginzburg landau equation in the interval e 1 e 1 e 0 with neumann boundary conditions perturbed by an additive white noise of strength sqrt varepsilon and reaction term being the derivative of a function which has two equal depth wells at 1 but is not symmetric when e 0 the equation has equilibrium solutions that are increasing and connect 1 with 1 we call them instantons and we study the evolution of the solutions of the perturbed equation in the limit e 0 when the initial datum is close to an instanton we prove that for times that may be of the order of e 1 the solution stays close to some instanton whose center suitably normalized converges to a brownian motion plus a drift this drift is known to be zero in the symmetric case and using a perturbative analysis we show that if the nonsymmetric part of the reaction term is sufficiently small it determines the sign of the drift

positive commutators in non equilibrium quantum statistical mechanics the method of positive commutators developed for zero temperature problems over the last twenty years has been an essential tool in the spectral analysis of hamiltonians in quantum mechanics we extend this method to positive temperatures i e to non equilibrium quantum statistical mechanics we use the positive commutator technique to give an alternative proof of a fundamental property of a certain class of large quantum systems called return to equilibrium this property says that equilibrium states are asymptotically stable if a system is slightly perturbed from its equilibrium state then it converges back to that equilibrium state as time goes to infinity

synchronization of interconnected systems with applications to biochemical networks an input output approach this paper provides synchronization conditions for networks of nonlinear systems the components of the network referred to as compartments in this paper are made up of an identical interconnection of subsystems each represented as an operator in an extended l2 space and referred to as a species the compartments are in turn coupled through a diffusion like term among the respective species the synchronization conditions are provided by combining the input output properties of the subsystems with information about the structure of the network the paper also explores results for state space models as well as biochemical applications the work is motivated by cellular networks where signaling occurs both internally through interactions of species and externally through intercellular signaling the theory is illustrated by providing synchronization conditions for networks of goodwin oscillators

robust stability of feedback systems a geometric approach using the gap metric a geometric framework for robust stabilization of infinite dimensional time varying linear systems is presented the uncertainty of a system is described by perturbations of its graph and is measured in the gap metric necessary and sufficient conditions for robust stability are generalized from the time invariant case an example is given to highlight an important difference between the obstructions which limit the size of a stabilizable gap ball in the time varying and time invariant cases several results on the gap metric and the gap topology are established that are central in a geometric treatment of the robust stabilizability problem in the gap in particular the concept of a graphable subspace is introduced in the paper subspaces that fail to be graphable are characterized by an index condition on a certain semi fredholm operator

robust stability analysis for feedback interconnections of time varying linear systems feedback interconnections of causal linear systems are studied in a continuous time setting the developments include a linear time varying ltv generalization of vinnicombe s nu gap metric and an integral quadratic constraint based robust l2 stability theorem for uncertain feedback interconnections of potentially open loop unstable systems these main results are established in terms of toeplitz wiener hopf and hankel operators and the fredholm index for a class of causal linear systems with the following attributes i a system graph i e subspace of l2 input output pairs admits normalized strong right i e image and left i e kernel representations and ii the corresponding hankel operators are compact these properties are first verified for stabilizable and detectable ltv state space models to initially motivate the abstract formulation and subsequently verified for frequency domain multiplication by constantly proper callier desoer transfer functions in analysis that confirms consistency of the developments with the time invariant theory to conclude the aforementioned robust stability theorem is applied in an illustrative example concerning the feedback interconnection of distributed parameter systems over a network with time varying gains less

a unifying framework for robust synchronization of heterogeneous networks via integral quadratic constraints a general framework for analysing robust synchronization in large scale heterogenous networks is proposed based on the theory of integral quadratic constraints iqcs dynamic agents are represented as linear time invariant single input single output systems the agents exchange information according to a sparse dynamical interconnection operator in order to achieve synchronization where their outputs are steered to the same possibly time varying signal the main technical hindrance to applying iqcs in this context lies with the presence of the marginally stable dynamics which define the trajectory to which the agents outputs synchronize it is shown that by working with conditions defined on modified signal spaces of interest and exploiting the graph structure underlying the connections between the dynamic systems iqc methods can be applied directly to synchronization analysis without recourse to loop transformations which may obscure the inherent structural properties of the multi agent networked systems decentralized and scalable conditions for synchronization are proposed within this setting the iqc framework is demonstrated to unify and generalize some of the existing results in the literature including certain nyquist type consensus certificates for time delay systems moreover it allows the role of feedback in robustness against uncertainty to be better manifested within the context of synchronization

optimal gaussian solutions of nonlinear stochastic partial differential equations we present a linearization procedure of a stochastic partial differential equation for a vector field x i t x t 0 x r d i l n t x i t x b i x t x d x i t x i f i t x here is the laplace beltrami operator inr d and f i t x is a gaussian random field with f i t x f j t x ij t t x x the procedure is a natural extension of the equivalent linearization for stochastic ordinary differential equations the linearized solution is optimal in the sense that the distance between true and approximate solutions is minimal when it is measured by the kullback leibler entropy the procedure is applied to the scalar valued ginzburg landau model in r1 withb 1 z z vz 3 stationary values of mean variance and correlation length are calculated they almost agree with exact ones if 1 24 2 1 4 d 1 1 3 c when c there appear quasistationary states fluctuating around one of the bottoms of the potentialu z b 1 z dz the second moment at the quasistationary states almost agrees with the exact one transient phenomena are also discussed half width at half maximum of a structure function decays liket 1 2 for small t the diffusion term 2 x accelerates the relaxation from the neighborhood of an unstable initial statex 0 x 0

synchronization and transient stability in power networks and nonuniform kuramoto oscillators motivated by recent interest for multiagent systems and smart grid architectures we discuss the synchronization problem for the network reduced model of a power system with nontrivial transfer conductances our key insight is to exploit the relationship between the power network model and a first order model of coupled oscillators assuming overdamped generators possibly due to local excitation controllers a singular perturbation analysis shows the equivalence between the classic swing equations and a nonuniform kuramoto model here nonuniform kuramoto oscillators are characterized by multiple time constants nonhomogeneous coupling and nonuniform phase shifts extending methods from transient stability synchronization theory and consensus protocols we establish sufficient conditions for synchronization of nonuniform kuramoto oscillators these conditions reduce to necessary and sufficient tests for the standard kuramoto model combining our singular perturbation and kuramoto analyses we derive

one boson scattering processes in the massive spin boson model abstract the spin boson model describes a two level quantum system that interacts with a second quantized boson scalar field recently the relation between the integral kernel of the scattering matrix and the resonance in this model has been established in 18 for the case of massless bosons in the present work we treat the massive case on the one hand one might rightfully expect that the massive case is easier to handle since in contrast to the massless case the corresponding hamiltonian features a spectral gap on the other hand it turns out that the non zero boson mass introduces a new complication as the spectrum of the complex dilated free hamiltonian exhibits lines of spectrum attached to every multiple of the boson rest mass energy starting from the ground and excited state energies this leads to an absence of decay of the corresponding complex dilated resolvent close to the real line which in 18 was a crucial ingredient to control the time evolution in the scattering regime with the new strategy presented here we provide a proof of an analogous formula for the scattering kernel as compared to the massless case and use the opportunity to provide the required spectral information by a mourre theory argument combined with a suitable application of the feshbach schur map instead of complex dilation

spectral theory of pauli fierz operators abstract we study spectral properties of pauli fierz operators which are commonly used to describe the interaction of a small quantum system with a bosonic free field we give precise estimates of the location and multiplicity of the singular spectrum of such operators applications of these estimates which will be discussed elsewhere concern spectral and ergodic theory of non relativistic qed our proof has two ingredients the feshbach method which is developed in an abstract framework and mourre theory applied to the operator restricted to the sector orthogonal to the vacuum

on the virial theorem in quantum mechanics we review the various assumptions under which abstract versions of the quantum mechanical virial theorem have been proved we point out a relationship between the virial theorem for a pair of operators h a and the regularity properties of the map we give an example showing that the statement of the virial theorem in cfks is incorrect

synchronization of networks of heterogeneous agents with common nominal behavior this paper deals with the problem of synchronization in networks of heterogeneous agents with common nominal behavior the agents are modeled as possibly nonlinear perturbed versions of a common siso nominal lti operator and they are interconnected via a sparse memoryless interconnection operator coherent with the communication graph underlying the network the network is said to synchronize if the outputs of the agents tend to align along given directions the most important case being consensus or agreement the paper provides a general result based on iqcs that ensures synchronization of the network with robustness w r t uncertainties in the interconnection and in each agent s dynamics scalability issues are discussed in the scenario where the interconnection operator is a constant normal matrix yielding the generalization of the very popular linear consensus algorithm the wide range of applicability of the proposed criterion is shown by providing synchronization conditions in some important examples whenever possible graphical criteria are proposed for checking the required conditions

consensus of heterogeneous lti agents a robust stability criterion is derived for a class of higher order consensus algorithms in distinction to the standard consensus problem where the dynamics of all individual agents are the same and of low dimension the criterion for reaching consensus now depends on the full spectrum of the communication graph and not only the largest nonzero eigenvalue the new consensus criterion is posed as a nyquist like criterion that provides a clear separation between the individual dynamics and the spectrum of the graph our results are illustrated on a formation control problem for a target encircling task

synchronization of networks of nonidentical euler lagrange systems with uncertain parameters and communication delays this paper addresses the problem of synchronizing networks of nonidentical nonlinear dynamical systems described by euler lagrange equations which are assumed fully actuated with their states available for measurement but with unknown parameters the only assumption made on the communication graph is that it is connected moreover the communication is subject to constant time delays which are also unknown the main result of the paper is the construction of an adaptive controller that achieves global full state synchronization i e the difference between the agents positions and velocities asymptotically converges to zero if a desired trajectory for all systems is given a slight modification to the proposed scheme achieves also full state synchronization simulations using a ten robot manipulator network are used to illustrate the performance of the proposed schemes

decay estimates and strichartz estimates of fourth order schrodinger operator abstract we study time decay estimates of the fourth order schrodinger operator h 2 v x in r d for d 3 and d 5 we analyze the low energy and high energy behaviour of resolvent r h z and then derive the jensen kato dispersion decay estimate and local decay estimate for e i t h p a c under suitable spectrum assumptions of h based on jensen kato type decay estimate and local decay estimate we obtain the l 1 l estimate of e i t h p a c in 3 dimension by ginibre argument and also establish the endpoint global strichartz estimates of e i t h p a c for d 5 furthermore using the local decay estimate and the georgescu larenas soffer conjugate operator method we prove the jensen kato type decay estimates for some functions of h

absence of singular continuous spectrum for certain self adjoint operators we give a sufficient condition for a self adjoint operator to have the following properties in a neighborhood of a pointe of its spectrum a its point spectrum is finite b its singular continuous spectrum is empty c its resolvent satisfies a class of a priori estimates

commutators c0 semigroups and resolvent estimates abstract we study the existence and the continuity properties of the boundary values on the real axis of the resolvent of a self adjoint operator h in the framework of the conjugate operator method initiated by mourre we allow the conjugate operator a to be the generator of a c 0 semigroup finer estimates require a to be maximal symmetric and we consider situations where the first commutator h i a is not comparable to h the applications include the spectral theory of zero mass quantum field models

a proof of the abstract limiting absorption principle by energy estimates abstract we give a proof in an abstract setting of various resolvent estimates including the limiting absorption principle starting from a mourre positive commutator estimate using standard energy estimates arguments

quantum electrodynamics of confined nonrelativistic particles abstract we consider a system of finitely many nonrelativistic quantum mechanical electrons bound to static nuclei the electrons are minimally coupled to the quantized electromagnetic field but we impose an ultraviolet cutoff on the electromagnetic vector potential appearing in covariant derivatives and the interactions between the radiation field and electrons localized very far from the nuclei are turned off for a class of hamiltonians we prove exponential localization of bound states establish the existence of a ground state and derive sufficient conditions for its uniqueness furthermore we show that excited bound states of the unperturbed system become unstable and turn into resonances when the electrons are coupled to the radiation field to this end we develop a novel renormalization transformation which acts directly on the space of hamiltonians

nonlinear parabolic stochastic differential equations with additive colored noise on bf r d times bf r _ a regulated stochastic quantization we prove the existence of solutions to the nonlinear parabolic stochastic differential equation partial mathord left vphantom partial partial t right kern nulldelimiterspace partial t delta varphi v varphi eta c for polynomialsv of even degree with positive leading coefficient and c a gaussian colored noise process onr d r when c is colored enough that the gaussian solution to the linear problem has holder continuous covariance the nongaussian processes are almost surely realized by continuous functions uniqueness regularity properties asymptotic perturbation expansions and nonperturbative fluctuation bounds are obtained for the infinite volume processes these equations are a cutoff version of the parisi wu stochastic quantization procedure forp d models and the results of this paper rigorously establish the nonperturbative nature of regularization via modification of the noise process in the limit c gaussian white noise we find that the asymptotic expansion and the rigorous bounds agree for processes corresponding to the regulated stochastic quantization of super renormalizable and small coupling strictly renormalizable scalar field theories and disagree for nonrenormalizable models

on the existence and uniqueness of ground states of a generalized spin boson model a generalization of the standard spin boson model is considered the hamiltonianh of the model with a coupling parameter racts in the tensor product h fbof a hilbert space h and the boson symmetric fock space fboverl2 r the existence and uniqueness of ground states ofh are investigated the degeneracy of the ground states is also discussed the results obtained arenonperturbative the methods used are those of constructive quantum field theory and the min max principle an exact asymptotic formula for the ground state energy ofh as is also established

regularity of eigenstates in regular mourre theory abstract the present paper gives an abstract method to prove that possibly embedded eigenstates of a self adjoint operator h lie in the domain of the kth power of a conjugate operator a conjugate means here that h and a have a positive commutator locally near the relevant eigenvalue in the sense of mourre the only requirement is c k 1 a regularity of h regarding integer k our result is optimal under a natural boundedness assumption of the multiple commutators we prove that the eigenstate dilated by exp i a is analytic in a strip around the real axis in particular the eigenstate is an analytic vector with respect to a natural applications are dilation analytic systems satisfying a mourre estimate where our result can be viewed as an abstract version of a theorem due to balslev and combes 1971 3 as a new application we consider the massive spin boson model

spectral theory of massless pauli fierz models we study the spectral theory of massless pauli fierz models using an extension of the mourre method we prove the local finiteness of point spectrum and a limiting absorption principle away from the eigenvalues for an arbitrary coupling constant in addition we show that the expectation value of the number operator is finite on all eigenvectors

positive commutators fermi golden rule and the spectrum of zero temperature pauli fierz hamiltonians abstract we perform the spectral analysis of a zero temperature pauli fierz system for small coupling constants under the hypothesis of fermi golden rule we show that the embedded eigenvalues of the uncoupled system disappear and establish a limiting absorption principle above this level of energy we rely on a positive commutator approach introduced by skibsted and pursued by georgescu gerard moller we complete some results obtained so far by derezinski jaksic on one side and by bach frohlich sigal soffer on the other side

large scale heterogeneous networks the davis wielandt shell and graph separation we consider a large scale network of interconnected heterogeneous dynamical components scalable stability conditions are derived that involve the input output properties of individual subsystems and the interconnection matrix the analysis is based on the davis wielandt shell a higher dimensional version of the numerical range with important convexity properties this can be used to allow heterogeneity in the agent dynamics while relaxing normality and symmetry assumptions on the interconnection matrix the results include small gain and passivity approaches as special cases with the three dimensional shell shown to be inherently connected with corresponding graph separation arguments

frequency analysis and norms of distributed spatially periodic systems we investigate several fundamental aspects of the theory of linear distributed systems with spatially periodic coefficients we develop a spatial frequency domain representation analogous to the lifted or frequency response operator representation for linear time periodic systems using this representation we introduce the notion of the h2 norm for this class of systems and provide algorithms for its computation a stochastic interpretation of the h2 norm is given in terms of spatially cyclostationary random fields and spectral correlation density operators when the periodic coefficients are viewed as feedback modifications of spatially invariant systems we show how they can stabilize or destabilize the original systems in a manner analogous to vibrational control or parametric resonance in time periodic systems two examples from physics are provided to illustrate the main results

a two space dimensional semilinear heat equation perturbed by gaussian white noise a two space dimensional heat equation perturbed by a white noise in a bounded volume is considered the equation is perturbed by a non linearity of the type f au where means wick re ordering with respect to the free solution a are small parameters u denotes a solution f is the fourier transform of a complex measure with compact support

delay robustness in consensus problems we investigate the robustness of consensus schemes for linear multi agent systems mas to feedback delays to achieve this we develop a unified framework that considers linear mas models with different feedback delays e g affecting only the neighbor s output or affecting both the agent s own and its neighbors output this framework has the advantage of providing scalable simple and accurate set valued conditions for consensus using these set valued conditions previous results on consensus in mas with delays can be recovered and generalized moreover we use them to derive conditions for the convergence rate of single integrator mas with feedback delays finally building on this framework we propose a scalable delay dependent design algorithm for consensus controllers for a large class of linear mas