gravity coupled with matter and the foundation of non commutative geometry we first exhibit in the commutative case the simple algebraic relations between the algebra of functions on a manifold and its infinitesimal length elementds its unitary representations correspond to riemannian metrics and spin structure whileds is the dirac propagatords x x d 1 whered is the dirac operator we extend these simple relations to the non commutative case using tomita s involutionj we then write a spectral action the trace of a function of the length element which when applied to the non commutative geometry of the standard model will be shown cc to give the sm lagrangian coupled to gravity the internal fluctuations of the non commutative geometry are trivial in the commutative case but yield the full bosonic sector of sm with all correct quantum numbers in this slightly non commutative case the group of local gauge transformations appears spontaneously as a normal subgroup of the diffeomorphism group

characterizations of input to state stability for infinite dimensional systems we prove characterizations of input to state stability iss for a large class of infinite dimensional control systems including some classes of evolution equations over banach spaces time delay systems ordinary differential equations ode and switched systems these characterizations generalize well known criteria of iss proved by sontag and wang for ode systems for the special case of differential equations in banach spaces we prove even broader criteria for iss and apply these results to show that under some mild restrictions the existence of a noncoercive iss lyapunov functions implies iss we introduce the new notion of strong iss siss that is equivalent to iss in the ode case but is strictly weaker than iss in the infinite dimensional setting and prove several criteria for the siss property at the same time we show by means of counterexamples that many characterizations which are valid in the ode case are not true for general infinite dimensional systems

zeta functions on a product of einstein manifolds and the multiplicative anomaly the global additive and multiplicative properties of laplace type operators acting on irreducible rank 1 symmetric spaces are considered the explicit form of the zeta function on product spaces and of the multiplicative anomaly is derived

functional determinants in higher derivative lagrangian theories motivated by the considerable success of alternative theories of gravity we consider the toy model of a higher derivative lagrangian theory which reduces in the gaussian approximation to the pais uhlenbeck oscillator studied in a recent paper by hawking and hertog phys rev d 65 103515 2002 its euclidean path integral is studied with a certain detail and a pedagogical derivation of the propagator which makes use of a theorem due to forman is consequently proposed

zeta function regularization the multiplicative anomaly and the wodzicki residue the multiplicative anomaly associated with the zeta function regularized determinant is computed for the laplace type operators v1 v1 and v2 v2 with v1 v2 constant in a d dimensional compact smooth manifold md making use of several results due to wodzicki and by direct calculations in some explicit examples it is found that the multiplicative anomaly is vanishing for d odd and for d 2 an application to the one loop effective potential of the o 2 self interacting scalar model is outlined

a generalization of the structured singular value and its application to model validation the structured singular value spl mu gives a means of assessing the stability and performance of a system under a class of norm bounded structured perturbations this paper introduces a generalization of spl mu which guarantees stability with respect to perturbations where some elements of the perturbation structure are bounded from above and some are bounded from below this generalization is used to solve the frequency domain robust control model validation problem model validation determines whether or not there exist a model in the model set and a bounded noise signal that are consistent with the experimental input output measurements a computable convex upper bound to the generalized spl mu problem is presented and the relationship between the generalized spl mu problem and the standard spl mu and model validation problems is discussed

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

compensating actuator and sensor dynamics governed by diffusion pdes for possibly unstable ode systems with actuator delay predictor based infinite dimensional feedback can compensate for actuator delay of arbitrary length and achieve stabilization we extend this concept to another class of pde ode cascades where the infinite dimensional part of the plant is of diffusive rather than convective type we derive predictor like feedback laws and observers with explicit gain kernels the gain kernels involve second order matrix exponentials of the system matrix of the ode plant which is the result of the second order in space character of the actuator sensor dynamics the construction of the kernel functions is performed using the continuum version of the backstepping method robustness to small perturbations in the diffusion coefficient is proved

the action functional in non commutative geometry we establish the equality between the restriction of the adler manin wodzicki residue or non commutative residue to pseudodifferential operators of order n on ann dimensional compact manifoldm with the trace which j dixmier constructed on the macaev ideal we then use the latter trace to recover the yang mills interaction in the context of non commutative differential geometry

forms on vector bundles over hyperbolic manifolds and the trace anomaly we study gauge theories based on abelian p forms on real compact hyperbolic manifolds an explicit formula for the trace anomaly corresponding to skew symmetric tensor fields is obtained by using zeta function regularization and the trace tensor kernel formula explicit exact and numerical values of the anomaly for p forms of order up to p 4 in spaces of dimension up to n 10 are then calculated

optimal robustness in the gap metric the application of the gap metric to robust stabilization of feedback systems is considered in particular a solution to the problem of robustness optimization in the gap metric is presented the problem of robust stabilization under simultaneous plant controller perturbations is addressed and the least amount of combined plant controller uncertainty measured by the gap metric that can cause instability of a nominally stable feedback system is determined included are a detailed summary of the main properties of the gap metric and the introduction of a dual metric called the t gap metric a key contribution of this study is to show that the problem of robustness optimization in the gap metric is equivalent to robustness optimization for normalized coprime factor perturbations this settles the question as to whether maximizing allowable coprime factor uncertainty corresponds to tolerating the largest ball of uncertainty in a well defined metric

local input to state stability characterizations and counterexamples abstract we show that a nonlinear locally uniformly asymptotically stable infinite dimensional system is automatically locally input to state stable liss provided the nonlinearity possesses some sort of uniform continuity with respect to external inputs also we prove that liss is equivalent to existence of a liss lyapunov function we show by means of a counterexample that if this uniformity is not present then the equivalence of local asymptotic stability and local iss does not hold anymore using a modification of this counterexample we show that in infinite dimensions a uniformly globally asymptotically stable at zero globally stable and locally iss system possessing an asymptotic gain property does not have to be iss in contrast to finite dimensional case

robust stabilization of normalized coprime factor plant descriptions with h sub infinity bounded uncertainty the problem of robustly stabilizing a family of linear systems is explicitly solved in the case where the family is characterized by h sub infinity bounded perturbations to the numerator and denominator of the normalized left coprime factorization of a nominal system this problem can be reduced to a nehari extension problem directly and gives an optimal stability margin all controllers satisfying a suboptimal stability margin are characterized and explicit state space formulas are given

foundations of a bicoprime factorization theory bicoprime factorizations bcfs are a generalization of the well known coprime factorizations commonly used in control theory however they have received negligible attention from the academic community so far this technical note lays the foundations of a bcf theory the theory is built from the ground up starting with the basic characteristics of such factorizations before moving on to state space parameterizations of bcfs and internal stability some advantages of bcfs are outlined including the possibility of reduced dimension internal stability tests an uncertainty structure induced by bcfs is also examined and the associated robust stability analysis tests provided in multiple instances it is shown how coprime factor results have their roots in the more abstract and more general bcfs

functional determinants for general sturm liouville problems simple and analytically tractable expressions for functional determinants are known to exist for many cases of interest we extend the range of situations for which these hold to cover systems of self adjoint operators of the sturm liouville type with arbitrary linear boundary conditions the results hold whether or not the operators have negative eigenvalues the physically important case of functional determinants of operators with a zero mode but where that mode has been extracted is studied in detail for the same range of situations as when no zero mode exists the method of proof uses the properties of generalized zeta functions the general form of the final results is the same for the entire range of problems considered

the graph metric for unstable plants and robustness estimates for feedback stability in this paper a graph metric is defined that provides a measure of the distance between unstable multivariable plants the graph metric induces a graph topology on unstable plants which is the weakest possible topology in which feedback stability is robust using the graph metric it is possible to derive estimates for the robustness of feedback stability without assuming that the perturbed and unperturbed plants have the same number of rhp poles if the perturbed and unperturbed systems have the same rhp poles then it is possible to obtain necessary and sufficient conditions for robustness with respect to a given class of perturbations as an application of these results the design of stabilizing controllers for unstable singularly perturbed systems is studied finally the relationship of the graph metric to the gap metric introduced by zames and el sakkary is studied in detail in particular it is shown that the robustness results of zames and el sakkary do not enable one to conclude the causality of the perturbed system whereas the present results do

frequency domain uncertainty and the graph topology a new metric on linear time invariant systems is defined this metric is no greater than the gap metric and is in fact the smallest metric for which a certain robust stabilization result holds unlike other known metrics which induce the graph topology it has a clear frequency response interpretation this allows questions regarding robustness in the face of parametric uncertainty to be considered in terms of this metric

construction of lyapunov functions for interconnected parabolic systems an iiss approach this paper is devoted to two issues the first one is to provide lyapunov based tools to establish integral input to state stability iiss and input to state stability iss for some classes of nonlinear parabolic equations the other one is to provide a stability criterion for interconnections of iiss parabolic systems the results addressing the former problem allow us to overcome obstacles arising in tackling the latter one the results for the latter problem are a small gain condition and a formula of lyapunov functions which can be constructed for interconnections whenever the small gain condition holds it is demonstrated that for interconnections of partial differential equations the choice of a right state and input spaces is crucial in particular for iiss subsystems which are not iss as illustrative examples stability of two highly nonlinear reaction diffusion systems is established by the proposed small gain criterion

functional determinants for radial operators we derive simple new expressions in various dimensions for the functional determinant of a radially separable partial differential operator thereby generalizing the one dimensional result of gel fand and yaglom to higher dimensions we use the zeta function formalism and the results agree with what one would obtain using the angular momentum cutoff method based on radial wkb the final expression is numerically equal to an alternative expression derived in a feynman diagrammatic approach but is considerably simpler

observer design for a class of nonlinear ode pde cascade systems abstract the problem of state observation is addressed for nonlinear systems that can be modeled by an ode pde series association the ode subsystem assumes a triangular structure while the pde element is of heat diffusion type the aim is to accurately estimate online the state vector of the ode subsystem and the distributed state of the pde element one major difficulty is that the state observation must only rely on the global system output i e the pde state at the terminal boundary in particular the connection point between the ode and the pde blocks is not accessible to measurements the observation problem is dealt with by designing a high gain type observer sufficient conditions involving the pde domain length are formally established that ensure the observer exponential convergence

control of homodirectional and general heterodirectional linear coupled hyperbolic pdes research on stabilization of coupled hyperbolic pdes has been dominated by the focus on pairs of counter convecting heterodirectional transport pdes with distributed local coupling and with controls at one or both boundaries a recent extension allows stabilization using only one control for a system containing an arbitrary number of coupled transport pdes that convect at different speeds against the direction of the pde whose boundary is actuated in this paper we present a solution to the fully general case in which the number of pdes in either direction is arbitrary and where actuation is applied on only one boundary to all the pdes that convect downstream from that boundary to solve this general problem we solve as a special case the problem of control of coupled homodirectional hyperbolic linear pdes where multiple transport pdes convect in the same direction with arbitrary local coupling our approach is based on pde backstepping and yields solutions to stabilization by both full state and observer based output feedback and trajectory tracking problems

iss in different norms for 1 d parabolic pdes with boundary disturbances for one dimensional parabolic partial differential equations with disturbances at both boundaries and distributed disturbances we provide input to state stability iss estimates in various norms due to the lack of an iss lyapunov functional for boundary disturbances the proof methodology uses i an eigenfunction expansion of the solution and ii a finite difference scheme the iss estimate for the sup norm leads to a refinement of the well known maximum principle for the heat equation finally the obtained results are applied to quasi static thermoelasticity models that involve nonlocal boundary conditions small gain conditions that guarantee the global exponential stability of the zero solution for such models are derived

stabilization of a system of n 1 coupled first order hyperbolic linear pdes with a single boundary input we solve the problem of stabilization of a class of linear first order hyperbolic systems featuring n rightward convecting transport pdes and one leftward convecting transport pde we design a controller which requires a single control input applied on the leftward convecting pde s right boundary and an observer which employs a single sensor on the same pde s left boundary we prove exponential stability of the origin of the resulting plant observer controller system in the spatial l2 sense

zeta functions formulas and applications abstract the existence conditions of the zeta function of a pseudodifferential operator and the definition of determinant thereby obtained are reviewed as well as the concept of multiplicative anomaly associated with the determinant and its calculation by means of the wodzicki residue exponentially fast convergent formulas valid in the whole of the complex plane and yielding the pole positions and residua that extend the ones by chowla and selberg for the epstein zeta function quadratic form and by barnes affine form are then given after briefly recalling the zeta function regularization procedure in quantum field theory some applications of these expressions in physics are described

exponential stability analysis of sampled data ode pde systems and application to observer design a small gain approach is presented for analyzing exponential stability of a class of dynamical hybrid systems the systems considered in the paper are composed of finite dimensional dynamics represented by a linear ordinary differential equation ode and infinite dimensional dynamics described by a parabolic partial differential equation pde exponential stability is established under conditions involving the maximum allowable sampling period masp this new stability result is shown to be useful in the design of sampled output exponentially convergent observers for linear systems that are described by an ode pde cascade the new stability result also proves to be useful in designing practical approximate observers involving no pdes

compensating a string pde in the actuation or sensing path of an unstable ode how to control an unstable linear system with a long pure delay in the actuator path this question was resolved using predictor or finite spectrum assignment designs in the 1970s here we address a more challenging question how to control an unstable linear system with a wave partial differential equation pde in the actuation path physically one can think of this problem as having to stabilize a system to whose input one has access through a string the challenges of overcoming string wave dynamics in the actuation path include their infinite dimension finite propagation speed of the control signal and the fact that all of their infinitely many eigenvalues are on the imaginary axis in this technical note we provide an explicit feedback law that compensates the wave pde dynamics at the input of an linear time invariant ordinary differential equation and stabilizes the overall system in addition we prove robustness of the feedback to the error in a priori knowledge of the propagation speed in the wave pde finally we consider a dual problem where the wave pde is in the sensing path and design an exponentially convergent observer

the gap metric robustness of stabilization of feedback systems in this paper we introduce the gap metric to study the robustness of the stability of feedback systems which may employ not necessarily stable open loop systems we elaborate on the computational aspects of the gap metric and provide upper bounds to the gap in cases where the exact formulas do not apply by admissible uncertainties we mean those which preserve closed loop stability and a specified small tolerance on the i o behavior of a feedback system we show that admissible uncertainties are precisely those which are constrained in the gap finally we conclude that any metric which preserves a continuous relationship between open loop systems and the corresponding stable feedback interconnections must have the topology of the gap metric

on general properties of eigenvalues and eigenfunctions of a sturm liouville operator comments on iss with respect to boundary disturbances for 1 d parabolic pdes in the paper iss with respect to boundary disturbances for 1 d parabolic pdes ieee transactions on automatic control vol 61 pp 3712 3724 2016 input to state stability properties are established for 1 d spatially varying parabolic partial differential equations pdes under certain assumptions imposed on eigenvalues and eigenfunctions of an associated sturm liouville operator a key assumption on the absolute convergence of an associated fourier series composed of the normalized eigenfunctions and inverse eigenvalues of the sturm liouville operator is analyzed in the present note general properties of the sturm liouville operator are carried out to demonstrate that such a key assumption becomes redundant for the underlying pdes with sign definite sufficiently smooth coefficients

a time domain approach to model validation in this paper we offer a novel approach to control oriented model validation problems the problem is to decide whether a postulated nominal model with bounded uncertainty is consistent with measured input output data our approach directly uses time domain input output data to validate uncertainty models the algorithms we develop are computationally tractable and reduce to generally nondifferentiable convex feasibility problems or to linear programming problems in special cases we give analytical solutions to these problems

existence and exponential stability of a damped wave equation with dynamic boundary conditions and a delay term abstract in this paper we consider a multi dimensional wave equation with dynamic boundary conditions related to the kelvin voigt damping and a delay term acting on the boundary if the weight of the delay term in the feedback is less than the weight of the term without delay or if it is greater under an assumption between the damping factor and the difference of the two weights we prove the global existence of the solutions under the same assumptions the exponential stability of the system is proved using an appropriate lyapunov functional more precisely we show that even when the weight of the delay is greater than the weight of the damping in the boundary conditions the strong damping term still provides exponential stability for the system

feedback stability under simultaneous gap metric uncertainties in plant and controller abstract the stability robustness of a feedback system is studied in this paper by assuming that the plant and the controller are subject to independent uncertainties and that the uncertainties are measured by the gap metric a fairly complete solution is obtained by exploring the trigonometric structure of the graphs of the plant and the controller

functional determinants and geometry in this paper we study the zeta function determinant in the context of elliptic boundary value problems our main technique is to relate the determinant of an operator or a ratio of determinants to the boundary values of the solutions of the operator this has the advantage of restricting attention to the solutions of the operator which do not depend on the boundary conditions and can often be written down explicitly rather than the eigenvalues which are usually difficult to work with in addition the problem is reduced to a calculation over the boundary of the manifold which is a closed manifold of dimension one less than the original manifold this has special significance in the case that the manifold is a finite interval in this case the boundary is a pair of points and the determinant of an ordinary differential operator is expressed in terms of the determinant of a finite matrix

distance measures for uncertain linear systems a general theory in this paper we propose a generic notion of distance between systems that can be used to measure discrepancy between open loop systems in a feedback sense under several uncertainty structures when the uncertainty structure is chosen to be four block or equivalently normalized coprime factor uncertainty then this generic distance measure reduces to the well known nu gap metric associated with this generic distance notion we also define a generic stability margin notion that allows us to give the distance measure a feedback interpretation by deriving generic robust stability and robust performance results the proposed distance notion and the corresponding results exploit a powerful generalization of the small gain theorem which handles perturbations in rfrl infin rather than only in rfrh infin when the uncertainty structure is fixed to one of the standard structures e g additive multiplicative inverse multiplicative coprime factor four block or any mixtures of the above we give a step by step procedure based on model validation ideas that shows how the generic notion of distance and the correspondingly generic winding number conditions can be reduced to simple formulae this work provides a unified framework that captures and embeds previous results in this area and also completes the picture by showing how other results of a similar nature can be obtained from the same framework the techniques used involve only basic linear algebra so they also provide a simplification of previous advanced proofs furthermore the various distance measures so created can be used for non conservative model embedding into the smallest uncertain family an illustrative example is also given that demonstrates the superior qualities above the nu gap metric of a particular distance measure obtained from this work in situations where the plant is lightly damped all systems considered in this paper are linear time invariant

iss with respect to boundary disturbances for 1 d parabolic pdes due to unbounded input operators in partial differential equations pdes with boundary inputs there has been a long held intuition that input to state stability iss properties and finite gains cannot be established with respect to disturbances at the boundary this intuition has been reinforced by many unsuccessful attempts as well as by the success in establishing iss only with respect to the derivative of the disturbance contrary to this intuition we establish such a result for parabolic pdes our methodology does not rely on the transformation of the boundary disturbance to a distributed input and the stability analysis is performed in time varying subsets of the state space the obtained results are used for the comparison of the gain coefficients of transport pdes with respect to inlet disturbances and for the establishment of the iss property with respect to control actuator errors for parabolic systems under boundary feedback control

spectrum and stability for elastic systems with global or local kelvin voigt damping in this paper we study the mathematical properties of a variational second order evolution equation which includes the equations modelling vibrations of the euler bernoulli and rayleigh beams with the global or local kelvin voigt k v damping in particular our results describe the semigroup setting the strong asymptotic stability and exponential stability of the semigroup the analyticity of the semigroup as well as characteristics of the spectrum of the semigroup generator under various conditions on the damping we also give an example to show that the energy of a vibrating string does not decay exponentially when the k v damping is distributed only on a subinterval which has one end coincident with one end of the string

almost complex and almost product einstein manifolds from a variational principle it is shown that the first order palatini variational principle for a generic nonlinear metric affine lagrangian depending on the symmetrized ricci square invariant leads to an almost product einstein structure or to an almost complex anti hermitian einstein structure on a manifold it is proved that a real anti hermitian metric on a complex manifold satisfies the kahler condition on the same manifold treated as a real manifold if and only if the metric is the real part of a holomorphic metric a characterization of anti kahler einstein manifolds and almost product einstein manifolds is obtained examples of such manifolds are considered

anomalies and analytic torsion on hyperbolic manifolds the global additive and multiplicative properties of the laplacian on j forms and related zeta functions are analyzed the explicit form of zeta functions on a product of closed oriented hyperbolic manifolds hd and of the multiplicative anomaly are derived we also calculate in an explicit form the analytic torsion associated with a connected sum of such manifolds

effective computation of traces determinants and ζ functions for sturm liouville operators abstract the principal aim in this paper is to develop an effective and unified approach to the computation of traces of resolvents and resolvent differences fredholm determinants functions and function regularized determinants associated with linear operators in a hilbert space in particular we detail the connection between fredholm and function regularized determinants concrete applications of our formalism to general i e three coefficient regular sturm liouville operators on bounded intervals with various separated and coupled boundary conditions and schrodinger operators on a half line are provided and further illustrated with an array of examples

systems of parabolic equations with continuous and discrete delays abstract in this paper we investigate the global existence and the dynamics of a coupled system of nonlinear parabolic equations where the nonlinear reaction function may depend on both continuous infinite or finite and discrete delays it is shown that if the reaction function is locally lipschitz continuous and the system possesses a pair of coupled upper and lower solutions then there exists a unique global solution to the system without any quasimonotone condition on the reaction function for systems with mixed quasimonotone reaction functions we use the monotone method to establish more dynamic property of the parabolic system in terms of the quasisolutions of the corresponding elliptic system this approach yields a global attractor of the parabolic system and under some additional conditions this attractor leads to the existence and asymptotic stability of a solution of the elliptic system application to three model problems in population dynamics and chemical reactions is given

asymptotics of the heat kernel on rank 1 locally symmetric spaces we consider the heat kernel and the zeta function associated with laplace type operators acting on a general irreducible rank 1 locally symmetric space x the set of minakshisundaram pleijel coefficients ak x k 0 in the short time asymptotic expansion of the kernel is calculated explicitly

input to state stability of a clamped free damped string in the presence of distributed and boundary disturbances this note establishes the exponential input to state stability eiss property for a clamped free damped string with respect to distributed and boundary disturbances while efficient methods for establishing iss properties for distributed parameter systems with respect to distributed disturbances have been developed during the last decades establishing iss properties with respect to boundary disturbances remains challenging one of the well known methods for well posedness analysis of systems with boundary inputs is the use of a lifting operator for transferring the boundary disturbance to a distributed one however the resulting distributed disturbance involves time derivatives of the boundary perturbation thus the subsequent iss estimate depends on its amplitude and may not be expressed in the strict form of iss properties to solve this problem we show for a clamped free damped string equation that the projection of the original system trajectories in an adequate riesz basis can be used to establish the desired eiss property

small gain stability analysis of certain hyperbolic parabolic pde loops abstract this work provides stability results in the spatial sup norm for hyperbolic parabolic loops in one spatial dimension the results are obtained by an application of the small gain stability analysis two particular cases are selected for the study because they contain challenges typical of more general systems to which the results are easily generalizable but at the expense of less pedagogical clarity and more notational clutter i the feedback interconnection of a parabolic pde with a first order zero speed hyperbolic pde with boundary disturbances and ii the feedback interconnection by means of a combination of boundary and in domain terms of a parabolic pde with a first order hyperbolic pde the first case arises in the study of the movement of chemicals underground and includes the wave equation with kelvin voigt damping as a subcase the second case arises when we apply backstepping to a pair of hyperbolic pdes that is obtained by ignoring diffusion phenomena moreover the second case arises in the study of parabolic pdes with distributed delays in the first case we provide sufficient conditions for iss in the spatial sup norm with respect to boundary disturbances in the second case we provide delay independent sufficient conditions for exponential stability in the spatial sup norm