conditions for the equivalence of internal and external stability for distributed parameter systems conditions are given on a controlled observed distributed parameter system under which external stability also known as input output stability implies internal stability which is the exponential stability of the underlying semigroup generator it is shown that when the system satisfies a general definition of stabilizability and detectability external stability is equivalent to internal stability unlike the case previous definitions of stabilizability and detectability a system which satisfies these does not necessarily have a spectrum decomposition therefore in order to check internal stability it is often sufficient to check the boundedness of the transfer function in the right half complex plane this is illustrated with a simple example

transfer functions of regular linear systems part ii the system operator and the lax phillips semigroup this paper is a sequel to a paper by the second author on regular linear systems 1994 referred to here as part i we introduce the system operator of a well posed linear system which for a finite dimensional system described by x ax bu y cx du would be the s dependent matrix s s a si c b d in the general case s s is an unbounded operator and we show that it can be split into four blocks as in the finite dimensional case but the splitting is not unique the upper row consists of the uniquely determined blocks a si and b as in the finite dimensional case but the lower row is more problematic for weakly regular systems which are introduced and studied here there exists a special splitting of s s where the right lower block is the feedthrough operator of the system using s 0 we give representation theorems which generalize those from part i to well posed linear systems and also to the situation when the initial time is we also introduce the lax phillips semigroup t induced by a well posed linear system which is in fact an alternative representation of a system used in scattering theory our concept of a lax phillips semigroup differs in several respects from the classical one for example by allowing an index r which determines an exponential weight in the input and output spaces this index allows us to characterize the spectrum of a and also the points where s s is not invertible in terms of the spectrum of the generator of for various values of the system is dissipative if and only if with index zero is a contraction semigroup

the differentiation matrix for daubechies based wavelets on an interval the differentiation matrix for a daubechies based wavelet basis defined on an interval is constructed it is shown that the differentiation matrix based on the currently available boundary constructions does not maintain the superconvergence encountered under periodic boundary conditions

representation of infinite dimensional neutral non autonomous control systems abstract this paper studies the well posedness of a class of non autonomous neutral control systems in banach spaces we prove that such systems are represented by absolutely regular non autonomous linear systems in the sense of schnaubelt r schnaubelt feedback for non autonomous regular linear systems siam j control optim 41 2002 1141 1165 this paper can be considered as the non autonomous version of the work presented in h bounit s hadd regular linear systems governed by neutral fdes j math anal appl 320 2006 836 858

on spl hscr sub spl infin control for dead time systems a mixed sensitivity spl hscr sub spl infin problem is solved for dead time systems it is shown that for a given bound on the spl hscr sub spl infin norm causal stabilizing controllers exist that achieve this bound if and only if a related finite dimensional riccati equation has a solution with a certain nonsingularity property in the case of zero time delay the riccati equation is a standard riccati equation and the nonsingularity condition is that the solution be nonnegative definite for nonzero time delay the nonsingularity condition is more involved but still allows us to obtain controllers all suboptimal controllers are parameterized and the central controller is shown to be a feedback interconnection of a finite dimensional system and a finite memory system both of which can be implemented some spl hscr sub spl infin problems are rewritten as pure rational spl hscr sub spl infin problems using a smith predictor parameterization of the controller

admissibility of unbounded control operators for linear systems described by dot x t ax t bu t where a generates a semigroup on the state space x and b is an unbounded operator some necessary as well as some sufficient conditions are given for b to be admissible i e for any t the state x t should be in x and should depend continuously on the input u in l p this approach begins with an axiomatic description of such a system in terms of a functional equation the results are applied to the wave equation on a bounded interval

on the average character degree and the average class size in finite groups let acd g and acs g denote the average irreducible character degree and the average conjugacy class size respectively of a finite group g the main objective of this paper is to find the lower bounds of acd g and acs g for a non r solvable group g where r is a prime we show that psl 2 r where r 5 is a non r solvable group with the smallest average character degree and the smallest average class size

stability radius and internal versus external stability in banach spaces an evolution semigroup approach in this paper the theory of evolution semigroups is developed and used to provide a framework to study the stability of general linear control systems these include autonomous and nonautonomous systems modeled with unbounded state space operators acting on banach spaces this approach allows one to apply the classical theory of strongly continuous semigroups to time varying systems in particular the complex stability radius may be expressed explicitly in terms of the generator of an evolution semigroup examples are given to show that classical formulas for the stability radius of an autonomous hilbert space system fail in more general settings upper and lower bounds on the stability radius are proven for banach space systems in addition it is shown that the theory of evolution semigroups allows for a straightforward operator theoretic analysis of internal stability as determined by classical frequency domain and input output operators even for nonautonomous banach space systems in particular for the nonautonomous setting internal stability is shown to be equivalent to input output stability for stabilizable and detectable systems for the autonomous setting an explicit formula for the norm of input output operator is given

well posedness of systems of linear elasticity with dirichlet boundary control and observation an open loop system of linear elasticity with dirichlet boundary control and collocated observation is considered the main result we obtained states that this system is well posed in the sense of salamon the result deduces the exponential stability of the closed loop system under proportional output feedback hence it answers positively an open question proposed by liu and krstic ima j appl math 65 2000 pp 109 121 moreover the well posedness together with the regularity of the open loop system in the sense of weiss that was obtained in our separate paper s g chai and b z guo feedthrough operator for linear elasticity system with boundary control and observation preprint school of computational and applied mathematics university of the witwatersrand south africa 2008 makes this infinite dimensional system parallel in many ways to a linear finite dimensional system in the framework of well posed and regular linear infinite dimensional systems

regular linear systems governed by neutral fdes the aim of this paper is to prove that a class of distributed parameter systems governed by neutral fdes provides regular linear systems employing the well established theory of representation transfer function and feedback of these later we then give new representations of the state and the output function of the neutral systems

new riccati equations for well posed linear systems we consider the classic problem of minimizing a quadratic cost functional for well posed linear systems over the class of inputs that are square integrable and that produce a square integrable output as is well known the minimum cost can be expressed in terms of a bounded nonnegative self adjoint operator x that in the finite dimensional case satisfies a riccati equation unfortunately the infinite dimensional generalization of this riccati equation is not always well defined we show that x always satisfies alternative riccati equations which are more suitable for algebraic and numerical computations

on invariance of p admissibility of control and observation operators to q type of perturbations of generator of c0 semigroup abstract in this paper it is proved in general setting that p admissibilities of control operators and observation operators are invariant to any q type of perturbations of generator of c 0 semigroups on banach space moreover some relations between the extensions of observation operators with respect to the original generator and the perturbed generator are also characterized so that the output can be expressed in the mild sense as an application the admissibility as well as the mild expressibility of output of a class of observation systems with time delay in state is deduced

regularity of a schrodinger equation with dirichlet control and colocated observation this paper studies the basic properties of the schrodinger equation defined on a bounded domain of r n n 2 with partial dirichlet control and colocated observation it is shown that the system is not only wellposed in the sense of d salamon but also regular in the sense of g weiss it is also shown that the corresponding feedthrough operator is zero 2005 elsevier b v all rights reserved

on the realization of periodic functions abstract in this article canonical and spectrally minimal infinite dimensional state space realizations for periodic functions are considered it is shown that the periodic functions having l 1 fourier coefficients are precisely those realizable by a riesz spectral system rss in which the system operator generates a periodic strongly continuous semigroup and the observation operator is bounded this realization can easily be converted to a canonical and spectrally minimal form it is shown how the use of rss and cesaro sums of fourier series allows the construction of a state space realization for a given periodic function merely integrable over its period simple finite dimensional approximations with error bounds are derived for the rss realization regular well posed linear systems wpls are used to construct a fuhrmann type realization for a given periodic function integrable over its period it is shown that the rss realizations and wpls realizations are precisely equally good at coping with the possible ill behavior of a given bounded periodic function integrable over its period but the wpls realization is not always spectrally minimal or canonical

toeplitz operators and h_ infty calculus let a be the generator of a strongly continuous exponentially stable semigroup on a hilbert space furthermore let the scalar function g be bounded and analytic on the left half plane i e g s in mathcal h infty by using the toeplitz operator associated to g we construct an infinite time admissible output operator g a if g is rational then this operator is bounded and equals the normal definition of g a although in general g a may be unbounded we always have that g a multiplied by the semigroup is a bounded operator for every positive time instant furthermore when there exists an admissible output operator c such that c a is exactly observable then g a is bounded for all g with g s in mathcal h infty i e there exists a bounded mathcal h infty calculus moreover we rediscover some well known classes of generators also having a bounded mathcal h infty calculus

admissible input elements for systems in hilbert space and a carleson measure criterion we study the control system dot x ax bu quad x in x quad u text scalar where a generates a semigroup on the hilbert space x but in general the control input element b notin x many boundary value control systems point control force situations etc can be studied in this context we define and analyze admissible input elements b and develop sufficient conditions for b to be admissible in terms of the carleson measure theorem of h p theory

model reduction by balanced truncation for systems with nuclear hankel operators we prove the h infinity error bounds for lyapunov balanced truncation and for optimal hankel norm approximation under the assumption that the hankel operator is nuclear this is an improvement of the result from glover curtain and partington siam j control optim 26 1998 pp 863 898 where additional assumptions were made the proof is based on convergence of the schmidt pairs of the hankel operator in a sobolev space we also give an application of this convergence theory to a numerical algorithm for model reduction by balanced truncation

the regularity of the wave equation with partial dirichlet control and colocated observation in this paper we analyze a multidimensional controlled wave equation on a bounded domain subject to partial dirichlet control and colocated observation by means of a partial fourier transform it is shown that the system is well posed and regular in the sense of d salamon and g weiss the corresponding feedthrough operator is found to be the identity operator on the input space

schur agler and herglotz agler classes of functions positive kernel decompositions and transfer function realizations abstract we discuss transfer function realization for multivariable holomorphic functions mapping the unit polydisk or the right polyhalfplane into the operator analogue of either the unit disk or the right halfplane schur herglotz functions over either the unit polydisk or the right polyhalfplane which satisfy the appropriate stronger contractive positive real part condition for the values of these functions on commutative tuples of strict contractions strictly accretive operators schur agler herglotz agler functions over either the unit polydisk or the right polyhalfplane as originally shown by agler the first case polydisk to disk can be solved via unitary extensions of a partially defined isometry constructed in a canonical way from a kernel decomposition for the function the lurking isometry method we show how a geometric reformulation of the lurking isometry method embedding of a given isotropic subspace of a kre n space into a lagrangian subspace the lurking isotropic subspace method can be used to handle the second two cases polydisk to halfplane and polyhalfplane to disk as well as the last case polyhalfplane to halfplane if an additional growth condition at is imposed for the general fourth case we show how a linear fractional transformation change of variable can be used to arrive at the appropriate symmetrized nonhomogeneous bessmertny long resolvent realization we also indicate how this last result recovers the classical integral representation formula for scalar valued holomorphic functions mapping the right halfplane into itself

solving inverse source problems using observability applications to the euler bernoulli plate equation the aim of this paper is to provide a general framework for solving a class of inverse source problems by using exact observability of infinite dimensional systems more precisely we show that if a system is exactly observable then a source term in this system can be identified by knowing its intensity and appropriate observations which often correspond to measurements of some boundary traces this abstract theory is then applied to obtain new identifiability results for a system governed by the euler bernoulli plate equation using a different methodology we show that exact observability can be used to identify both the locations and the intensities of combinations of point sources in the plate equation

dynamic stabilization of regular linear systems we consider a general class of infinite dimensional linear systems called regular linear systems for which convenient representations are known to exist both in time and in frequency domain for this class of systems we investigate the concepts of stabilizability and detectability in particular their invariance under feedback and their relationship to exponential stability we introduce two concepts of dynamic stabilization the first formulated as usual with the plant and the controller connected in feedback and the second with two feedback loops even for finite dimensional systems the second concept stabilization with an internal loop in the controller is more general we argue that the more general concept is the natural one and we derive sufficient conditions under which an observer based stabilizing controller with an internal loop can be constructed

brief every stabilizing dead time controller has an observer predictor based structure this paper considers the stabilization problem for linear time invariant systems with a single delay h in the feedback loop two parametrizations of all stabilizing controllers are derived in terms of a state space realization of the rational part of the plant these parametrizations have simple structures and clear interpretations the first is in the form of a generalized dead time compensator smith predictor and the second is in the observer predictor form applications of the proposed parametrization to the h 2 optimization and the robust stabilization problems are discussed

funnel control for the boundary controlled heat equation we consider an output regulation problem for a single input single output system with dynamics described by the heat equation on some bounded domain omega subset mathbb r d with sufficiently smooth boundary the input is formed by neumann boundary control and the output is the surface integral of the state at the boundary we show that the funnel controller can be applied to this system in order to track a given output reference signal within a prespecified performance funnel

addendum to paper on the commuting probability in finite groups by r m guralnick and g r robinson j algebra 300 2 2006 509 528

optimal control on the doubly infinite continuous time axis and coprime factorizations we study the problem of existence of weak right or left or strong coprime factorizations in h infinity over the right half plane of an analytic function defined in some subset of the right half plane we give necessary and sufficient conditions for the existence of such coprime factorizations in terms of an optimal control problem over the doubly infinite continuous time axis in particular we show that an equivalent condition for the existence of a strong coprime factorization is that both the control and the filter algebraic riccati equation of an arbitrary realization that need not be well posed have a solution in general unbounded and not even densely defined and that a coupling condition involving these two solutions is satisfied the proofs that we give are partly based on corresponding discrete time results which we have recently obtained

root locus and boundary feedback design for a class of distributed parameter systems in this paper a fairly complete parallel of the finite dimensional root locus theory is presented for quite general nonconstant coefficient even order ordinary differential operators on a finite interval with control and output boundary conditions representative of a choice of colocated point actuators and sensors root locus design methods for linear distributed parameter systems have also been studied for some time and the primary difficulties in rigorously interpreting root locus conclusions for distributed parameter systems are well known first the transfer function of a distributed parameter system may not be meromorphic at infinity so that many of the standard rouche arguments required even in the lumped case to determine the asymptotic behavior of the root loci are not generally valid another difficulty is that the infinitesimal generator in the state space model for a closed loop system may not be selfadjoint accretive or even satisfy the spectrum determined growth condition thus regardless of whether the root loci interpreted as closed loop eigenvalues lie in the open left half plane additional analysis would be required to conclude that the closed loop system would be asymptotically stable formulating the systems in the classical format of a boundary control problem the asymptotic analysis of the root loci can be based on the pioneering work by birkhoff on eigenfunction expansions for boundary value problems work that predated and indeed motivated the development of spectral theory in hilbert space birkhoff s work also contains an asymptotic expansion of eigenfunctions in the spatial variable generalizing the earlier sturm liouville theory for second order operators by further extending this general asymptotic analysis to also include expansions in the gain parameter a rigorous treatment of the open and closed loop transfer functions and of the corresponding return difference equation can be presented the asymptotic analysis of the return difference equation forms the basis for both our rigorous formulation of the basic problem and its solution

well posedness and regularity of naghdi s shell equation under boundary control and observation abstract a system of naghdi s shell equation with dirichlet boundary control and collocated observation is considered results on the associated nonhomogeneous boundary value problem are presented based on these results it is shown that the system is well posed in the sense of d salamon and regular in the sense of g weiss the expression of the corresponding feedthrough operator is explicitly found by means of riemannian geometric method and partial fourier transform these properties make this partial differential control system parallel in many ways finite dimensional ones in the general framework of well posed and regular infinite dimensional systems

the linear quadratic control problem for infinite dimensional systems with unbounded input and output operators this paper establishes a general semigroup framework for solving quadratic control problems with infinite dimensional state space and unbounded input and output operators

a finite dimensional robust controller for systems in the cd algebra robust multivariable controllers for stable infinite dimensional systems in the callier desoer cd algebra are discussed in particular the following robust regulation problem is solved given reference and disturbance signals which are linear combinations of signals of the form t sup j sin spl omega sub k t spl phi sub k j spl ges 0 k 0 spl middot spl middot spl middot n find a low order finite dimensional controller so that the outputs asymptotically track the reference signals asymptotically reject the disturbance signals and the closed loop system is stable and robust with respect to a class of perturbations in the plant the proposed controller consists of a positive scalar gain spl epsi and certain polynomial matrices k sub k s for k 0 spl middot spl middot spl middot n the main result of the paper shows that the matrices k sub k s must satisfy certain stability conditions involving the values of the plant transfer function only at the reference and disturbance signal frequencies spl omega sub k for k 0 spl middot spl middot spl middot n the controller has unstable poles on the imaginary axis the behavior of these poles as a function of the scalar parameter spl epsi in the form of a puiseux series is given

low gain integral control of well posed linear infinite dimensional systems with input and output nonlinearities abstract time varying low gain integral control strategies are presented for asymptotic tracking of constant reference signals in the context of exponentially stable well posed linear infinite dimensional single input single output systems subject to globally lipschitz nondecreasing input and output nonlinearities it is shown that applying error feedback using an integral controller ensures that the tracking error is small in a certain sense provided that a the steady state gain of the linear part of the system is positive b the reference value r is feasible in an entirely natural sense and c the positive gain function t k t is ultimately sufficiently small and not of class l 1 under a weak restriction on the initial data it is shown that a b and c ensure asymptotic tracking if additionally the impulse response of the linear part of the system is a finite signed borel measure the global lipschitz assumption on the output nonlinearity may be considerably relaxed

a fast adaptive wavelet collocation algorithm for multidimensional pdes a fast multilevel wavelet collocation method for the solution of partial differential equations in multiple dimensions is developed the computational cost of the algorithm is independent of the dimensionality of the problem and is o n where n is the total number of collocation points the method can handle general boundary conditions the multilevel structure of the algorithm provides a simple way to adapt computational refinements to local demands of the solution high resolution computations are performed only in regions where singularities or sharp transitions occur numerical results demonstrate the ability of the method to resolve localized structures such as shocks which change their location and steepness in space and time the present results indicate that the method has clear advantages in comparison with well established numerical algorithms

robustness measures for linear systems with application to stability radii of hurwitz and schur polynomials in this paper we consider robustness measures stability radii for system matrices which are subjected to structured real and complex perturbations of the form a a bdc where b c are given matrices our object is twofold a to present a number of new results mainly concerning the real stability radius and its differences from the complex one b to give an overview of our approach to the robustness analysis of linear state space systems including basic properties and characterizations of the complex stability radius applying the results to the special case where a is in companion form and b 0 0 0 1 t we are able to determine stability radii for hurwitz and schur polynomials under arbitrary complex and real affine perturbations of the coefficient vector computable formulae are obtained and illustrated by several examples

time varying and adaptive integral control of infinite dimensional regular linear systems with input nonlinearities closing the loop around an exponentially stable single input single output regular linear system subject to a globally lipschitz nondecreasing actuator nonlinearity and compensated by an integral controller with time dependent gain k t is shown to ensure asymptotic tracking of a constant reference signal r provided that a the steady state gain of the linear part of the system is positive b the reference value r is feasible in an entirely natural sense and c the function t mapsto k t monotonically decreases to zero at a sufficiently slow rate this result forms the basis of a simple adaptive control strategy that ensures asymptotic tracking under conditions a and b

exponential stability of coupled beams with dissipative joints a frequency domain approach two examples of coupled euler bernoulli beams with a dissipative joint are considered the joint placements that lead to exponential stability for these systems are characterized the technique used shows input output stability of a related controlled observed system and then shows that in these examples input output stability implies exponential stability in the first example the energy dissipation arises from a discontinuity in the shear at the joint in the second example the energy dissipation arises from a discontinuity in the bending moment at the joint the analysis of this system involves a complete spectrum analysis of the zero dynamics of the associated controlled observed system

transfer functions of regular linear systems part i characterizations of regularity we recall the main facts about the representation of regular linear systems essentially that they can be described by equations of the form x t ax t bu t y t cx t du t like finite dimensional systems but now a b and c are in general unbounded operators regular linear systems are a subclass of abstract linear systems we define transfer functions of abstract linear systems via a generalization of a theorem of foures and segal we prove a formula for the transfer function of a regular linear system which is similar to the formula in finite dimensions the main result is a simple to state but hard to prove necessary and sufficient condition for an abstract linear system to be regular in terms of its transfer function other conditions equivalent to regularity are also obtained the main result is a consequence of a new tauberian theorem which is of independent interest

coprime factorization for regular linear systems mild sufficient conditions are given for the existence of a doubly coprime factorization of the transfer function of a regular linear system as well as formulae for such a factorization the results are illustrated by two examples of delay systems one of which has infinitely many unstable poles

exponential stabilization of a rayleigh beam using collocated control we consider a hinged elastic beam described by the rayleigh beam equation on the interval 0 pi we assume the presence of two sensors one measures the angular velocity of the beam at a point xi epsiv 0 pi and the other measures the bending curvature of the beam at the same point if xi 0 or xi pi then the second output is not needed the corresponding operator semigroup is unitary on a suitable hilbert state space these two measurements are advantageous because they make the open loop system exactly observable regardless of the point xi we design the actuators and the feedback law in order to exponentially stabilize this system using the theory of collocated static output feedback developed in our recent paper we design the actuators such that they are collocated meaning that b c where b is the control operator and c is the observation operator it turns out that if xi epsiv 0 pi then the actuators cause a discontinuity of the bending exactly at this is the price in this example of having collocated actuators and sensors this obliges us to use an extension of to define the output signal in terms of the left and right limit of the bending at xi we prove that for all static output feedback gains in a suitable finite range the closed loop system is well posed and exponentially stable this follows from the general theory in our paper whose main points are recalled here

wave extrapolation in the spatial wavelet domain with application to poststack reverse time migration a wavelet transformation is performed over each of the spatial coordinates of the scalar wave equation this transformed equation is solved directly with a finite difference scheme for both homogeneous and smooth inhomogeneous media wavefield extrapolation is performed completely in the spatial wavelet domain without transforming back into the space domain at each time step the wavelet coefficients are extrapolated rather than the wavefield itself the numerical solution of the scalar wave equation in the spatial wavelet domain is closely related to the finite difference method because of the compact support of the wavelet bases poststack reverse time migration is implemented as an application the resolution spaces of the wavelet transform provide a natural framework for multigrid analysis migrated images are constructed from various resolution spaces

optimal control of systems with a unitary semigroup and with colocated control and observation abstract we solve the quadratic optimal control problem on an infinite time interval for a class of linear systems whose state space is a hilbert space and whose operator semigroup is unitary the difficulty is that the systems in this class having unbounded control and observation operators may be ill posed we show that there is a surprisingly simple solution to the problem the optimal feedback turns out to be output feedback our approach is to use a change of variables which transforms the system into a one which according to recent research is known to be conservative we show that under a mild assumption the transfer function of this conservative system is inner and then it follows that the optimal control of this conservative system is trivial we give an example with the wave equation on an n dimensional domain with neumann control and dirichlet observation of the velocity

lack of time delay robustness for stabilization of a structural acoustics model in this paper we consider a natural robustness question for a model for structural acoustics this model which has been of great interest in recent years is represented by a wave equation in r 2 coupled to a kelvin voigt beam the coupling is natural physically and is represented mathematically by highly unbounded operators we assume that the observation consists of point evaluation of the beam position the beam velocity and the wave velocity we are interested in the effect of arbitrarily small delays in the feedback loop on a controller that uses these observations we show that it is not possible to construct a dynamic stabilizer of a very general form including static feedback such that the stabilization is robust with respect to delays in the feedback loop in order to do this we need to carefully analyze the input to output map finally we relate these results to already existing numerical results obtained for a galerkin approximation of the system

a wavelet optimized very high order adaptive grid and order numerical method wavelets detect information at different scales and at different locations throughout a computational domain furthermore wavelets can detect the local polynomial content of computational data numerical methods are most efficient when the basis functions of the method are similar to the data present by designing a numerical scheme in a completely adaptive manner around the data present in a computational domain one can obtain optimal computational efficiency this paper extends the numerical wavelet optimized finite difference wofd method to arbitrarily high order so that one obtains in effect an adaptive grid and adaptive order numerical method which can achieve errors equivalent to errors obtained with a spectrally accurate numerical method

transfer functions for infinite dimensional systems in this paper we study three definitions of the transfer function for an infinite dimensional system the first one defines the transfer function as the expression c si a 1 b d in the second definition the transfer function is defined as the quotient of the laplace transform of the output and input with initial condition zero in the third definition we introduce the transfer function as the quotient of the input and output when the input and output are exponentials we show that these definitions always agree on the right half plane bounded to the left by the growth bound of the underlying semigroup but that they may differ elsewhere

tauberian theorems for the laplace stieltjes transform let a 0 oo c be a function of locally bounded variation with a 0 0 whose laplace stieltjes transform g z fo e z da t is absolutely convergent for re z 0 let e be the singular set of g in ir and suppose that 0 e various estimates for lim supt la t g o l are obtained in particular a t g 0 as t oo if i e is null ii sup sup e aysda s oo yee t o o iii lim lim sup sup a s a t 0 610 t oo t 6 s t this result contains tauberian theorems for laplace transforms power series and dirichlet series

a wavelet optimized adaptive multi domain method the formulation and implementation of wavelet based methods for the solution of multi dimensional partial differential equations in complex geometries is discussed utilizing the close connection between daubechies wavelets and finite difference methods on arbitrary grids we formulate a wavelet based collocation method well suited for dealing with general boundary conditions and nonlinearities to circumvent problems associated with completely arbitrary grids and complex geometries we propose to use a multi domain formulation in which to solve the partial differential equation with the ability to adapt the grid as well as the order of the scheme within each subdomain besides supplying the required geometric flexibility the multi domain formulation also provides a very natural load balanced data decomposition suitable for parallel environments the performance of the overall scheme is illustrated by solving two dimensional hyperbolic problems

robustness and continuity of the spectrum for uncertain distribution parameter systems we analyze the spectrum of a class of abstract partial differential equations with boundary feedback control if we interpret the feedback as a perturbation then we obtain a robustness radius for spectrum placement we obtain continuity of the spectrum using this radius and rouche s theorem we exploit extensively recent developments in the representation theory for abstract linear systems especially regular systems we also analyze ill posed systems by interchanging the roles of input and output we apply these results to a wave equation and an ill posed beam equation

adaptive regulator design for a class of infinite dimensional second order systems in this paper we consider adaptive regulator design for undamped infinite dimensional second order systems in the case of collocated input and output operators the adaptive regulator is constructed by the concept of adaptive pd control proportional plus derivative control and the estimation mechanism of the unknown parameters for the disturbances in the controlled system the convergence of the system state to zero will be guaranteed

on the commuting probability in finite groups for bernd fischer on the occasion of his 70th birthdayintroduction when gis a nite group we may endow g gwith the structureof a probability space by assigning the uniform distribution as was pointed outby w h gustafson 10 the probability that a randomly chosen pair of elementsof gcommute is then

some spectral properties of operator valued positive real functions abstract we consider operator valued positive real functions h and show that the intersections of the point continuous and residual spectra of h s with the imaginary axis do not depend on s in particular if h is positive real and h z is invertible for some z in the open right half plane then h s is invertible for all s in the open right half plane furthermore we prove that the eigenspace of h s corresponding to an imaginary eigenvalue does not depend on s it is also shown that the intersection of the numerical range of h s with the imaginary axis is independent of s finally we prove that under suitable assumptions application of a sufficiently positive real static output feedback to a positive real transfer function leads to a strictly positive real closed loop system

exact controllability of a class of nonlinear distributed parameter systems using back and forth iterations we investigate the exact controllability of a nonlinear plant described by the equation x t ax t bu t bnn x t t where t 0 here a is the infinitesimal generator of a strongly cont

quadratic optimal control of stable well posed linear systems we consider the infinite horizon quadratic cost minimization problem for a stable time invariant well posed linear system in the sense of salamon and weiss and show that it can be reduced to a spectral factorization problem in the control space more precisely we show that the optimal solution of the quadratic cost minimization problem is of static state feedback type if and only if a certain spectral factorization problem has a solution if both the system and the spectral factor are regular then the feedback operator can be expressed in terms of the riccati operator and the riccati operator is a positive self adjoint solution of an algebraic riccati equation this riccati equation is similar to the usual algebraic riccati equation but one of its coefficients varies depending on the subspace in which the equation is posed similar results are true for unstable systems as we have proved elsewhere

hankel norm approximation for well posed linear systems the sub optimal hankel norm approximation problem is solved for a well posed linear system with generating operators a b c and transfer function g satisfying some mild assumptions in the special case of the sub optimal nehari problem an explicit parameterization of all solutions is obtained in terms of the system parameters a b c and g 0

well posedness of a structural acoustics control model with point observation of the pressure abstract we consider a controlled and observed partial differential equation pde which describes a structural acoustics interaction physically this pde describes an acoustic chamber with a flexible chamber wall the control is applied to this flexible wall and the class of controls under consideration includes those generated by piezoceramic patches the observation we consider is point measurements of acoustic pressure inside the cavity mathematically the model consists of a wave equation coupled through boundary trace terms to a structurally damped plate or beam equation and the point controls and observations for this system are modeled by highly unbounded operators we analyze the map from the control to the observation since the properties of this map are central to any control design which is based upon this observation we also show there exists an appropriate state space x so that if the initial state is in x and the control is in l2 then the state evolves continuously in x and the observation is in l2 the analysis of this system entails a microlocal analysis of the wave component of the system and the use of pseudodifferential machinery

well posedness and exact controllability of fourth order schrodinger equation with boundary control and collocated observation in this paper we study the well posedness and exact controllability of a system described by a fourth order schrodinger equation on a bounded domain of mathbb r n n geqslant 2 with boundary control and collocated observation the neumann boundary control problem is first discussed it is shown that the system is well posed in the sense of d salamon this implies the exponential stability of the closed loop system under proportional output feedback control the well posedness result is then generalized to the dirichlet boundary control problem in particular in order to conclude feedback stabilization from well posedness we discuss the exact controllability with the dirichlet boundary control which is similar to the neumann boundary control case in addition we show that both systems are regular in the sense of g weiss and their feedthrough operators are zero

low gain adaptive stabilization of infinite dimensional second order systems in this paper low gain adaptive stabilization of infinite dimensional undamped second order systems is considered in the case where the input and output operators are collocated the systems have an infinite number of poles and zeros on the imaginary axis the adaptive stabilizer is constructed by a low gain adaptive pi controller proportional plus integral controller an energy like function and a multiplier function are introduced and low gain adaptive stabilization of the linear second order systems is analyzed

higher order wavelet reconstruction differentiation filters and gibbs phenomena an orthogonal wavelet basis is characterized by its approximation order which relates to the ability of the basis to represent general smooth functions on a given scale it is known though perhaps not widely known that there are ways of exceeding the approximation order i e achieving higher order error in the discretized wavelet transform and its inverse the focus here is on the development of a practical formulation to accomplish this first for 1d smooth functions then for 1d functions with discontinuities and then for multidimensional here 2d functions with discontinuities it is shown how to transcend both the wavelet approximation order and the 2d gibbs phenomenon in representing electromagnetic fields at discontinuous dielectric interfaces that do not simply follow the wavelet basis grid

time varying regular bilinear systems the theory of infinite dimensional systems introduced and developed by salamon trans amer math soc 300 1987 pp 383 431 and weiss siam j control optim 27 1989 pp 527 545 has been recently applied for various systems see e g the jacob partington survey current trends in operator theory and its applications birkhauser boston 2004 in the same spirit schnaubelt in his recent work siam j control optim 41 2002 pp 1141 1165 has extended their approach to the time varying linear systems in this paper we follow this spirit and prove that one can extend the results recently obtained for infinite dimensional time invariant bilinear systems to the time varying setting in particular we show that the absolute regularity and detectability assumptions ensure the existence of an observer for this kind of systems

discrete time and sampled data low gain control of infinite dimensional linear systems in the presence of input hysteresis we introduce a general class of causal dynamic discrete time nonlinearities which have certain monotonicity and lipschitz continuity properties in particular the discretizations of a large class of continuous time hysteresis operators obtained by applying the standard sampling and hold operations belong to this class it is shown that closing the loop around a power stable linear infinite dimensional discrete time single input single output system subject to an input nonlinearity from the class under consideration and compensated by a discrete time integral controller guarantees asymptotic tracking of constant reference signals provided that a the positive integrator gain is sufficiently small and b the reference value is feasible in a very natural sense we apply this result in the development of sampled data low gain integral control for exponentially stable regular linear infinite dimensional continuous time systems subject to input hysteresis

robustness with respect to delays for exponential stability of distributed parameter systems in this paper we address the question of whether the open loop exponential growth rate of a linear system can be improved by a feedback in such a way that this improvement is robust with respect to small delays in the feedback loop when the input operator is admissible and the class of possible feedbacks consists of compact operators we find that if a feedback can improve the exponential growth rate then it can do so robustly furthermore we find that if the control space is finite dimensional and a bounded feedback cannot be found to improve exponential stability then a large class of unbounded feedbacks cannot improve the exponential growth rate robustly even if such feedbacks can improve the exponential growth rate in the absence of delays

robust stability of input output systems with initial conditions we consider the development of a general nonlinear input output theory which encompasses systems with initial conditions systems are defined in a set theoretic manner from input output pairs on a doubly infinite time axis and a general construction of the initial conditions is given in terms of an equivalence class of trajectories on the negative time axis input output operators are then defined for given initial conditions and a suitable notion of input output stability on the positive time axis with initial conditions is given this notion of stability is closely related to the iss ios concepts of sontag a fundamental robust stability theorem is derived which represents a generalization of the input output operator robust stability theorem of georgiou and smith to include the case of initial conditions this includes a suitable generalization of the nonlinear gap metric some applications are given to show the utility of the robust stability theorem

a general realization theorem for matrix valued herglotz nevanlinna functions abstract new special types of stationary conservative impedance and scattering systems the so called non canonical systems involving triplets of hilbert spaces and projection operators are considered it is established that every matrix valued herglotz nevanlinna function of the form v z q lz r 1 t z t 1 t 2 d t can be realized as a transfer function of such a new type of conservative impedance system in this case it is shown that the realization can be chosen such that the main and the projection operators of the realizing system satisfy a certain commutativity condition if and only if l 0 it is also shown that v z with an additional condition namely l is invertible or l 0 can be realized as a linear fractional transformation of the transfer function of a non canonical scattering f system in particular this means that every scalar herglotz nevanlinna function can be realized in the above sense moreover the classical livsic systems brodskii livsic operator colligations can be derived from f systems as a special case when f i and the spectral measure d t is compactly supported the realization theorems proved in this paper are strongly connected with and complement the recent results by ball and staffans

internal model based tracking and disturbance rejection for stable well posed systems in this paper we solve the tracking and disturbance rejection problem for infinite dimensional linear systems with reference and disturbance signals that are finite superpositions of sinusoids we explore two approaches both based on the internal model principle in the first approach we use a low gain controller and here our results are a partial extension of results by hamalainen and pohjolainen in their papers the plant is required to have an exponentially stable transfer function in the callier desoer algebra while in this paper we only require the plant to be well posed and exponentially stable these conditions are sufficiently unrestrictive to be verifiable for many partial differential equations in more than one space variable our second approach concerns the case when the second component of the plant transfer function from control input to tracking error is positive in this case we identify a very simple stabilizing controller which is again an internal model but which does not require low gain we apply our results to two problems involving systems modeled by partial differential equations the problem of rejecting external noise in a model for structure acoustics interactions and a similar problem for two coupled beams

controlled functional differential equations and adaptive tracking abstract adaptive tracking control of a class n of single input single output systems described by nonlinear functional differential equations is considered the control objective is that of tracking by the system output of reference signals of class r absolutely continuous and bounded with essentially bounded derivative a n r universal servomechanism in the form of an adaptive error feedback strategy incorporating gains of nussbaum type is developed which for every system of class n and every reference signal of class r ensures either i practical tracking in the sense that prespecified asymptotic tracking accuracy quantified by 0 is assured or ii asymptotic tracking in the sense that the tracking error approaches zero the first case i is achievable by continuous feedback the second case ii necessitates discontinuous feedback both cases are developed within a framework of functional differential inclusions

universal commutator relations bogomolov multipliers and commuting probability abstract let g be a finite p group we prove that whenever the commuting probability of g is greater than 2 p 2 p 2 p 5 the unramified brauer group of the field of g invariant functions is trivial equivalently all relations between commutators in g are consequences of some universal ones the bound is best possible and gives a global lower bound of 1 4 for all finite groups the result is attained by describing the structure of groups whose bogomolov multipliers are nontrivial and bogomolov multipliers of all of their proper subgroups and quotients are trivial applications include a classification of p groups of minimal order that have nontrivial bogomolov multipliers and are of nilpotency class 2 a nonprobabilistic criterion for the vanishing of the bogomolov multiplier and establishing a sequence of bogomolov s absolute minimal factors which are 2 groups of arbitrarily large nilpotency class thus providing counterexamples to some of bogomolov s claims in relation to this we fill a gap in the proof of triviality of bogomolov multipliers of finite almost simple groups

equivalence of input output stability and exponential stability abstract for a large class of well posed infinite dimensional systems we show that if their state space realisation is exponentially stabilizable and detectable then exponential stability of the system operator is equivalent to input output stability

generalized sampled data stabilization of well posed linear infinite dimensional systems we consider well posed linear infinite dimensional systems the outputs of which are sampled in a generalized sense using a suitable weighting function under certain natural assump tions on the system the weighting function and the sampling period we show that there exists a generalized hold function such that unity sampled data feedback renders the closed loop system exponentially stable in the state space sense as well as l 2 stable in the input output sense to illustrate our main result we describe an application to a structurally damped euler bernoulli beam

infinite dimensional linear systems with unbounded control and observation a functional analytic approach abstract the object of these notes is to develop a unifying framework for the functional analytic representation of infinite dimensional linear systems with unbounded input and output operators on the basis of the general approach new results are derived on the wellposedness of feedback systems and on the linear quadratic control problem the implications of the theory for large classes of functional and partial differential equations are discussed in detail keywords representation of infinite dimensional systems semigroups boundary control feedback linear quadratic control

exponential stabilization of well posed systems by colocated feedback we consider well posed linear systems whose state trajectories satisfy dot x ax bu where u is the input and a is an essentially skew adjoint and dissipative operator on the hilbert space x this means that the domains of a and a are equal and a a q where q geq 0 is bounded on x the control operator b is possibly unbounded but admissible and the observation operator of the system is b such a description fits many wave and beam equations with colocated sensors and actuators and it has been shown for many particular cases that the feedback u ka y v with ka 0 stabilizes the system strongly or even exponentially here y is the output of the system and v is the new input we show by means of a counterexample that if b is sufficiently unbounded then such a feedback may be unsuitable the closed loop semigroup may even grow exponentially our counterexample is a simple regular system with feedthrough operator zero however we prove that if the original system is exactly controllable and observable and if ka is sufficiently small then the closed loop system is exponentially stable

discrete time low gain control of uncertain infinite dimensional systems using a frequency domain analysis it is shown that the application of a feedback controller of the form k z 1 or kz z 1 where k spl isin r to a power stable infinite dimensional discrete time system with square transfer function matrix g z will result in a power stable closed loop system which achieves asymptotic tracking of arbitrary constant reference signals provided that i all the eigenvalues of g 1 have positive real parts and ii the gain parameter k is positive and sufficiently small moreover if g 1 is positive definite we show how the gain parameter gain x can be tuned adaptively the resulting adaptive tracking controllers are universal in the sense that they apply to any power stable system with g 1 0 in particular they are not based on system identification or plant parameter estimation algorithms nor is the injection of probing signals required finally we apply these discrete time results to obtain adaptive sampled data low gain controllers for the class of regular systems a rather general class of infinite dimensional continuous time systems for which convenient representations are known to exist both in state space and in frequency domain we emphasize that our results guarantee not only asymptotic tracking at the sampling instants but also in the sampling interval

a physically motivated class of scattering passive linear systems we introduce a class of scattering passive linear systems motivated by examples from mathematical physics the state space of the system is x h oplus e where h and e are hilbert spaces we also have a hilbert space e 0 which is dense in e with continuous embedding and e 0 is the dual of e 0 with respect to the pivot space e the input space is the same as the output space and it is denoted by u the semigroup generator has the structure a big begin smallmatrix 0 l l g frac 1 2 k k end smallmatrix big where l in cal l e 0 h and k in cal l e 0 u are such that big begin smallmatrix l k end smallmatrix big with domain e 0 is closed as an unbounded operator from e to h oplus u the operator g in cal l e 0 e 0 is such that rm re langle gw 0 w 0 rangle leq 0 for all w 0 in e 0 the observation operator is c left begin matrix 0 k end matrix right the control operator is b c and the output equatio

adaptive low gain integral control of multivariable well posed linear systems the principle of low gain integral control for finite dimensional systems is well known more recently low gain integral control results have been obtained for classes of infinite dimensional systems in this paper we show that integral control with a simple and natural adaptation of the integrator gain achieves tracking of constant reference signals for every exponentially stable multivariable well posed infinite dimensional linear system whose steady state gain matrix has its spectrum in the open right half plane our results considerably extend improve and simplify previous work by the authors siam j control optim 35 1997 pp 78 116

well posedness and controllability of a class of coupled linear systems we consider coupled systems consisting of an infinite dimensional part and a finite dimensional part connected in feedback such as for example the well known scole system a beam with a rigid body attached at one end the external world interacts with the coupled system via the finite dimensional part which receives the external input and sends out the output the infinite dimensional part is assumed to be such that it becomes well posed and strictly proper when connected in cascade with an integrator under several assumptions we derive well posedness and exact controllability results for such coupled systems the first main result concerns the case when the input signal of the finite dimensional part is the difference between the external input and the feedback signal the second main result allows a more general structure for the finite dimensional part we also prove a result for the approximate controllability of coupled systems