estimates for riesz kernels of eigenfunction expansions of elliptic differential operators on compact manifolds let 2 and be the eigenvalues and an orthonormal system of eigenvectors of a second order elliptic differential operator on a compact n dimensional manifold m the riesz means of order of an integrable function on m are defined by r x m 1 2 2 x y y d y in this paper we study the kernels and the operator norms of he operators r on lp m 1 p 2 n 1 n 3 we also prove that if 1 p 2 n 1 n 3 and np n 1 2 then the operators r are of weak type p p uniformly with respect to

feedback boundary control of linear hyperbolic systems with relaxation we consider boundary stabilization for one dimensional systems of linear hyperbolic partial differential equations with relaxation structure such equations appear in many applications by combining weighted lyapunov functions the structure is used to derive new stabilization results the result is illustrated with an application to boundary stabilization of water flows in open canals

on the asymptotics of the bochner riesz kernel on compact lie groups suppose g is an n dimensional compact connected lie group and for 0 let dr r 0 be the bochner riesz kernel on g we prove that the lebesgue constants dr 1 are of the same order as r n 1 2 as r tends to infinity 0 n 1 2 while dr 1 log r if n 1 2 as a convergence we find best conditions for the norm convergence of the bochner riesz means on the spaces l1 g and c g

vertical versus conical square functions we study the difference between vertical and conical square functions in the abstract and also in the specific case where the square functions come from an elliptic operator trans amer math soc 364 2012 no 10 5469 5489

brief paper boundary control with integral action for hyperbolic systems of conservation laws stability and experiments a strict lyapunov function for boundary control with integral actions of hyperbolic systems of conservation laws that can be diagonalised with riemann invariants is presented the time derivative of this lyapunov function can be made strictly negative definite by an appropriate choice of the boundary conditions and the integral control gains previous stability results are extended to guarantee the local convergence of the state towards a desired set point furthermore the control can be implemented as a feedback of the state measured only at the boundaries the control design method is illustrated with a hydraulic application namely the level and flow regulation in a reach of the sambre river and in the micro channel of valence respectively through simulations and experimentations

stability analysis of coupled linear ode hyperbolic pde systems with two time scales abstract this paper is concerned with a class of coupled ode pde systems with two time scales the fast constant time scale is modeled by a small positive perturbation parameter first we state a general sufficient stability condition for such systems this condition is also sufficient for the stability of the reduced and boundary layer subsystems however counterexamples illustrate that the converse is not true next we study the stability of such systems by taking into account the fact that the perturbation parameter is sufficiently small for linear ode coupled with fast hyperbolic pde systems the stability of both subsystems implies the stability of the full system on the other hand a counterexample shows that the full system can be unstable even though the two subsystems are stable for a pde coupled with fast ode system numerical simulations on academic examples are proposed moreover an application to boundary control of a gas flow transport system is used to illustrate the theoretical result

sharp estimates for lebesgue constants on compact lie groups suppose g is a n dimensional compact connected semisimple lie group and dr is the spherical dirichlet kernel on g we prove the existence of a positive constant k such that dr 1 kr n 1 2 this complements the known result dr 1 hr n 1 2 we also prove that for a polyhedral dirichlet kernel dn the above inequalities hold with np in place of r n 1 2 p is the number of positive roots of g

pi boundary control of linear hyperbolic balance laws with stabilization of arz traffic flow models abstract this paper investigates the proportional integral pi boundary feedback control for the linear hyperbolic systems of balance laws which control and output measures are located at the boundaries we address the issue of feedback stabilization by means of pi boundary controllers by constructing a new weighted lyapunov function the sufficient conditions in terms of matrix inequalities are developed for the exponential stability of closed loop systems these results are illustrated by the linearized aw rascle zhang arz traffic flow model we design a pi boundary controller to stabilize the oscillations of the traffic parameters of a freeway segment and evaluate the performance with numerical simulations

stability of linear density flow hyperbolic systems under pi boundary control we consider a class of density flow systems described by linear hyperbolic conservation laws which can be monitored and controlled at the boundaries these control systems are open loop unstable and subject to unmeasured flow disturbances we address the issue of feedback stabilization and disturbance rejection under pi boundary control explicit necessary and sufficient stability conditions in the frequency domain are provided

tent spaces associated with semigroups of operators abstract we study tent spaces on general measure spaces we assume that there exists a semigroup of positive operators on l p satisfying a monotone property but do not assume any geometric metric structure on the semigroup plays the same role as integrals on cones and cubes in euclidean spaces we then study bmo spaces on general measure spaces and get an analogue of fefferman s h 1 bmo duality theory we also get a h 1 bmo duality inequality without assuming the monotone property all the results are proved in a more general setting namely for noncommutative l p spaces

boundary feedback control in networks of open channels this article deals with the regulation of water flow in open channels modelled by saint venant equations by means of a riemann invariants approach we deduce stabilizing control laws for a single horizontal reach without friction the stability condition is extended to a general class of hyperbolic systems which can describe canal networks with more general topologies a control law design based on this condition is illustrated with a simple case study two reaches in cascade the proof of the main stability theorem is based on a previous result from li ta tsien concerning the existence and decay of classical solutions of hyperbolic systems

control of 2 2 linear hyperbolic systems backstepping based trajectory generation and pi based tracking abstract we consider the problems of trajectory generation and tracking for general 2 2 systems of first order linear hyperbolic pdes with anti collocated boundary input and output we solve the trajectory generation problem via backstepping the reference input which generates the desired output incorporates integral operators acting on advanced and delayed versions of the reference output with kernels which were derived by vazquez krstic and coron for the backstepping stabilization of 2 2 linear hyperbolic systems we apply our approach to a wave pde with indefinite in domain and boundary damping for tracking the desired trajectory we employ a pi control law on the tracking error of the output we prove exponential stability of the closed loop system under the proposed pi control law when the parameters of the plant and the controller satisfy certain conditions by constructing a novel non diagonal lyapunov functional we demonstrate that the proposed pi control law compensates in the output the effect of in domain and boundary disturbances we illustrate our results with numerical examples

finite time stabilization of 2 times2 hyperbolic systems on tree shaped networks we investigate the finite time boundary stabilization of a one dimensional first order quasilinear hyperbolic system of diagonal form on 0 1 the dynamics of both boundary controls are governed by a finite time stable ode the solutions of the closed loop system issuing from small initial data in lip 0 1 are shown to exist for all times and to reach the null equilibrium state in finite time when only one boundary feedback law is available a finite time stabilization is shown to occur roughly in a twice longer time the above feedback strategy is then applied to the saint venant system for the regulation of water flows in a network of canals

adding integral action for open loop exponentially stable semigroups and application to boundary control of pde systems the article deals with the output feedback regulation of exponentially stable systems by an integral controller we propose appropriate lyapunov functionals to prove exponential stability of the closed loop system an example of parabolic partial differential equation pde systems and an example of hyperbolic systems are worked out to show how exponentially stabilizing integral controllers are designed the proof is based on a novel lyapunov functional construction that employs the forwarding techniques

pi regulation of a reaction diffusion equation with delayed boundary control the general context of this work is the feedback control of an infinite dimensional system so that the closed loop system satisfies a fading memory property and achieves the setpoint tracking of a given reference signal more specifically this paper is concerned with the proportional integral pi regulation control of the left neumann trace of a one dimensional reaction diffusion equation with a delayed right dirichlet boundary control in this setting the studied reaction diffusion equation might be either open loop stable or unstable the proposed control strategy goes as follows first a finite dimensional truncated model that captures the unstable dynamics of the original infinite dimensional system is obtained via spectral decomposition the truncated model is then augmented by an integral component on the tracking error of the left neumann trace after resorting to the artstein transformation to handle the control input delay the pi controller is designed by pole shifting stability of the resulting closed loop infinite dimensional system consisting of the original reaction diffusion equation with the pi controller is then established thanks to an adequate lyapunov function in the case of a time varying reference input and a time varying distributed disturbance our stability result takes the form of an exponential input to state stability iss estimate with fading memory finally another exponential iss estimate with fading memory is established for the tracking performance of the reference signal by the system output in particular these results assess the setpoint regulation of the left neumann trace in the presence of distributed perturbations that converge to a steady state value and with a time derivative that converges to zero numerical simulations are carried out to illustrate the efficiency of our control strategy

stabilization of ode with hyperbolic equation actuator subject to boundary control matched disturbance in this paper we consider stabilisation for a cascade of ode and first order hyperbolic equation with external disturbance flowing to the control end the active disturbance rejection control adr

design of integral controllers for nonlinear systems governed by scalar hyperbolic partial differential equations the paper deals with the control and regulation by integral controllers for the nonlinear systems governed by scalar quasi linear hyperbolic partial differential equations both the control input and the measured output are located on the boundary the closed loop stabilization of the linearized model with the designed integral controller is proved first by using the method of spectral analysis and then by the lyapunov direct method based on the elaborated lyapunov function we prove local exponential stability of the nonlinear closed loop system with the same controller the output regulation to the set point with zero static error by the integral controller is shown upon the nonlinear system numerical simulations by the preissmann scheme are carried out to validate the robustness performance of the designed controller to face to unknown constant disturbances

pi controllers for 1 d nonlinear transport equation in this paper we introduce a method to obtain necessary and sufficient stability conditions for systems governed by one dimensional nonlinear hyperbolic partial differential equations with closed loop integral controllers when the linear frequency analysis cannot be used anymore we study the stability of a general nonlinear transport equation where the control input and the measured output are both located on the boundaries the principle of the method is to extract the limiting part of the stability from the solution using a projector on a finite dimensional space and then use a lyapunov approach this paper improves a result of trinh andrieu and xu and gives an optimal condition for the design of the controller the results are illustrated with numerical simulations where the predicted stable and unstable regions can be clearly identified

functional calculus for generators of symmetric contraction semigroups we prove that every generator of a symmetric contraction semigroup on a finite measure space admits for 1 p a hormander type holomorphic functional calculus on lp in the sector of angle p arcsin 1 2 p the obtained angle is optimal

admissible decomposition for spectral multipliers on gaussian l p l p this paper concerns harmonic analysis of the ornstein uhlenbeck operator l on the euclidean space we examine the method of decomposing a spectral multiplier phi l into three parts according to the notion of admissibility which quantifies the doubling behaviour of the underlying gaussian measure gamma we prove that the above mentioned admissible decomposition is bounded in l p gamma for 1 p le 2 in a certain sense involving the gaussian conical square function the proof relates admissibility with e nelson s hypercontractivity theorem in a novel way

harmonic analysis on semigroups wherever t 0 and 1 p so e m stein st 3 developed a littlewoodpaley theory for such semigroups with some additional hypotheses r r coifman r rochberg and g weiss crw pointed out that some of stein s estimates could be obtained more simply by the use of transference techniques which were developed by coifman and weiss cw our object is to push these transference methods to present an alternative and simpler approach to some of the principal results of stein s book we shall also use these results to prove estimates for the wave equation

on the modelling and stabilization of flows in networks of open canals in this paper we present a model for the controlled flow of a fluid through a network of canals using a coupled system of st venant equations we then generalize in a variety of ways recent results of coron d andrea novel and bastin concerning the stabilizability around equilibrium of the flow through a single channel this work is based on the theory of quasilinear hyperbolic systems and in particular on a delicate result of li ta tsien concerning the existence and decay of global classical solutions

boundary feedback control of linear hyperbolic systems application to the saint venant exner equations distributed parameter systems modeled by hyperbolic partial differential equations are considered in this paper the dynamic models include a source term and heterodirectional velocities a boundary control problem is introduced and it is first shown that it is well posed in the sense of tucsnak and weiss under appropriate assumptions coupling the boundary conditions and the source term then a sufficient exponential stability condition is derived using operator theory this condition is written in terms of linear matrix inequalities that are numerically tractable and that allow an optimization program connections with another classical stability condition are given this approach is applied to the saint venant exner equation describing the dynamics of the water level of the water flow and of the sediment inside of a channel the effect of the friction and of the slope are taken into account in the application model

bmo and h1 for the ornstein uhlenbeck operator in this paper we develop a theory of singular integral operators acting on function spaces over the measured metric space rd where denotes the euclidean distance and the gauss measure our theory plays for the ornstein uhlenbeck operator the same role that the classical calderon zygmund theory plays for the laplacian on rd where is the lebesgue measure our method requires the introduction of two new function spaces the space bmo of functions with bounded mean oscillation and its predual the atomic hardy space h1 we show that if p is in 2 then lp is an intermediate space between l2 and bmo and that an inequality of john nirenberg type holds for functions in bmo then we show that if m is a bounded operator on l2 and the schwartz kernels of m and of its adjoint satisfy a local integral condition of hormander type then m extends to a bounded operator from h1 to l1 from l to bmo and on lp for all p in 1 as an application we show that certain singular integral operators related to the ornstein uhlenbeck operator which are unbounded on l1 and on l turn out to be bounded from h1 to l1 and from l to bmo

numerical boundary feedback stabilisation of non uniform hyperbolic systems of balance laws in this paper numerical boundary stabilisation of a non uniform hyperbolic system of balance laws is studied for the numerical discretisation of the balance laws a first order explicit upwind sc

l p norms of certain kernels on the n dimensional torus in this paper we study a class of kernels fr which generalize the bochner riesz kernels on the n dimensional torus our main result consists in upper estimates for the lp norms of fr as r tends to infinity as a consequence we prove a convergence theorem for means of functions belonging to suitable besov spaces 1 throughout this paper we identify the n dimensional torus tn n 2 with the n dimensional cube qn x e r n 1 2 0 we form the means fr g t f r m g m exp 2 7imt 1 m where t e tn m ranges on the integer lattice zn and denotes the convolution in tn in several cases one is able to estimate the l 1 tn norm of the kernel fr t e f r 1m exp 2 srimt 2 m and if 0 e s and f o 1 deduce convergence results for the means 1 as r x oc when g belongs to certain classes of periodic functions see e g 1 3 7 8 12 for instance if f x 1 i x12 8 when lxl 1 then fr k a is the familiar bochner riesz kernel in this case k i babenko cf 1 has shown that iikriil is of the same order as r n 1 2 as r tends to infinity n 2 0 8 n 1 2 while stein proved the estimate iikr il log r if 8 n 1 2 9 recently yudin 12 obtained the estimate lf o r n1 2 if f is the characteristic function of a closed balanced set c whose boundary has finite upper minkowski measure the method of yudin which also yields estimates for the lp tn norms of fr uses jackson approximation theorem which in tum involves estimates for the l2 r n modulus of continuity of f such a method can be adapted to a more general situation as we show in this paper in 2 4 we consider kernels of the type 2 associated with functionsf whose derivatives of certain orders are controlled by not necessarily positive received by the editors may 13 1980 1980 mathematics subject classification primary 42b99 secondary 46e35

boundary observers for linear and quasi linear hyperbolic systems with application to flow control in this paper we consider the problem of boundary observer design for one dimensional first order linear and quasi linear strict hyperbolic systems with n rightward convecting transport pdes by means of lyapunov based techniques we derive some sufficient conditions for exponential boundary observer design using only the information from the boundary control and the boundary conditions we consider static as well as dynamic boundary controls for the boundary observer design the main results are illustrated on the model of an inviscid incompressible flow

lyapunov based boundary control for a class of hyperbolic lotka volterra systems this paper considers a boundary feedback control problem for two first order nonlinearly coupled hyperbolic partial differential equations with lotka volterra type coupling boundary control action is used on one equation to drive the state at the end of the spatial domain to a desired constant reference value static and dynamic boundary controllers are designed based on a special lyapunov functional that is related to an entropy function the time derivative of the entropy function is made strictly negative by an appropriate choice of boundary conditions a unique classical solution is shown to exist globally in time and asymptotic exponential convergence to the desired steady state solution is shown in the norm the boundary control design is illustrated with simulations

regularity and lyapunov stabilization of weak entropy solutions to scalar conservation laws we consider the problem of lyapunov boundary stabilization of the weak entropy solution to a scalar conservation law with strictly convex flux in one dimension of space around a uniform equilibrium we show that for a specific class of boundary conditions the solution to the initial boundary value problem for an initial condition with bounded variations can be approximated arbitrarily closely in the l 1 norm by a piecewise smooth solution with finitely many discontinuities the constructive method we present designs explicit boundary conditions in this class which guarantee lyapunov stability of the weak entropy solution to the initial boundary value problem we show how the greedy control obtained by maximizing the decrease of the natural lyapunov function may fail to asymptotically stabilize and a brute force control generates unbounded variation of traces we then design a stabilizing control which avoid oscillations and propose a nonlocal technique depending on time and the whole initial datum which optimizes the convergence time controllers performance is illustrated on numerical benchmarks using the godunov scheme