nonlinear scattering theory at low energy abstract we study the scattering theory of a conservative nonlinear one parameter group of operators on a hilbert space x relative to a group of linear unitary operators under certain hypotheses the scattering operator carries a neighborhood of 0 in x into x the theory is designed to apply to the semilinear schrodinger and klein gordon equations

a boundedness criterion for generalized calderon zygmund operators in 1965 a p calderon showed the l2 boundedness of the so called first calderon commutator this is one of the first examples of a non convolution operator associated to a kernel which satisfies certain size and smoothness properties comparable to those of the kernel of the hilbert transform these properties together with the l2 boundedness imply the lp boundedness for all p s in 1 oo many operators in analysis such as certain classes of pseudo differential operators and the cauchy integral operator on a curve are associated with kernels having these properties for these operators one of the major questions is if they are bounded on l2 we are going to give necessary and sufficient conditions for such an operator to be bounded on l2 they are essentially that the images of the function 1 under the actions of the operator and its adjoint both lie in bmo in the case of the aforementioned first commutator this can be checked by an integration by parts in the first section we present some basic notions and state the theorem which is proved in sections 2 and 3 in section 4 we show how to recover some classical results in sections 5 and 6 we construct a functional calculus for small perturbations of a and in section 7 we show a connection between the theory of calderon zygmund operators and kato s conjecture it is a pleasure to express our thanks to r r coifman and y meyer for suggesting many elegant simplifications in our proofs and most of the applications we also wish to thank stephen semmes for several pertinent remarks

normal forms and quadratic nonlinear klein gordon equations on construit une transformation qui change l equation a non linearite cubique que l on sait resoudre a 3 1 dimensions on demontre que la solution du probleme original a le meme comportement asymptotique que l equation de klein gordon non lineaire

np hard graph problems and boundary classes of graphs any graph problem which is np hard in general graphs becomes polynomial time solvable when restricted to graphs in special classes when does a difficult problem become easy to answer this question we study the notion of boundary classes in the present paper we define this notion in its most general form describe several approaches to identify boundary classes and apply them to various graph problems

state and output feedback boundary control for a coupled pde ode system abstract this note is devoted to stabilizing a coupled pde ode system with interaction at the interface first a state feedback boundary controller is designed and the system is transformed into an exponentially stable pde ode cascade with an invertible integral transformation where pde backstepping is employed moreover the solution to the resulting closed loop system is derived explicitly second an observer is proposed which is proved to exhibit good performance in estimating the original coupled system and then an output feedback boundary controller is obtained for both the state and output feedback boundary controllers exponential stability analyses in the sense of the corresponding norms for the resulting closed loop systems are provided the boundary controller and observer for a scalar coupled pde ode system as well as the solutions to the closed loop systems are given explicitly

boundary stabilization of a coupled wave ode system with internal anti damping in this article the backstepping method is employed to stabilize a coupled wave ode system with internal anti damping by means of decoupling them into a stable cascaded wave ode system at the same time the existence of the kernels in backstepping transformation and inverse transformation is proved by iterative method

singular perturbation approximation of linear hyperbolic systems of balance laws this technical note deals with a class of linear hyperbolic systems of balance laws with multiple time scales the scale of time constants is modeled by a perturbation parameter this parameter is introduced in both dynamics and boundary conditions the solution of the full system is approximated by that of the reduced subsystem when the perturbation parameter is small enough lyapunov technique is used to prove it the main result is illustrated by an academic example moreover the boundary control synthesis to a gas flow transport model is shown based on singular perturbation approach

heat equation with memory lack of controllability to rest abstract we prove that the one dimensional heat equation with memory cannot be controlled to rest for large classes of memory kernels and controls the approach is based on the application of the theory of interpolation in paley wiener spaces

blow up for quasi linear wave equations in three space dimensions we consider equations of the form u phi x t u u u 1a for a function u u x 1 x n t u x t here denotes the d alembertian square frac partial 2 partial t 2 vartriangle 1b and u represents the vector of first derivatives u that of second derivatives of u with respect to the x k and t on linearizing 1a shall go over into the classical wave equation u 0 this means that and its first derivatives with respect to u u u shall vanish for u u u 0

closed form boundary state feedbacks for a class of 1 d partial integro differential equations in this paper a problem of boundary stabilization of a class of linear parabolic partial integro differential equations p i des in one dimension is considered using the method of backstepping avoiding spatial discretization required in previous efforts the problem is formulated as a design of an integral operator whose kernel is required to satisfy a hyperbolic p i de the kernel p i de is then converted into an equivalent integral equation and by applying the method of successive approximations the equation s well posedness and the kernel s smoothness are established it is shown how to extend this approach to design optimally stabilizing controllers an adaptation mechanism is developed to reduce the conservativeness of the inverse optimal controller and the performance bounds are derived for a broad range of physically motivated special cases feedback laws are constructed explicitly and the closed loop solutions are found in closed form a numerical scheme for the kernel p i de is proposed its numerical effort compares favorably with that associated with operator riccati equations

planar formulae and their uses we define the set of planar boolean formulae and then show that the set of true quantified planar formulae is polynomial space complete and that the set of satisfiable planar formulae is np complete using these results we are able to provide simple and nearly uniform proofs of np completeness for planar node cover planar hamiltonian circuit and line geometric connected dominating set and of polynomial space completeness for planar generalized geography the np completeness of planar node cover and planar hamiltonian circuit and line were first proved elsewhere m r garey and d s johnson the rectilinear steiner tree is np complete siam j appl math 32 1977 pp 826 834 and m r garey d s johnson and r e tarjan the planar hamilton circuit problem is np complete siam j comp 5 1976 pp 704 714

remark on the global existence and large time asymptotics of solutions for the quadratic nls abstract we study the global in time existence of small solutions to the nonlinear schrodinger equation with quadratic interactions 0 1 i t u 1 2 u u 2 t 0 x r 4 u 0 x u 0 x x r 4 we prove that if the initial data u 0 satisfy smallness conditions in the weighted sobolev norm then the solution of the cauchy problem 0 1 exists globally in time furthermore we prove the existence of the usual scattering states and find the large time asymptotics of the solutions

boundary stabilization of a 1 d wave equation with in domain antidamping we consider the problem of boundary stabilization of a 1 d one dimensional wave equation with an internal spatially varying antidamping term this term puts all the eigenvalues of the open loop system in the right half of the complex plane we design a feedback law based on the backstepping method and prove exponential stability of the closed loop system with a desired decay rate for plants with constant parameters the control gains are found in closed form our design also produces a new lyapunov function for the classical wave equation with passive boundary damping

brief paper lyapunov exponential stability of 1 d linear hyperbolic systems of balance laws explicit boundary dissipative conditions are given for the exponential stability in l 2 norm of one dimensional linear hyperbolic systems of balance laws email protected email protected email protected x 0 over a finite interval when the matrix m is marginally diagonally stable the result is illustrated with an application to boundary feedback stabilisation of open channels represented by linearised saint venant exner equations

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

minimum weight triangulation is np hard a triangulation of a planar point set s is a maximal plane straight line graph with vertex set s in the minimum weight triangulation mwt problem we are looking for a triangulation of a given point set that minimizes the sum of the edge lengths we prove that the decision version of this problem is np hard using a reduction from planar 1 in 3 sat the correct working of the gadgets is established with computer assistance using dynamic programming on polygonal faces as well as the skeleton heuristic to certify that certain edges belong to the minimum weight triangulation

controllability and observability of a well posed system coupled with a finite dimensional system we consider coupled systems consisting of a well posed and strictly proper hence regular subsystem and a finite dimensional subsystem connected in feedback the external world interacts with the coupled system via the finite dimensional part which receives the external input and sends out the output under several assumptions we derive well posedness regularity exact or approximate controllability and exact or approximate observability results for such coupled systems

almost global existence to nonlinear wave equations in three space dimensions on considere l existence a t grand de solutions de petite amplitude pour des equations d onde d ordre 2 non lineaires a 4 dimensions d espace temps

some non existence results for the semilinear schrodinger equation without gauge invariance we consider the cauchy problem for the semilinear schrodinger equation nls i t u u p t x 0 t rd u 0 x f x x rd where u u t x is a c valued unknown function c 0 p 1 0 f f x is a c valued given function and t is a maximal existence time of the solution our first aim in the present paper is to prove a large data blow up result for nls in hs critical or hs subcritical case p ps 1 4 d 2s for some s 0 more precisely we show that in the case 1 0 such that for any 0 the following estimates t c and limt t 0 u t hs if1 p ps u ltr 0 t b 2s ifp ps hold where c 0 are constants independent of and r is an admissible pair see theorem 2 3 our second aim is to prove non existence local weak solution for nls in the hs supercritical case p ps which means that if p ps then there exists an hs function f such that if there exist t 0 and a weak solution u to nls on 0 t then 0 see theorem 2 5

tikhonov theorem for linear hyperbolic systems abstract a class of linear hyperbolic systems of conservation laws with multiple time scales which are modeled by a perturbation parameter is considered in this paper by setting the perturbation parameter to zero two subsystems the reduced subsystem standing for the slow dynamics and the boundary layer subsystem representing the fast dynamics are computed it is first proved that the exponential stability of the full system implies the stability of both subsystems secondly a counterexample is given to indicate that the stability of the two subsystems does not ensure the full system s stability moreover a new tikhonov theorem for this class of infinite dimensional systems is stated the solution of the full system can be approximated by that of the reduced subsystem and this is proved by lyapunov techniques an application to boundary feedback stabilization of gas transport model is used to illustrate the results

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

small size eps nets for axis parallel rectangles and boxes we show the existence of varepsilon nets of size o left frac 1 varepsilon log log frac 1 varepsilon right for planar point sets and axis parallel rectangular ranges the same bound holds for points in the plane and fat triangular ranges and for point sets in boldsymbol r 3 and axis parallel boxes these are the first known nontrivial bounds for these range spaces our technique also yields improved bounds on the size of varepsilon nets in the more general context considered by clarkson and varadarajan for example we show the existence of varepsilon nets of size o left frac 1 varepsilon log log log frac 1 varepsilon right for the dual range space of fat regions and planar point sets where the regions are the ground objects and the ranges are subsets stabbed by points plugging our bounds into the technique of bronnimann and goodrich or of even rawitz and shahar we obtain improved approximation factors computable in expected polynomial time by a randomized algorithm for the hitting set or the set cover problems associated with the corresponding range spaces

global existence of small amplitude solutions to nonlinear klein gordon equations in four space time dimensions consider the following class of nonlinear wave equations square text u u text text f u u u n k g where t 2 1 2 2 2 3 2 is the d alembertian of the 4 dimensional minkowski space time and f a smooth function of u u t x and its first and second partial derivates vanishing together with its first derivatives at u u u 0 in other words f is at least quadratic in u u u in a neighborhood of the trivial solution u 0 we subject u to the pure initial value problem text u mkern 1mu varepsilon text f x u t varepsilon g x at t 0 i v p with f g c 0 3 and e 0 a small parameter

singular perturbation analysis of a coupled system involving the wave equation this article considers a system coupling an ordinary differential equation with a wave equation through its boundary data the existence of a small parameter in the wave equation as a factor multiplying the time derivative suggests the idea of applying a singular perturbation method to get the stability of the full system by analyzing the stability of some appropriate subsystems given by the method however for infinite dimensional systems it is known that in some cases this method does not work indeed one cannot be sure of the stability of the full system even if the given subsystems are stable in this article we prove that the singular perturbation method works for the system under study using this strategy we get the stability of the system and a tikhonov theorem which is the first of this kind for systems involving the wave equation simulations are performed to show the applicability of our results

initial boundary value problem for a singularly perturbed parabolic equation in case of exchange of stability the theorem on passage to the limit is proved for a singularly perturbed parabolic problem in case when the degenerate equation has two intersecting roots

lyapunov stability analysis of a string equation coupled with an ordinary differential system this paper considers the stability problem of a linear time invariant system in feedback with a string equation a new lyapunov functional is proposed using augmented states that enriches and encompasses the classical functionals of the literature it results in tractable stability conditions expressed in terms of linear matrix inequalities this methodology follows from the application of the bessel inequality to the projections over the legendre polynomials numerical examples illustrate the potential of our approach through three scenario a stable ode perturbed by the pde an unstable open loop ode and an unstable closed loop ode stabilized by the pde

new upper bounds in klee s measure problem new upper bounds for the measure problem of klee are given which significantly improve the previous bounds for dimensions greater than two an o n d 2 log n n time space upper bound is obtained and used to compute the measure of a set of n boxes in euclidean d space the solution is based on a new data structure which is called an orthogonal partition tree this structure has other applications as well

uniform decay estimates and the lorentz invariance of the classical wave equation on etudie le comportement asymptotique des solutions de u 0 ou t 2 1 2 n 2 pour des conditions initiales u 0 u t g x en t 0 avec g reguliere a support compact dans r n

computing coverage kernels under restricted settings abstract given a set b of d dimensional boxes i e axis aligned hyperrectangles a minimum coverage kernel is a subset of b of minimum size covering the same region as b computing it is np hard but as for many similar np hard problems e g box cover and orthogonal polygon covering the problem becomes solvable in polynomial time under restrictions on b we show that computing minimum coverage kernels remains np hard even when restricting the graph induced by the input to a highly constrained class of graphs alternatively we present two polynomial time approximation algorithms for this problem one deterministic with an approximation ratio within o log n and one randomized with an improved approximation ratio within o lg opt with high probability

global existence of small solutions to nonlinear evolution equations abstract we study the global existence and asymptotic behaviour of small solutions of a large class of nonlinear partial differential equations if the nonlinear terms are of high degree the solutions will be asymptotic to solutions of the linear equation

boundary feedback stabilization of an unstable heat equation in this paper we study the problem of boundary feedback stabilization for the unstable heat equation ut x t uxx x t a x u x t this equation can be viewed as a model of a heat conducting rod in which not only is the heat being diffused mathematically due to the diffusive term uxx but also the destabilizing heat is generating mathematically due to the term a u with a 0 we show that for any given continuously differentiable function a and any given positive constant l we can explicitly construct a boundary feedback control law such that the solution of the equation with the control law converges to zero exponentially at the rate of l this is a continuation of the recent work of boskovic krstic and liu ieee trans automat control 46 2001 pp 2022 2028 and balogh and krstic european j control 8 2002 pp 165 176

backstepping observers for a class of parabolic pdes in this paper we design exponentially convergent observers for a class of parabolic partial integro differential equations p i des with only boundary sensing available the problem is posed as a problem of designing an invertible coordinate transformation of the observer error system into an exponentially stable target system observer gain output injection function is shown to satisfy a well posed hyperbolic pde that is closely related to the hyperbolic pde governing backstepping control gain for the state feedback problem for several physically relevant problems the observer gains are obtained in closed form the observer gains are then used for an output feedback design in both collocated and anti collocated setting of sensor and actuator the order of the resulting compensator can be substantially lowered without affecting stability explicit solutions of a closed loop system are found in particular cases

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

nonlinear control of a coupled pde ode system modeling a switched power converter with a transmission line we consider an infinite dimensional system modelling a boost converter connected to a load via a transmission line the governing equations form a system coupling the telegraph partial differential equation with the ordinary differential equations model ing the converter the coupling is given by the boundary conditions and the nonlinear controller we introduce we design a nonlinear saturating control law using a lya punov function for the averaged model of the system the main results give the well posedness and stability properties of the obtained closed loop system