on the controllability of some equations of sobolev galpern type abstract in this paper we deal with the controllability problem for some sobolev type equations we show that the equations cannot be driven to zero if the control region is strictly supported within the domain nevertheless we also prove that it is possible to control the equations using controls which have a moving support under some assumptions on their movement

a note on a class of observability problems for pdes abstract the question of observability arises naturally in the analysis of control problems if the solution of a pde initial boundary value problem is known to be zero in a part of the domain does this guarantee it is zero everywhere the most popular techniques to establish such results are based on local unique continuation results holmgren s theorem or carleman estimates the purpose of this note is to draw attention to a class of problems where the observed region is bounded by characteristics and local unique continuation fails nevertheless observability may hold a problem of this nature arose in recent work by the author on control of viscoelastic flows m renardy are viscoelastic flows under control or out of control system control lett 54 2005 1183 1193 in this note we first analyze a simple example which shares the same essential features specifically we consider the problem u x t u for spatially periodic solutions we show that observability holds for data given on the line x 0 we shall show however that there is no observability estimate we shall then show how the methods used in the more elementary example can be extended to the case of viscoelastic shear flows

null controllability of a system of linear thermoelasticity we consider a linear system of thermoelasticity in a compact c infin n dimensional connected riemannian manifold this system consists of a wave equation coupled to a heat equation when the boundary of the manifold is non empty dirichlet boundary conditions are considered we study the controllability properties of this system when the control acts in the hyperbolic equation and not in the parabolic one and has its support restricted to an open subset of the manifold we show that if the control time and the support of the control satisfy the geometric control condition for the wave equation this system of thermoelasticity is null controllable more precisely any finite energy solution can be driven to zero at the control time an analogous result is proved when the control acts on the parabolic equation finally when the manifold has no boundary the null controllability of the linear system of three dimensional thermoelastic ity is proved

local exact controllability to the trajectories of the 1 d kuramoto sivashinsky equation abstract we are concerned with the boundary controllability to the trajectories of the kuramoto sivashinsky equation by using a carleman estimate we obtain the null controllability of the linearized equation around a given solution from a local inversion theorem we get the local controllability to the trajectories of the nonlinear system

on the controllability of the improved boussinesq equation the improved boussinesq equation is studied in this paper control properties for this equation posed on a bounded interval are first considered when the control acts through the dirichlet boundary condition the linearized system is proved to be approximately but not spectrally controllable in a second part the equation is posed on the one dimensional torus and distributed moving controls are considered under some condition on the velocity at which the control moves exact controllability results for both linear and nonlinear improved boussinesq equations are obtained applying the moment method and a fixed point argument

controllability and stabilizability of the linearized compressible navier stokes system in one dimension in this paper we consider the one dimensional compressible navier stokes system linearized about a constant steady state q 0 0 with q 0 0 we study the controllability and stabilizability of this linearized system we establish that the linearized system is null controllable for regular initial data by an interior control acting everywhere in the velocity equation we prove that this result is sharp by showing that the null controllability cannot be achieved by a localized interior control or by a boundary control acting only in the velocity equation on the other hand we show that the system is approximately controllable we also show that the system is not stabilizable with a decay rate e omega t for omega omega 0 where omega 0 is an accumulation point of the real eigenvalues of the linearized operator

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

approximate controllability for linearized compressible barotropic navier stokes system in one and two dimensions abstract in this paper we consider compressible barotropic navier stokes equations in one and two dimensions linearized around a constant steady state with dirichlet boundary conditions we explore the controllability of this linearized system using a control only for the velocity equation we prove that the system with homogeneous dirichlet boundary conditions is approximately controllable by a localized interior control when time is sufficiently large

feedback stabilization of a fluid structure model we study a system coupling the incompressible navier stokes equations in a 2d rectangular type domain with a damped euler bernoulli beam equation where the beam is a part of the upper boundary of the domain occupied by the fluid due to the deformation of the beam the fluid domain depends on time we prove that this system is exponentially stabilizable locally about the null solution with any prescribed decay rate by a feedback control corresponding to a force term in the beam equation the feedback is applied on the whole structure and it is determined via a riccati equation by solving an infinite time horizon control problem for the linearized model a crucial step in this analysis consists of showing that this linearized system can be rewritten thanks to an analytic semigroup of which the infinitesimal generator has a compact resolvent

null controllability for the parabolic equation with a complex principal part abstract the paper is devoted to a study of the null controllability for the semilinear parabolic equation with a complex principal part for this purpose we establish a key weighted identity for partial differential operators i t j k 1 n k a j k j with real functions and by which we develop a universal approach based on global carleman estimate to deduce not only the desired explicit observability estimate for the linearized complex ginzburg landau equation but also all the known controllability observability results for the parabolic hyperbolic schrodinger and plate equations that are derived via carleman estimates

mobile point controls versus locally distributed ones for the controllability of the semilinear parabolic equation it is well known now that a rather general semilinear parabolic equation with globally lipschitz nonlinear term is both approximately and exactly null controllable in l2 omega when governed in a bounded domain by the locally distributed controls in this paper we intend to show that in fact in one space dimension omega 0 1 the very same results can be achieved by employing em at most two mobile point controls with support on the curves properly selected within an arbitrary subdomain of qt 0 1 0 t we will show that such curves can be described by a certain differential inequality and the explicit examples are provided we also discuss some extensions of our main results to the superlinear terms and to the case of several dimensions

fredholm s equations for subwavelength focusing subwavelength focusing sf is a very useful tool that can be carried out with the use of left hand materials for optics that involve the range of the microwaves many recent works have described a successful alternative procedure using time reversal methods the advantage is that we do not need devices which require the complicated manufacture of left hand materials nevertheless the theoretical mathematical bases are far from complete because before now we lacked an adequate easy to apply frame in this work we give for a broad class of discrete systems a solid support for the theory of electromagnetic sf that can be applied to communications and nanotechnology the very central procedure is the development of vector matrix formalism vmf based on exploiting both the inhomogeneous and homogeneous fredholm s integral equations in cases where the last two kinds of integral equations are applied to some selected discrete systems to this end we first establish a generalized newmann series for the fourier transform of the green s function in the inhomogeneous fredholm s equation of the problem then we go from an integral operator equation to a vector matrix algebraic one in this way we explore the inhomogeneous case and later on also the very interesting one about the homogeneous equation thus on the one hand we can relate in a simple manner the arriving electromagnetic signals with those at their sources and we can use them to perform a sf on the other hand we analyze the homogeneous version of the equations finding resonant solutions that have analogous properties to their counterparts in quantum mechanical scattering that can be used in a proposed very powerful way in communications also we recover quantum mechanical operator relations that are identical for classical electromagnetics finally we prove two theorems that formalize the relation between the theory of fredholm s integral equations and the vmf we present here

null controllability of the structurally damped wave equation with moving control we investigate the internal controllability of the wave equation with structural damping on the one dimensional torus we assume that the control is acting on a moving point or on a moving small interval with a constant velocity we prove that the null controllability holds in some suitable sobolev space and after a fixed positive time independent of the initial conditions

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

automorphisms with finite reidemeister number in residually finite groups abstract let g be a residually finite group if g admits an automorphism of prime order p with finite reidemeister number then g is virtually nilpotent of class bounded by a function of p

actuator design for parabolic distributed parameter systems with the moment method in this paper we model and solve the problem of designing in an optimal way actuators for parabolic partial differential equations settled on a bounded open connected subset omega of r n we optimize not only the location but also the shape of actuators by finding what is the optimal distribution of actuators in omega over all possible such distributions of a given measure using the moment method we formulate a spectral optimal design problem which consists of maximizing a criterion corresponding to an average over random initial data of the largest l 2 energy of controllers since we choose the moment method to control the pde our study mainly covers one dimensional parabolic operators but we also provide several examples in higher dimensions we consider two types of controllers either internal controls modeled by characteristic functions or lumped controls that are tensorized functions in time and space under appropriate spectral assumptions we prove existence and uniqueness of an optimal actuator distribution and we provide a simple computation procedure numerical simulations illustrate our results

an algorithm for calculating the nielsen number on surfaces with boundary let f m m be a self map of a hyperbolic surface with boundary the nielsen number n f depends only on the induced map f of the fundamental group which can be viewed as a free group on n generators al an we determine conditions for fixed points to be in the same fixed point class and if these conditions are enough to determine the fixed point classes we say that f is w characteristic we define an algebraic condition on the f ai and show that most nmaps satisfy this condition and that all maps which satisfy this condition are w characteristic if f is wcharacteristic we present an algorithm for calculating n f and prove that the inequality jl f x m l n f x m holds where l f deno tes the lefschetz number of f and x m the euler characteristic of m thus answering in part a question of jiang and guo

reachability problems for a class of integro differential equations abstract we study control problems for some integro differential equations using the hilbert uniqueness method to do that we follow a harmonic analysis approach our results can be applied to concrete examples in viscoelasticity theory

on the controllability of the linearized benjamin bona mahony equation we study the boundary controllability properties of the linearized benjamin bona mahony equation left begin array ll u t u xxt u x 0 x in 0 1 t 0 u t 0 0 u 1 t f t t 0 end array right we show that the equation is approximately controllable but not spectrally controllable no finite linear combination of eigenfunctions other than zero is controllable next we prove a finite controllability result and we estimate the norms of the controls needed in this case

approximate controllability of the fitzhugh nagumo equation in one dimension abstract the fitzhugh nagumo fhn equation is a simplified model of a nerve axon we explore the controllability of this model using a localized interior control only for the first equation the linearized system is not null controllable using a localized interior control since the spectrum of the linearized system has an accumulation point though it is approximate controllable we show that the solution of the fhn equation fails to be globally approximate controllable in a given time but it is possible to move from any steady state to any other steady state arbitrarily close after some appropriate time by a localized interior control provided that both steady states are in the same connected component of the set of steady states finally we make some additional remarks and comments and we mention some open questions for our system for the sake of completeness we give the details of the existence uniqueness and uniform bound of the solution in appendix

approximate controllability of a reaction diffusion system abstract an open loop control for a system coupling a reaction diffusion system and an ordinary differential equation is proposed in this study we use a flatness like property indeed the solution can be expressed in terms of an infinite series depending on a flat output its derivatives and its integrals this series is shown to be convergent if the flat output is gevrey of order 1 a 2 approximate controllability of the system is then proved

a structurally damped plate equation with dirichlet neumann boundary conditions abstract we investigate sectoriality and maximal regularity in l p l q sobolev spaces for the structurally damped plate equation with dirichlet neumann clamped boundary conditions we obtain unique solutions with optimal regularity for the inhomogeneous problem in the whole space in the half space and in bounded domains of class c 4 it turns out that the first order system related to the scalar equation on r n is sectorial only after a shift in the operator on the half space one has to include zero boundary conditions in the underlying function space in order to obtain sectoriality of the shifted operator and maximal regularity for the case of homogeneous boundary conditions we further show that the semigroup solving the problem on bounded domains is exponentially stable

minimal time for the null controllability of parabolic systems the effect of the condensation index of complex sequences let a d a be a diagonalizable generator of a c0 semigroup of contractions on a complex hilbert space x b l c y y being some suitable extrapolation space of x and u l2 0 t c under some assumptions on the sequence of eigenvalues k k 1 c of a d a we prove the existence of a minimal time t0 0 depending on bernstein s condensation index of and on b such that y ay bu is null controllable at any time t t0 and not null controllable for t t0 as a consequence we solve controllability problems of various systems of coupled parabolic equations in particular new results on the boundary controllability of one dimensional parabolic systems are derived these seem to be difficult to achieve using other classical tools

strongly damped wave problems bootstrapping and regularity of solutions abstract the aim of the article is to present a unified approach to the existence uniqueness and regularity of solutions to problems belonging to a class of second order in time semilinear partial differential equations in banach spaces our results are applied next to a number of examples appearing in literature which fall into the class of strongly damped semilinear wave equations the present work essentially extends the results on the existence and regularity of solutions to such problems previously these problems have been considered mostly within the hilbert space setting and with the main part operators being selfadjoint in this article we present a more general approach involving sectorial operators in reflexive banach spaces

null controllability properties of the wave equation with a second order memory term abstract we study the internal controllability of a wave equation with memory in the principal part defined on the one dimensional torus t r 2 z we assume that the control is acting on an open subset t t which is moving with a constant velocity c r 1 0 1 the main result of the paper shows that the equation is null controllable in a sufficiently large time t and for initial data belonging to suitable sobolev spaces its proof follows from a careful analysis of the spectrum associated with our problem and from the application of the classical moment method

some controllability results for linear viscoelastic fluids we analyze the controllability properties of systems which provide a description at first approximation of a kind of viscoelastic fluid we consider linear maxwell fluids first we establish the large time approximate finite dimensional controllability of the system with distributed or boundary controls supported by arbitrary small sets then we prove the large time exact controllability of fluids of the same kind with controls supported by suitable large sets the proofs of these results rely on classical arguments in particular the approximate controllability result is implied by appropriate unique continuation properties while exact controllability is a consequence of observability inverse inequalities we also discuss questions concerning the controllability of viscoelastic fluids and some related open problems

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

are viscoelastic flows under control or out of control we discuss the controllability of viscoelastic shear flows both linear and nonlinear results are reviewed an interesting aspect of the problem is that the nonlinear problem introduces new stress components which are automatically zero in the linear case it is therefore a challenging problem to determine to what extent these stresses are controllable

controllability of the linear one dimensional wave equation with inner moving forces we address the null inner controllability of the one dimensional wave equation with supports of the controls that may vary with respect to the time variable we consider theoretical and numerical aspects assuming a geometric optic condition on the support of the controls we first prove a generalized observability inequality for the homogeneous wave equation implying the controllability we then introduce and prove the well posedness of a mixed formulation that characterizes the controls of minimal square integrable norm a numerical approximation of this formulation based on a conformal space time finite element method is then presented several numerical experiments are finally reported

diffusion phenomena for the wave equation with structural damping in the lp lq framework abstract in this paper we study diffusion phenomena for the wave equation with structural damping u t t u 2 a u t 0 u 0 x u 0 x u t 0 x u 1 x with a 0 and 0 1 2 we show that the solution u behaves like the solution v to v t 1 2 a 1 v 0 v 0 x v 0 x for suitable choice of initial data v 0 more precisely we derive l p l q decay estimates for the difference u v and its time and space derivatives where 1 p q possibly not on the conjugate line satisfying some additional condition related to in particular we show that under suitable assumptions on p q a double diffusion phenomenon appears that is the difference u v behaves like the solution to v t 2 a v 0 v 0 x v 0 x for a suitable choice of initial data v 0

time optimal boundary controls for the heat equation the fact that the time optimal controls for parabolic equations have the bang bang property has been recently proved for controls distributed inside the considered domain interior control the main result in this article asserts that the boundary controls for the heat equation have the same property at least in rectangular domains this result is proved by combining methods from traditionally distinct fields the lebeau robbiano strategy for null controllability and estimates of the controllability cost in small time for parabolic systems on one side and a remez type inequality for muntz spaces and a generalization of turan s inequality on the other side

analysis of a viscoelastic spring mass model in this paper we consider a linear wave equation with strong damping and dynamical boundary conditions as an alternative model for the classical spring mass damper ode our purpose is to compare analytically these two approaches to the same physical system we take a functional analysis point of view based on semigroup theory spectral perturbation analysis and dominant eigenvalues

approximate controllability results for viscoelastic flows with infinitely many relaxation modes abstract we prove results on approximate controllability for linear viscoelastic flows with a localized distributed control in the momentum balance equation the constitutive law is a multimode maxwell or jeffreys model with an infinite number of relaxation modes

on the control of viscoelastic jeffreys fluids abstract this paper is devoted to analyzing the control of vicoelastic fluids of the jeffreys kind also known as oldroyd models we will present the interesting problems with special emphasis in the difficulties that they involve then we will consider appropriate linear approximations and we will establish some partial approximate finite dimensional controllability results in an arbitrarily small time with distributed or boundary controls supported by arbitrarily small sets the proofs rely on some specific unique continuation properties which are implied by the structure of the solutions

bounds on functions biorthogonal to sets of complex exponentials control of damped elastic systems abstract in this article we obtain explicit bounds on the norms of biorthogonal functions to sets of complex exponentials exp k t where the k belong to a sector of the positive real axis the bounds obtained are uniform within a class of sequences which exhibit similar growth and separation properties the dependence of the bounds upon the growth and separation parameters is carefully examined the main theorem then leads to some results regarding the solvability of moment problems of the form 0 t f t exp k t dt c k which arise in control problems associated with damped or dissipative systems as an application we establish an exact null controllability result for boundary control of a rectangular structurally damped plate

some controllability results for the n dimensional navier stokes and boussinesq systems with n 1 scalar controls in this paper we deal with some controllability problems for systems of the navier stokes and boussinesq kind with distributed controls supported in small sets our main aim is to control n dimensional systems n 1 scalar unknowns in the case of the navier stokes equations with n 1 scalar control functions in a first step we present some global carleman estimates for suitable adjoint problems of linearized navier stokes and boussinesq systems in this way we obtain null controllability properties for these systems then we deduce results concerning the local exact controllability to the trajectories we also present global null controllability results for some truncated approximations of the navier stokes equations

unique continuation for some evolution equations this work is concerned with unique continuation results for some evolution equations essentially of parabolic type to be more precise let l be an evolution operator acting on functions defined on some connected open set b of r r x r l is said to have the unique continuation property if every solution u of lu 0 which vanishes one some nonempty open set u i of fl vanishes in the horizontal component of o in 0 i e in x t e 0 3x x t e co the first general result of this type was established by mizohata 6 in the case where l is a second order parabolic operator with smooth coefficients however mizohata s proof is somewhat delicate it is based in particular on an extended class of calderon zygmund operators and does not lead obviously to a weakening of the smoothness of the coefficients on the other hand it is important for the applications to deal with nonsmooth coefficients since l is often obtained as the linearization of some nonlinear operator at a nonnecessarily smooth solution we refer to lions 4 and kernevez and lions 3 for applications to control theory and to 1 l for applications to transversality techniques let us now describe the content of this work in the first section we prove a unique continuation theorem when l is a second order parabolic equation our proof is simple and based on the derivation of a carleman estimate which is reminiscent of the classical carleman estimates for second order elliptic operators see 121 this carleman inequality allows the weakening of the 118 0022 0396187 3 00

null controllability of the linearized compressible navier stokes system in one dimension in this paper we consider the one dimensional compressible navier stokes equations linearized around a constant steady state q0 v0 q0 0 v0 0 with periodic boundary conditions in the interval i2 0 2 we explore the controllability of this linearized system using a control only for the velocity equation we prove that the linearized system with homogeneous periodic boundary conditions is null controllable in h 1 per i2 l 2 i2 by a localized interior control when time is sufficiently large where h 1 per i2 denotes the sobolev space of periodic functions with mean value zero we show null controllability of the system by proving an observability inequality with the help of two types of ingham inequality

control problems for weakly coupled systems with memory abstract we investigate control problems for wave petrovsky coupled systems in the presence of memory terms by writing the solutions as fourier series we are able to prove ingham type estimates and hence reachability results our findings have applications in viscoelasticity theory and linear acoustic theory

a complex analytic approach to the problem of uniform controllability of a transport equation in the vanishing viscosity limit abstract we revisit a result by coron and guerrero stating that the one dimensional transport diffusion equation u t m u x e u x x 0 in 0 t 0 l controlled by the left dirichlet boundary value is zero controllable at a bounded cost as e 0 when t 4 3 l m if m 0 and when t 57 2 l m if m 0 by a completely different method relying on complex analysis we prove that this still holds when t 4 2 l m if m 0 and when t 6 1 l m if m 0

analyticity and exponential stability of semigroups for the elastic systems with structural damping in banach spaces abstract in this paper we consider a second order evolution equation in a banach space which can model an elastic system with structural damping new forms of the corresponding first order evolution equation are introduced and their well posed property is proved by means of the operator semigroup theory we give sufficient conditions for analyticity and exponential stability of the associated semigroups

null controllability of a pseudo parabolic equation with moving control abstract in this paper we study the internal controllability of the pseudo parabolic equation on the one dimensional torus our control function is acting on a moving small interval with a constant velocity with this moving distributed control we obtain the system is null controllable for given data with certain regularity

unique continuation and control for the heat equation from an oscillating lower dimensional manifold we consider the linear heat equation with dirichlet boundary conditions in a bounded domain of r n n geq 1 and with a control acting on a lower dimensional time dependent manifold of dimension k leq n 1 we analyze the approximate controllability problem this problem is equivalent to a suitable uniqueness or unique continuation property of solutions of the heat equation without control more precisely it consists of proving that the unique solution of the dirichlet problem vanishing on the time dependent manifold is identically zero this uniqueness problem however does not fit in the class of classical cauchy problems and therefore the existing tools based on power series expansions carleman inequalities and doubling properties do not seem to apply we give sufficient conditions on the time dependent manifold for this uniqueness property to hold the techniques we employ combine the fourier series representation and the time analyticity of solutions and allow us to reduce the problem to a uniqueness question for the eigenfunctions of the laplacian we then apply well known results on the nodal sets of these eigenfunctions we also analyze the asymptotic behavior of the control when the time oscillation of the manifold supporting the control increases when the frequency of oscillation tends to infinity we prove that the controls converge to an approximate control for the same heat equation but on a manifold of dimension k 1 that is independent of time this is done under suitable time periodicity assumptions on the original manifold and confirms the fact that increasing time oscillations of the support of the control increases the efficiency of the control mechanism

controllability and observability theory of certain parabolic integrodifferential equations the exact null controllability of linear and nonlinear parabolic integrodifferential equations with space periodic boundary conditions are studied carleman estimate for the linearized problem providing the observability results is fundamental to the analysis and by duality also provides exact global controllability local exact controllability for the nonlinear problem is obtained using the kakutani fixed point theorem

controllability and asymptotic stabilization of the camassa holm equation we investigate the problems of exact controllability and asymptotic stabilization of the camassa holm equation on the circle by means of a distributed control the results are global and in particular the control prevents the solution from blowing up

carleman estimate for an adjoint of a damped beam equation and an application to null controllability abstract in this article we consider a control problem of a linear euler bernoulli damped beam equation with potential in dimension one with periodic boundary conditions we derive a new carleman estimate for an adjoint of the equation under consideration then using a well known duality argument we obtain explicitly the control function which can be used to drive the solution trajectory of the control problem to zero state

lack of controllability of the heat equation with memory abstract we consider a model for the heat equation with memory which has infinite propagation speed like the standard heat equation we prove that in spite of this for every t 0 there exist square integrable initial data which cannot be steered to hit zero at time t using square integrable controls we show that the counterexample we present complies with the restrictions imposed by the second principle of thermodynamics

boundary control of a linear thermoelastic beam abstract we show in this paper that the motions of a linear thermoelastic beam may be controlled exactly to zero in a finite time by a single boundary control that acts on one end of the beam the optimal time of controllability depends upon the moment of inertia parameter of the beam and becomes arbitrarily small if this parameter is omitted as in the euler bernoulli beam theory

unique continuation property and control for the benjamin bona mahony equation on a periodic domain abstract we consider the benjamin bona mahony bbm equation on the one dimensional torus t r 2 z we prove a unique continuation property ucp for small data in h 1 t with nonnegative zero means next we extend the ucp to certain bbm like equations including the equal width wave equation and the kdv bbm equation applications to the stabilization of the above equations are given in particular we show that when an internal control acting on a moving interval is applied in the bbm equation then a semiglobal exponential stabilization can be derived in h s t for any s 1 furthermore we prove that the bbm equation with a moving control is also locally exactly controllable in h s t for any s 0 and globally exactly controllable in h s t for any s 1 in a sufficiently large time depending on the h s norms of the initial and terminal states

controllability results for the moore gibson thompson equation arising in nonlinear acoustics abstract we show that the moore gibson thomson equation t t t y t t y c 2 y b t y k t t y 2 t u is controlled by a force that is supported on an moving subset t of the domain satisfying a geometrical condition using the concept of approximately outer invertible map a generalized implicit function theorem and assuming that c 2 b 0 the local null controllability in the nonlinear case is established moreover the analysis of the critical value 0 for the linear equation is included

controllability for one dimensional nonlinear wave equations with degenerate damping abstract internal exact controllability for the nonlinear wave equation in one space dimension y t t y x x g y y t h is studied where g is a nonnegative function for the case lim sup s g s ln s we obtain the global exact controllability for the equation with dirichlet boundary condition the proof is based on the combination of fixed point arguments and explicit observability estimates for the linearized wave equation with a potential that depends on both x and t for the case g s s we only get a local exact controllability by means of banach fixed point theorem

approximate controllability of the semilinear heat equation this article is concerned with the study of approximate controllability for the semilinear heat equation in a bounded domain when the control acts on any open and nonempty subset of or on a part of the boundary in the case of both an internal and a boundary control the approximate controllability in l p for 1 p l in the case of the interior control we also prove approximate controllability in c 0 the proof combines a variational approach to the controllability problem for linear equations and a fixed point method we also prove that the control can be taken to be of quasi bang bang form

local exact controllability for the one dimensional compressible navier stokes equation in this paper we deal with the isentropic compressible navier stokes equation in one space dimension and we adress the problem of the boundary controllability for this system we prove that we can drive initial conditions which are sufficiently close to some constant states to those constant states this is done under some natural hypotheses on the time of control and on the regularity on the initial conditions

a generalized internal control for the wave equation in a rectangle abstract when is a rectangle in r 2 and a disk inside for each t 0 sufficiently large we establish the exact controllability of the evolution system y y h t x in 0 t y 0 on 0 t by means of a generalized control h t x localized on 0 t in other terms for any prescribed initial state of finite energy we find h t x a kind of analytic functional in x depending smoothly on t supported on 0 t such that y t y t 0

completeness and basis properties of complex exponentials this paper is concerned with what might be termed the fine structure of the completeness and basis properties of complex exponentials we give new criteria for two sequences to have the same excess in the sense of paley and wiener a result that illuminates and supplements a well known completeness criterion of levinson and new examples and counterexamples pertaining to riesz bases

local stabilization of the compressible navier stokes system around null velocity in one dimension abstract in this paper we study the exponential stabilization of the one dimensional compressible navier stokes system in a bounded interval 0 locally around a constant steady state 0 0 by a localized distributed control acting only in the velocity equation we determine a linear feedback law able to stabilize a nonlinear transformed system coming back to the original nonlinear system we obtain a nonlinear feedback law able to stabilize locally this nonlinear system to the best of our knowledge the results of the paper are the first ones providing feedback control laws stabilizing compressible fluid flows

pseudospectra of exponential matrix polynomials this paper extends a result of graillat which was based on rump s theorem to show that pseudospectra of matrix polynomials expressed in other bases and also some nonlinear matrix functions are unaffected by drawing the matrix coefficients from certain structured families we show by example that this behaviour is not universal this result is of interest because computation of structured pseudospectra is much more expensive in general than computation of unstructured pseudospectra and in the cases covered by rump s theorem there is no point going to the extra effort the paper also contains some examples that may be of interest to the community including an analytical solution of a nested exponential equation that shows why exponential polynomial decomposition algorithms may be of interest

one channel time reversal in chaotic cavities theoretical limits one channel time reversal experiments in closed chaotic cavities produce excellent but not perfect time reversed focusing this paper investigates such experiments by a simple eigenmode analysis of the system it shows that the process is even for long reversed signals subject to an information loss during recording and re emission which prevents perfect time reversal this fact can be expressed by a simple equation called the cavity equation which states that the signal of a one channel time reversal is equal to the signal of a perfect time reversal after convolution with the backscattering impulse response of the reversal point i e from this point to this point the latter convolution describes the introduction of the loss of information furthermore arguments are presented suggesting that one third of the total energy of the time reversed wave field is actually contained in the refocusing wavefront the predictions are verified by numerical finite difference time domain simulations

exact boundary controllability for quasi linear hyperbolic systems using a result on the existence and uniqueness of the semiglobal c1 solution to the mixed initial boundary value problem for first order quasi linear hyperbolic systems with general nonlinear boundary conditions we establish the exact boundary controllability for quasi linear hyperbolic systems if the c1 norm of initial and final states is small enough

completeness of systems of complex exponentials and the lambert w functions we study some of the properties of the solution system e i nt of the delay differential equation y t ay t 1 we first establish some general results on the stability of the completeness of exponential systems in l 2 and then show that the solution system above is always complete but is not an unconditional basis in l 2 1 2 1 2

global existence in critical spaces for compressible navier stokes equations we investigate global strong solutions for isentropic compressible fluids with initial data close to a stable equilibrium we obtain the existence and uniqueness of a solution in a functional setting invariant by the scaling of the associated equations more precisely the initial velocity has the same critical regularity index as for the incompressible homogeneous navier stokes equations and one more derivative is needed for the density we point out a smoothing effect on the velocity and a l 1 decay on the difference between the density and the constant reference state the proof lies on uniform estimates for a mixed hyperbolic parabolic linear system with a convection term

on the excess of complex exponential systems in l2 a a abstract in this article we give new criteria for two complex sequences to have the same excess in the sense of paley and wiener in l 2 a a as a result we prove that given any positive integer q a real number 0 1 2 and complex numbers 0 0 n n q i log n q n 1 the exponential system t k e i t n k 0 1 q 1 n has excess 0 in l 2

ingham type inequalities for complex frequencies and applications to control theory abstract for complex valued sequences n n 1 of the form n a n i b n with a n r and b n 0 we prove inequalities of the form 0 t n p x n e i t n 2 d t n p x n 2 1 b n for all sequences x n with n 1 x n 2 1 b n we apply these to prove exact null controllability for a class of hinged beam equations with mild internal damping with either boundary control or internal control

on computing the trace of the kernel of the homogeneous fredholm s equation a method for computing the trace of the kernel of the homogeneous fredholm s equation for resonant states arising from nonlocal potentials is proposed we show that this integral formulation is convergent

non structural controllability of linear elastic systems with structural damping this paper proves that any initial condition in the energy space for the plate equation with square root damping 2 u on a smooth bounded domain with hinged boundary conditions 0 can be steered to zero by a square integrable input function u supported in arbitrarily small time interval 0 t and subdomain as t tends to zero for initial states with unit energy norm the norm of this u grows at most like exp cp tp for any real p 1 and some cp 0 indeed this fast controllability cost estimate is proved for more general linear elastic systems with structural damping and non structural controls satisfying a spectral observability condition moreover under some geometric optics condition on the subdomain allowing to apply the control transmutation method this estimate is improved into p 1 and the dependence of cp on the subdomain is made explicit these results are analogous to the optimal ones known for the heat flow

stable observation of additive superpositions of partial differential equations abstract we address the problem of observing the solutions of a given partial differential equation pde submitted to unknown additive perturbations fulfilling another pde possibly of different nature the problem under consideration is motivated by the issue of averaged control and observation in the latter one aims at determining the initial datum of a dynamics submitted to unknown perturbations by means of measurements done on the average of solutions with respect to the unknown parameters we do it in two different cases first we assume that a given dynamics is perturbed in an additive manner by a state governed by a pde but not necessarily of the same type in particular we consider the additive superposition of multi physics solutions of wave and heat type for instance assuming the pdes under consideration have constant coefficients we prove that the by now well known observability properties of pde mainly heat and wave equations from open subsets of the domain where the dynamics evolves are stable under these additive perturbations the proof is rather systematic and easy to apply it consists in composing all but one the observable one pde operators so to reduce the problem to the consideration of a single equation and applying its known observability properties the same proof leads to very interesting open problems for systems involving pdes with variable coefficients where commutators lead to non local lower order perturbations we then consider a particular case of a pde depending on a parameter in a continuous manner we show that the average is governed by the superposition of two distinct solutions corresponding to the extremal values of the parameter and experiencing an extra regularizing effect this allows showing that the averaged observability property holds for the continuous average but to the price of losing a finite number of derivatives this opens a wide range of interesting problems to be explored

characterization of domains of fractional powers of certain operators arising in elastic systems and applications with x a hilbert space where a the elastic operator and b the damping operator are closed operators in x with dense domains q a and 9 b respectively moreover a is throughout assumed to be self adjoint and strictly positive stimulated by two conjectures raised in crl the most general of which referred to the case lx f described below we have already studied the operators and sb defined respectively by

exact semi internal control of an euler bernoulli equation this paper proves exact semi internal controllability of an euler bernoulli equation with variable coefficients the basic principle of the proof is the hilbert uniqueness method for this a multiplier technique and a unique continuation property have been used the proof of the unique continuation property is the main feature of this work