on newton s method and halley s method for the principal pth root of a matrix if a is a matrix with no negative real eigenvalues and all zero eigenvalues of a are semisimple the principal pth root of a can be computed by newton s method or halley s method with a preprocessing procedure if necessary we prove a new convergence result for newton s method and discover an interesting property of newton s method and halley s method in terms of series expansions we explain how the convergence of newton s method and halley s method can be improved when the eigenvalues of a are known or when a is a singular matrix we also prove new results on pth roots of m matrices and h matrices and consider the application of newton s method and halley s method to find the principal pth roots of these special matrices ams classification 65f30 41a58 15a48

a residual recurrence for halley s method for the matrix pth root abstract in this note we prove a residual relation for halley s method for finding the principal p th root of a matrix

renormalization group and asymptotics of solutions of nonlinear parabolic equations we present a general method for studying long time asymptotics of nonlinear parabolic partial differential equations the method does not rely on a priori estimates such as the maximum principle it applies to systems of coupled equations to boundary conditions at infinity creating a front and to higher possibly fractional differential linear terms we present in detail the analysis for nonlinear diffusion type equations with initial data falling off at infinity and also for data interpolating between two different stationary solutions at infinity in an accompanying paper 5 the method is applied to systems of equations where some variables are slaved such as the complex ginzburg landau equation c 1994 john wiley sons inc

uniform stabilization of the fourth order schrodinger equation abstract we study both boundary and internal stabilization problems for the fourth order schrodinger equation in a smooth bounded domain of r n we first consider the boundary stabilization problem by introducing suitable dissipative boundary conditions we prove that the solution decays exponentially in an appropriate energy space in the internal stabilization problem by assuming that the damping term is effective on a neighborhood of a part of the boundary we prove the exponential decay of the l 2 energy of the solution both results are established by using multiplier techniques and compactness uniqueness arguments

controllability for neutral stochastic functional differential inclusions with infinite delay in abstract space abstract in this paper sufficient conditions are given for the controllability of a class of partial stochastic functional differential inclusions with infinite delay in an abstract space with the help of the leray schauder nonlinear alternative an example is provided to illustrate the theory

existence of bounded solutions of a second order system with dissipation in this article we study the following second order system of ordinary differential equations with dissipationu cu dau khu pt u rn t r where c d and k are positive constants h rn rn is a locally lipschitz function and p r rn is a continuous and bounded function a is a n n matrix whose eigenvalues are positive under these conditions we prove that for some values of c d and k this system has a bounded solution which is exponentially asymptotically stable moreover if p t is almost periodic then this bounded solution is also almost periodic these results are applied to the spatial discretization of very well known second order partial differential equations

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

a porous media equation with absorption ii uniqueness of the very singular solution abstract we prove the uniqueness of the very singular solution for an equation of the form formula where 1 2 n m p m 2 n the solution we find is of the form formula where f is the unique nontrivial solution of an ordinary differential equation formula with conditions f 0 on 0 f 0 0 and lim n 2 p m f 0

the cubic fourth order schrodinger equation fourth order schrodinger equations have been introduced by karpman and shagalov to take into account the role of small fourth order dispersion terms in the propagation of intense laser beams in a bulk medium with kerr nonlinearity in this paper we investigate the cubic defocusing fourth order schrodinger equation i tu 2u u 2u 0 in arbitrary space dimension rn for arbitrary initial data we prove that the equation is globally well posed when n 8 and ill posed when n 9 with the additional important information that scattering holds true when 5 n 8

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

feedthrough operator for linear elasticity system with boundary control and observation we are concerned with the feedthrough operator for the open loop system of linear elasticity with dirichlet boundary control and collocated observation this system has been known to be regular in the sense of g weiss in this paper the analytic expression of the corresponding feedthrough operator is presented by means of differential geometry and fourier transform methods

constrained controllability in banach spaces the aim of this paper is to study null controllability of the linear infinite dimensional control problem dot x ax bu where the control u is constrained to lie in a convex weakly compact subset omega of the control space with 0 in omega a necessary and sufficient condition for a particular initial state to be omega null controllable within a fixed finite time t is given the result is extended to the case omega convex cone with vertex at 0 applications to the one dimensional heat and wave equation are given

survey paper well posed systems the lti case and beyond this survey is an introduction to well posed linear time invariant lti systems for non specialists we recall the more general concept of a system node classical and generalized solutions of system equations criteria for well posedness the subclass of regular linear systems some of the available linear feedback theory motivated by physical examples we recall the concepts of impedance passive and scattering passive systems conservative systems and systems with a special structure that belong to these classes we illustrate this theory by examples of systems governed by heat and wave equations we develop local and global well posedness results for lti systems with nonlinear in particular bilinear feedback by extracting the abstract idea behind various proofs in the literature we apply these abstract results to derive well posedness results for the burgers and navier stokes equations

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

approximate controllability of a system of parabolic equations with delay abstract in this paper we give necessary and sufficient conditions for the approximate controllability of the following system of parabolic equations with delay z t x t d z l z t b u t x t 0 r z 0 x t 0 r z 0 x 0 x x z s x s x s 0 x where is a bounded domain in r n d is an n n nondiagonal matrix whose eigenvalues are semi simple with nonnegative real part the control u l 2 0 r u l 2 0 r l 2 r m and b l u z with u l 2 r m z l 2 r n the standard notation z t x defines a function from 0 to r n with x fixed by z t x s z t s x s 0 here 0 is the maximum delay which is supposed to be finite we assume that the operator l l 2 0 z z is linear and bounded and 0 z l 2 0 z to this end first we reformulate this system into a standard first order delay equation secondly the semigroup associated with the first order delay equation on an appropriate product space is expressed as a series of strongly continuous semigroups and orthogonal projections related with the eigenvalues of the laplacian operator a 2 this representation allows us to reduce the controllability of this partial differential equation with delay to a family of ordinary delay equations finally we use the well known result on the rank condition for the approximate controllability of delay system to derive our main result

controllability of laguerre and jacobi equations in this paper we study the controllability of the controlled laguerre equation and the controlled jacobi equation for each case we find conditions which guarantee when such systems are approximately controllable on the interval 0 t 1 moreover we show that these systems can never be exactly controllable

infinite dimensional linear systems a distributional approach we introduce the concept of a distributional resolvent linear system and solve the linear quadratic optimal control problem for this class of systems the class of distributional resolvent linear systems includes all linear time invariant systems that have been studied in the control literature we show connections with integrated semigroups and distribution semigroups and we consider some examples of systems described by partial differential equations that could not be handled by the existing theory

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

controllability and stability of a second order hyperbolic system with collocated sensor actuator a second order hyperbolic system with collocated sensor actuator is considered the semigroup generation is shown for the closed loop system under the feedback of a generic unbounded observation operator the equivalence between the exponential stability of the closed loop system and exact controllability of the open loop system is established in the general framework of well posed linear systems finally the conditions are weakened for the diagonal semigroups with finite dimensional inputs example of beam equation is presented to display the application

convergence and convergence rate of the balanced realization truncations for infinite dimensional discrete time systems abstract in this paper we study approximation via balanced realizations for a large class of infinite dimensional discrete time linear systems we give properties of the truncated system and prove that the approximation is l 2 convergent in the case of nuclear systems we prove convergence in nuclear norm and give an estimation of the l convergence rate

a very singular solution of the porous media equation with absorption turns out to play an important role in the description of both the short and the long time behaviour of positive solutions of eq 1 1 for the short time behaviour it provides together with the one parameter family of singular solutions i of problem i a complete description of the possible types of singular behaviour at the point 0 o 121 for the long time behaviour of solutions u x t of the cauchy problem for 1 1 1 2 it provides the asymptotic form when the initial function decays sufficiently fast as 1x1 co t p u x t w x t 0 ast co uniformly in r 7 81 in this paper we shall show that if

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

rank condition for the controllability of a linear time varying system in this paper we study the controllability of the following linear time varying control system x a t x bu t where a t a t a and a is a scalar function which is in general not continuous and satisfies the only condition 0t a s ds 0 t 0 a and b are constant matrices of dimension n x n and n x m respectively under this condition we prove that this system is controllable on 0 t iff the usual rank condition holds rank b ab an 1b n

characterization of extremal controls for infinite dimensional time varying systems in this paper we prove some properties of attainable sets for time varying infinite dimensional linear control systems with time varying constrained controls and target sets we also characterize the extremal controls and give necessary and sufficient conditions for the normality of the system moreover we prove an existence theorem for the time optimal control establishing two maximum principles

a balanced canonical form for discrete time minimal systems using characteristic maps abstract this paper presents a balanced canonical form for the class of discrete time minimal systems the main tool is to establish a bijection between the set of minimal systems and the class of minimal discrete time asymptotically stable systems of the same dimension this characteristic map is shown to preserve system equivalence and balancing the canonical form for discrete time minimal systems is then constructed by mapping the system to its discrete time asymptotically stable counterpart via the characteristic map transforming the resulting system to lyapunov balanced canonical form and returning to the original system class by means of the inverse characteristic map

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

universal self similarity of porous media equation with absorption the critical exponent case abstract in this paper we study the large time behavior of non negative solutions to the cauchy problem of u t u m u q in r n 0 where m 1 and q q c m 2 n is a critical exponent for non negative initial value u x 0 u 0 x l 1 r n we show that the solution converges if u 0 x 1 x k is bounded for some k n to a unique fundamental solution of u t u m independent of the initial value with additional logarithmic anomalous decay exponent in time as t

bilinear transformation of infinite dimensional state space systems and balanced realizations of nonrational transfer functions the bilinear transform maps the open right half plane to the open unit disk and is therefore a suitable tool for carrying over results for continuous time systems to discrete time systems and vice versa corresponding state space formulae are widely used and well understood for the case of finite dimensional systems in this paper infinite dimensional generalizations of these formulae are studied for a general class of infinite dimensional state space systems in particular it is shown that reachability and observability are carried over and that the reachability and observability gramians are preserved under this transformation young showed that a wide class of nonrational discrete time transfer functions admit a balanced state space representation it is shown that this result carries over to the continuous time situation via the bilinear transformation

wellposedness for the fourth order nonlinear schrodinger equations abstract we study the local smoothing effects and wellposedness of cauchy problem for the fourth order nonlinear schrodinger equations in 1d i t u x 4 u p x u 2 x u 2 t x r where p is a polynomial excluding constant or linear terms

lqg balancing for continuous time infinite dimensional systems in this paper we study the existence of linear quadratic gaussian lqg balanced realizations for continuous time infinite dimensional systems lqg balanced realizations are those for which the optimal cost operator for the system and its dual system are equal and diagonal the class of systems we consider is that of distributional resolvent linear systems which includes well posed linear systems as a subclass we prove the existence of lqg balanced realizations under a finite cost condition for both the system and its dual system we also show that an lqg balanced realization of a well posed transfer function is well posed we further show that approximately controllable and observable lqg balanced realizations are unique up to a unitary state space transformation finally we show that the spectrum of the product of the optimal cost operator of a system and its dual system is independent of the particular realization our method of proof shows the connections with coprime factorizations lyapunov balanced realizations and discrete time systems the main reason for studying lqg balanced realizations is that truncated lqg balanced realizations provide a good approximation of the original system we show that under certain conditions this is also true in the infinite dimensional case by proving an error bound in the gap metric

exponential decay rates for the solutions of euler bernoulli equations with boundary dissipation occurring in the moments only abstract we shall derive uniform decay rates for the total energy of the solutions to the euler bernoulli equations wherein energy dissipation occurs on the boundary the main feature which distinguishes this paper from other related works is the fact that the boundary feedback is acting through the moments only the key technical element responsible for the proof of the main result is a new regularity estimate for the solutions to the nonhomogeneous schrodinger equation

realisation and approximation of linear infinite dimensional systems with error bounds the class of linear infinite dimensional systems with finite dimensional inputs and outputs whose impulse response h satisfies h in l 1 cap l 2 0 infty mathbb c p times m and induces a nuclear hankel operator is said to be of nuclear type for this class of systems it is shown that balanced or output normal realisations always exist and their truncations converge to the original system in various topologies furthermore explicit l infty bounds on the transfer function errors l 1 and l 2 bounds on the impulse response errors and hilbert schmidt and nuclear bounds on the hankel operator errors are obtained these truncations also generate an approximating sequence to the optimal hankel norm approximations to the original system and various error bounds of these approximants are deduced

on stability equivalence between dynamic output feedback and static output feedback for a class of second order infinite dimensional systems we consider stabilization for a class of abstract second order infinite dimensional systems with collocated control and observation we show that the closed loop system under a proportional direct output feedback control is asymptotically stable if and only if the closed loop system under some dynamic output feedback control is asymptotically stable a hautus test is developed to ensure the asymptotic stability two types of dynamic output feedback controls are investigated the results are applied to some coupled wave heat equations where the heat system is considered as a controller of the wave system this study provides a different view in the study of the coupled systems described by partial differential equations

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

exact controllability stabilization and perturbations for distributed systems exact controllability is studied for distributed systems of hyperbolic type or for petrowsky systems like plate equations the control is a boundary control or a local distributed control exact controllability consists in trying to drive the system to rest in a given finite time the solution of the problems depends on the function spaces where the initial data are taken and also depends on the function space where the control can be chosen a systematic method named hum for hilbert uniqueness method is introduced as the terminology indicates it is based on uniqueness results classical or new and on hilbert spaces constructed in infinitely many ways by using uniqueness a number of applications are indicated having a general method for exact controllability implies having a general method for stabilization this leads to new and complicated nonlinear riccati s type pdes to be compared with direct methods when available in the last part of the paper we consider how all this behaves fo

monotonicity and positivity of coefficients of power series expansions associated with newton and halley methods for the matrix pth root abstract if a is a matrix with all eigenvalues in the disk z 1 1 the principal pth root of a can be computed by newton s method or halley s method the study of newton s method and halley s method for the matrix pth root can be done through a study of power series expansions of some sequences of scalar functions in this paper we prove monotonicity results for the coefficients in the power series expansions for both newton s method and halley s method and for all integer p 2 the sign patterns of these coefficients can be seen directly from the monotonicity results we then use these monotonicity results and their consequences to obtain some new error estimates in the matrix case to obtain a monotonic convergence result when a is a nonsingular m matrix and to obtain a structure preserving result when a is a nonsingular m matrix or a real nonsingular h matrix with positive diagonal entries

a robust lqg controller design for dps we review the popular linear gaussian quadratic lqg controller design for dps and its shortcomings in particular its applicability is essentially restricted to exponentially stabilizable and detectable dps with finite rank bounded input and output operators as an alternative we propose a new practical control design that is applicable to dps with unbounded finite rank input and output operators provided that the generator has finitely many unstable eigenvalues and satisfies the spectrum determined growth assumption

exact controllability of the suspension bridge model proposed by lazer and mckenna abstract in this paper we give a sufficient condition for the exact controllability of the following model of the suspension bridge equation proposed by lazer and mckenna in a c lazer p j mckenna large amplitude periodic oscillations in suspension bridges some new connections with nonlinear analysis siam rev 32 1990 537 578 w t t c w t d w x x x x k w p t x u t x f t w u t x 0 x 1 w t 0 w t 1 w x x t 0 w x x t 1 0 t r where t 0 d 0 c 0 k 0 the distributed control u l 2 0 t 1 l 2 0 1 p r 0 1 r is continuous and bounded and the non linear term f 0 t 1 r r r is a continuous function on t and globally lipschitz in the other variables i e there exists a constant l 0 such that for all x 1 x 2 u 1 u 2 r we have f t x 2 u 2 f t x 1 u 1 l x 2 x 1 u 2 u 1 t 0 t 1 to this end we prove that the linear part of the system is exactly controllable on 0 t 1 then we prove that the non linear system is exactly controllable on 0 t 1 for t 1 small enough that is to say the controllability of the linear system is preserved under the non linear perturbation k w p t x f t w u t x

large time behaviour of the solutions of a semilinear parabolic equation in rn abstract the asymptotic behaviour as t tends to of the solution of u t u u u p 1 0 in r n r p 1 was studied it was proved that the behaviour depends strongly on the sign of n 2 n p and also on the rate of decay of the admissible initial data u 0 x as x tends to

the hilbert schmidt property of feedback operators in this article we prove new sufficient conditions under which the feedback operator associated with the linear quadratic regulator control design for distributed parameter systems is nuclear or hilbert schmidt examples illustrating the main results are also given

source type solutions of the porous media equation with absorption the fast diffusion case in this paper we consider the cauchy problem u t u m u p on s t r n 0 t u x 0 0 x r n 0 where n 1 p 1 and 0 m 1

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

large time behavior for the fast diffusion equation with critical absorption we study the large time behavior of nonnegative solutions to the cauchy problem for a fast diffusion equation with critical zero order absorption partial t u delta u m u q 0 quad quad hbox in 0 infty times real n with m c n 2 n 1

explicit convergence regions of newton s method and chebyshev s method for the matrix pth root abstract for a square matrix with all eigenvalues in a suitable region in the complex plane its principal pth root exists and is the limit of the convergent sequence generated by newton s method or chebyshev s method starting from the identity matrix such a region is called a convergence region for newton s method or chebyshev s method in this paper we obtain explicit convergence regions that significantly expand existing ones

holder estimates for local solutions of some doubly nonlinear degenerate parabolic equations abstract we prove interior and boundary holder continuity for bounded weak soluttions of a class of quasilinear parabolic equations whose prototype is u t div u m 1 du p 2 du

inverse m matrices abstract this is an attempt at a comprehensive expository study of those nonnegative matrices which happen to be inverses of m matrices and is aimed at an audience conversant with basic ideas of matrix theory a theme is the parallels anddifferences between the class of m matrices and the class of inverse m matrices among the primary tools used are diagonal multiplications the neumann expansion and the form of the inverse of a partitioned matrix some of the results seem not to have appeared before and several unsolved problems are mentioned

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

coprime factorization and dynamic stabilization of transfer functions it is known that a matrix valued transfer function p has a stabilizing dynamic controller q i e begin smallmatrix i q p i end smallmatrix 1 in rm h infty iff p has a right or left coprime factorization we show that the same result is true in the operator valued case thus the standard youla bongiorno parameterization applies to every dynamically stabilizable function we then derive further equivalent conditions one of them being that p has a stabilizing controller with internal loop this and some others are new even in the scalar valued case we also establish certain related results for example we extend the classical results on coprime factorization and partial feedback measurement feedback stabilization to nonrational transfer functions all our results apply in both discrete and continuous time settings except that in the latter it is not clear whether the controller q can always be chosen so that it is continuous time proper holomorphic and bounded on a right half plane unless e g p z rightarrow0 as operatorname re z rightarrow infty

classification of singular solutions of a nonlinear heat equation on considere toutes les solutions non negatives de l equation de la chaleur non lineaire avec absorption u t u m u p dans q r n 0 t pour n 1 m p 1 avec des conditions initiales u x 0 0 pour x 0

robustly stabilizing controllers with respect to left coprime factor perturbations for infinite dimensional linear systems we solve the problem of robust stabilization with respect to left coprime factor perturbations for a large class of irrational transfer functions with a realization in terms of unbounded operators the key condition is that the associated dynamical system and its dual should satisfy a finite cost condition so that two optimal cost operators exist we obtain an explicit parameterization of infinitely many robustly stabilizing controllers in terms of these optimal cost operators and the generating operators of the realization

asymptotic behaviour of nonlinear parabolic equations with critical exponents a dynamical systems approach we investigate the large time behaviour of solutions to the nonlinear heat conduction equation with absorption ut u 1 u in q rn 0 e with n 1 0 and critical absorption exponent 1 2n the initial function u x 0 0 is assumed to be integrable nonnegative and compactly supported we prove that u converges as t to a unique self similar function which is a contracted version of one of the asymptotic profiles of the nonabsorptive problem ut u 1 the same for any initial data the cornerstone of the proof is a result about limits of infinite dimensional asymptotical dynamical systems combining this result with an asymptotic evaluation of the mass function as well as typical pde estimates gives the behaviour of e for large times similar unusual asymptotic behaviour is obtained for the equation ut div du du u with same conditions on and u x 0 and critical value for 1 2 n

the dual quasisemigroup and controllability of evolution equations we study the notion of dual quasisemigroups of bounded linear operators as a generalization of that for strongly continuous semigroup and prove some properties similar to the dual of a semigroup among other things we prove that for reflexive banach spaces the dual quasisemigroup is strongly continuous on 0 this allows us to extend some recent criteria of controllability to a general class of evolution equations in reflexive banach spaces

the cauchy problem for ut δum when 0 m 1 on considere le probleme de cauchy pour l equation de diffusion non lineaire u t u u m 1 0 sur 0 r n u 0 u 0 quand 0 m 1 on montre qu il existe une solution temporelle globale pour une fonction localement integrable u 0 on montre l unicite dans cette classe

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

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

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

well posedness of cauchy problem for the fourth order nonlinear schrodinger equations in multi dimensional spaces abstract we study the well posedness of cauchy problem for the fourth order nonlinear schrodinger equations i t u e u 2 u p x u 2 x u 2 t r x r n where e 1 0 1 n 2 denotes the spatial dimension and p is a polynomial excluding constant and linear terms

stability of solitons described by nonlinear schrodinger type equations with higher order dispersion abstract steady solitons described by the fourth order nonlinear schrodinger type equations in one two and three dimensions with power law nonlinearities are studied conditions of existence and stability of such solitons are found numerical results demonstrating soliton existence stability and instability are described they agree with the developed theory

a representation of all solutions of the control algebraic riccati equation for infinite dimensional systems we obtain a representation of all self adjoint solutions of the control algebraic riccati equation associated to the infinite dimensional state linear system a b c under the following assumptions a generates a c0 group the system is output stabilizable strongly detectable and the dual riccati equation has an invertible self adjoint non negative solution

the suboptimal nehari problem for well posed linear systems we solve the suboptimal nehari problem for a transfer function that has a state space realization as a system stable input output and input output stable well posed linear system we obtain an explicit solution in terms of the state space parameters