invited paper survey of quantitative feedback theory qft abstract qft is an engineering design theory devoted to the practical design of feedback control systems the foundation of qft is that feedback is needed in control only when plant p parameter and or disturbance d uncertainties sets p p d d exceed the acceptable a system performance uncertainty set a a the principal properties of qft are as follows 1 the amount of feedback needed is tuned to the p d a sets if a exceeds p d feedback is not needed at all 2 the simplest modelling is used a command disturbance and sensor noise inputs and b the available sensing points and the defined outputs no special controllability test is needed in either linear or non linear plants it is inherent in the design procedure there is no observability problem because uncertainty is included the number of independent sensors determines the number of independent loop transmissions li the functions which provide the benefits of feedback 3 the simplest mathematical tools h

abstract quasilinear parabolic equations on considere des problemes paraboliques quasi lineaires abstraits p 0 u t a t u t u t f t u t t 0 u o u 0 ou u t est une fonction a valeurs dans un espace de banach x on donne des theoremes d existence locale et d unicite pour des solutions classiques et strictes de p 0

holder classes with boundary conditions as interpolation spaces on etudie la caracterisation de certains espaces d interpolation reels entre le domaine d un operateur differentiel elliptique avec des conditions aux limites generales et l espace de banach de fonctions continues dans lequel le domaine est plonge

transmission problems in a thin layer set in the framework of the holder spaces resolution and impedance concept abstract we consider a family of singular transmission problems depending on some small positive parameter set in the juxtaposition of two rectangular domains they are written in the form of abstract elliptic equations and the study here is given in the holder spaces completing in this way the work in lp cases given in a favini r labbas k lemrabet s maingot study of the limit of transmission problems in a thin layer by the sum theory of linear operators rev mat complut 18 1 2005 143 176 in this first part we present a new approach for the resolution of these problems by using the concept of impedance operator this method is different of the one performing a rescaling in the thin layer see a favini r labbas k lemrabet s maingot study of the limit of transmission problems in a thin layer by the sum theory of linear operators rev mat complut 18 1 2005 143 176 it leads to obtain direct and simplified problems we use the dunford calculus and some techniques similar to that in r labbas problemes aux limites pour une equation differentielle abstraite de type elliptique these d etat universite de nice 1987 a favini r labbas s maingot h tanabe a yagi unified study of elliptic problems in holder spaces c r math acad sci paris 134 2005 g dore a favini r labbas k lemrabet s maingot a transmission problem in a thin layer part i sharp estimates in press to prove existence uniqueness results and some specific estimates on the impedance operator this study will allow us in a forthcoming work to obtain respectively optimal regularities and the limit problem when 0

stationary navier stokes equations with critically singular external forces existence and stability results abstract we show the unique existence of solutions to stationary navier stokes equations with small singular external forces belonging to a critical space to the best of our knowledge this is the largest critical space that is available up until now for this kind of existence this result can be viewed as the stationary counterpart of the existence result obtained by h koch and d tataru for the free non stationary navier stokes equations with initial data in bmo 1 the stability of the stationary solutions in such spaces is also obtained by a series of sharp estimates for resolvents of a singularly perturbed operator and the corresponding semigroup

causality and stability of linear systems described by partial differential operators definitions are suggested for the concepts of causality and stability of linear systems described by partial differential operators with constant coefficients these definitions are modelled after the one dimensional case the operators having causal and stable fundamental solutions are characterized algebraically and nyquist type criteria are established

nonlinear analytic semiflows in this paper a local existence and regularity theory is given for nonlinear parabolic initial value problems x t f x t and quasilinear initial value problems x t a x t x t f x t this theory extends the theory of daprato and grisvard of 1979 and shows how various properties like analyticity of solutions can be derived as a direct corollary of the existence theorem

small time gaussian behavior of symmetric diffusion semi groups this work is involved with the short time asymptotics of diffusion semigroups in a general setting a generalization of fang s version of varadhan s formula is proven for general dirichlet spaces that are local and conservative the intrinsic metric appearing in the formula is characterized by pointwise distance for canonical dirichlet spaces on loop groups

logarithmic sobolev inequalities on loop groups let g be a compact lie group denote by m the brownian bridge measure on the loop group y g c 0 1 g g 0 g 1 e the finite energy subgroup of y determines in a natural way a gradient operation for functions on y the following logarithmic sobolev inequality is proven f2 log f dm gradf y 2 v y f y 2 dm f 2log f wherein f denotes the l2 m norm and v is a potential which is quadratic in the associated lie algebra valued brownian motion the inequality is derived by a method of inheritance from the known inequality for the g valued brownian motion

on a class of retarded partial differential equations on developpe des resultats recents sur des equations d evolution abstraites afin de les appliquer a un type d equations dans des espaces de banach avec un retard discret et continu ces equations peuvent etre considerees comme la version abstraite d equations integrodifferentielles partielles a retard

global solutions of abstract quasilinear parabolic equations abstract local and global existence and uniqueness for strict solutions of abstract quasilinear parabolic equations are studied applications to quasilinear parabolic partial differential equations are also given

quasilinear systems with memory kernels we consider a quasilinear integrodifferential system in non normal form such a system is a generalization of a phase field model with memory and includes as a particular case the system describing the combustion of a material with memory in this paper we study both the direct and the inverse problems our fundamental tools are the theory of analytic semigroups optimal regularity results and fixed point arguments

analysis of fully discrete approximations to parabolic problems with rough or distribution valued initial data we consider fully discrete approximations to a parabolic initial boundary value problem with rough or distribution valued initial data in two space dimensions for discretization in time and space we apply single step methods and the standard galerkin method with piecewise linear test functions respectively for spatial discretization of the initial condition we are however forced to use more involved constructions our main result is stability and error estimates of the discrete solutions

stability in a two dimensional free boundary combustion model we prove rigorously the stability of the planar travelling wave solution to a free boundary system in r2 arising as a model in combustion theory for appropriate values of the reduced lewis number after changements of coordinates and unknowns the problem is reduced to the stability of the null solution of a fixed boundary parabolic system the difficulties consist in the fact that the final system is fully nonlinear and the spectrum of the linearized system contains the halffine 0 they are overcome setting the problem in a suitably weighted holder space

uncertain partially non causal distributed feedback systems abstract there is given a constrained plant p described by a partial differential equation pde in two variables t and x owing to uncertainty p belongs to a given set there are assigned tolerances on the system response to command and or disturbances to be achieved over the defined t and x intervals for all p in previous treatment was confined to systems causal in both t and x this work extends quantitative synthesis to systems causal in l and non causal in x frequency response methods in two complex variables are presented which are generalizations of quantitative synthesis techniques for lumped systems the principal treatment is to a t in 0 x in and b t in 0 x in 0 consideration is also given to uncertain non linear pde plants

integration on loop groups ii heat equation for the wiener measure abstract let g be a compact semi simple lie group and l g the loop group over g on l g is defined a family t l t 0 of quasi invariant wiener measures we construct on l g a laplace operator l such that t l are the measures generated by the process associated to l modified by a feynman kac density

a new numerical method for backward parabolic problems in the maximum norm setting a new method for solving numerically backward parabolic problems is proposed as usual for this kind of ill posed problems it is assumed that an a priori bound for the solution is available the algorithm consists of two basic steps first a standard forward integration is performed in order to approximate the solution at suitable future time levels second a holomorphic recovery procedure is carried out providing the required approximations for the preceding times the analysis is valid in the maximum norm setting and rigorous estimates are derived among other advantages the method can also be applied to nonlinear problems and it produces a continuous output some numerical illustrations are presented

one point feedback approach to distributed linear systems abstract the disturbance attenuation problem is studied for linear distributed plants described by partial differential operators and by using one lumped i e acting at a single point feedback loop a result emphasizing the trade off between stability and disturbance attenuation is proven also the effect of the feedback loop location on disturbance attenuation is pointed out finally a synthesis procedure is derived valid also for plants having bounded uncertainty in the gain the procedure is illustrated by two significant examples in a one dimensional bounded domain

abstract inverse problem and application abstract in this paper we consider the determination of the unknown function 0 t x x is a banach space in the abstract cauchy problem u t au t t and u 0 x from an additional given data first we establish an existence and uniqueness theorem when i 1 n h i y i where h i 1 i n are unknown numerical functions and y i x 1 i n are known this theorem is then used to derive a nonuniqueness result for the class of functions which are not written in the form i 1 m k i z i k i 0 t c and z i x we also show the continuous dependence of the inverse problem solution on the additional data finally an application to the parabolic equation is indicated

direct and inverse problems for a phase field model with memory we consider a semilinear integrodifferential system in non normal form such a system is a generalization of the one that arises in the phase field theory with memory we prove an abstract existence and uniqueness theorem and a continuous dependence result for the direct problem reformulating the direct problem in a suitable way we prove that the identification problem admits a unique solution

asymptotic expansion for layer solutions of a singularly perturbed reaction diffusion system for a singularly perturbed n dimensional system of reactiondiffusion equations assuming that the 0th order solutions possess boundary and internal layers and are stable in each regular and singular region we construct matched asymptotic expansions for formal solutions in all the regular boundary internal and initial layers to any desired order in e the formal solution shows that there is an invariant manifold of wave front like solutions that attracts other nearby solutions we also give conditions for the wave front like solutions to converge slowly to stationary solutions on that manifold

regular solutions for time dependent abstract integrodifferential equations with singular kernel abstract results of existence uniqueness and regularity for strict and classical solutions of linear nonautonomous integrodifferential initial value problems are proved using maximal regularity properties for the autonomous nonintegral case

on the solutions of the first boundary value problem for the linear parabolic equations on etudie le premier probleme aux valeurs limites pour une equation parabolique lineaire d ordre 2 sous l hypothese que le terme inhomogene est continu en espace et en temps et continu de holder seulement par rapport aux variables d espace

existence and regularity for a class of integrodifferential equations of parabolic type the method we use rests on the inversion of the formal laplace transform of r and allows one to analyze the regularity of the solution u problem 1 1 has been studied by several authors using various methods see for instance l 18 211 the laplace transform approach presented here has been used in s 7 9 14 153 in our previous papers s 7 93 we have in fact analyzed a more general situation while here we take full 36 0022 247x 85 3 00

a cameron martin type quasi invariance theorem for brownian motion on a compact riemannian manifold abstract let m be a compact manifold without boundary o be a fixed base point in m g be a riemannian metric on m and be a g compatible covariant derivative on tm the tangent space to m assume the torsion t of satisfies the skew symmetry condition g t x y y 0 for all vector fields x and y on m for example take to be the levi civita covariant derivative on m g also let denote the wiener measure on w0 m c 0 1 m 0 o and let h s denote stochastic parallel translation relative to along the path w0 m up to time s 0 1 given a c1 function h 0 1 t0m it is shown that the differential equation t h t h with initial condition 0 id w m w m has a solution r maps w m w m the measurable maps from w m to w m this function is a flow on w m i e for all t r t t furthermore t has the quasi invariance property the law t of t with respect to the wiener measure is equivalent to for all t r this result is used to prove an integration by parts formula for the h derivative hf defined by h f d dt 0 f t where f is a c2 cylinder function on w m

quasilinear evolution equations and parabolic systems it is shown that general quasilinear parabolic systems possess unique maximal classical solutions for sufficiently smooth initial values provided the boundary conditions are time independent moreover it is shown that in the autonomous case these equations generate local semi flows on appropriate sobolev spaces our results apply in particular to the case of prescribed boundary values dirichlet boundary conditions

on the abstract cauchy problem of parabolic type in spaces of continuous functions on etudie l existence l unicite et les proprietes de regularite des solutions du probleme de cauchy u t au r f t o t t u o u 0 ou a d a ce e est un operateur lineaire clos dans l espace de banach e

yang mills theory and the segal bargmann transform we use a variant of the segal bargmann transform to study canonically quantized yang mills theory on a space time cylinder with a compact structure group k the non existent lebesgue measure on the space of connections is approximated by a gaussian measure with large variance the segal bargmann transform is then a unitary map from the l 2 space over the space of connections to a holomorphicl 2 space over the space of complexified connections with a certain gaussian measure this transform is given roughly bye t1a 2 followed by analytic continuation here 1a is the laplacian on the space of connections and is the hamiltonian for the quantized theory on the gauge trivial subspace consisting of functions of the holonomy around the spatial circle the segal bargmann transform becomes e t1 k 2 followed by analytic continuation where 1k is the laplacian for the structure group k this result gives a rigorous meaning to the idea that 1a reduces to 1k on functions of the holonomy by letting the variance of the gaussian measure tend to infinity we recover the standard realization of the quantized yang mills theory on a space time cylinder namely 1 1k is the hamiltonian and l 2 k is the hilbert space as a byproduct of these considera tions we find a new one parameter family of unitary transforms from l 2 k to certain holomorphicl 2 spaces over the complexification of k this family of transformations interpolates between the two previously known unitary transformations our work is motivated by results of landsman and wren and uses probabilistic techniques similar to those of gross and malliavin

holder regularity in variational parabolic non homogeneous equations abstract we prove optimal holder regularity results for the solutions to second order variational parabolic equations with holder continuous coefficients and non homogeneous boundary data using analytic semigroups methods

logarithmic sobolev inequalities for pinned loop groups abstract letgbe a connected compact type lie group equipped with anadg invariant inner product on the lie algebra g ofg given this data there is a well known left invariant h1 riemannian structure on l l g the infinite dimensional group of continuous based loops ing using this riemannian structure we define and construct a heat kernel t g0 associated to the laplace beltrami operator on l g heret 0 g0 l g and t g0 is a certain probability measure on l g for fixedg0 l g andt 0 we use the measure t g0 and the riemannian structure on l g to construct a classical pre dirichlet form the main theorem of this paper asserts that this pre dirichlet form admits a logarithmic sobolev inequality

on the equivalence of measures on loop space let k be a simply connected compact lie group equipped with an ad k invariant inner product on the lie algebra of k given this data there is a well known left invariant h 1 riemannian structure on l k the infinite dimensional group of continuous based loops in k as well as a heat kernel t k 0 associated with the laplace beltrami operator on l k here t 0 k 0 l k and t k 0 is a certain probability measure on l k in this paper we show that 1 e is equivalent to pinned wiener measure on k on s0 x t t 0 s 0 the algebra generated by truncated loops up to time s 0

quantitative synthesis of feedback systems with distributed uncertain plants there is given a constrained plant p described by a partial differential equation p d o in two variables t and x owing to uncertainty p belongs to a given sot p tliero are assigned tolerances on the system response y t xx in rosponso to command input r t x to be achieved over the defined t x intervals for all pe p frequency response methods in two complex variables s1 and s2 are used to synthesize the system those methods are two variable generalizations of the quantitative design techniques developed for lumped uncertain plants detailed design examples are given for two cases 1 linear constant coefficient p d e plant defined on semi infinite t x with zero boundary conditions 2 similar to 1 except for definition on finite x the latter case is converted to a problem on two semi infinite variables t and x with coefficients which are functions of a synthesis technique for lumped time varying plants is adapted for this now problem this method may be used for the general l

bounded solutions of linear periodic abstract parabolic equations on etudie le comportement asymptotique de l operateur d evolution parabolique g t s associe a une famille de generateurs de sous groupes analytiques a t t r dans un espace de banach general sous l hypothese de periodicite a t a t t on trouve de nouveaux resultats de regularite maximale pour les solutions bornees d equations paraboliques non homogenes abstraites dans des intervalles de temps non bornes

recovering a leading coefficient and a memory kernel in first order integro differential operator equations abstract we are concerned with the identification of the scalar functions a and k in the convolution first order integro differential equation u t a t au t k bu t f t 0 t t k v t 0 t k t s v s ds in a banach space x where a and b are linear closed operators in x a being the generator of an analytic semigroup of linear bounded operators taking advantage of two pieces of additional information we can recover under suitable assumptions and locally in time both the unknown functions a and k the results so obtained are applied to an n dimensional integro differential identification problem in a bounded domain in r n

global existence via ginzburg landau formalism and pseudo orbits of ginzburg landau approximations the so called ginzburg landau formalism applies for parabolic systems which are defined on cylindrical domains which are close to the threshold of instability and for which the unstable fourier modes belong to non zero wave numbers this formalism allows to describe an attracting set of solutions by a modulation equation here the ginzburg landau equation if the coefficient in front of the cubic term of the formally derived ginzburg landau equation has negative real part the method allows to show global existence in time in the original system of all solutions belonging to small initial conditions inl another aim of this paper is to construct a pseudo orbit of ginzburg landau approximations which is close to a solution of the original system up tot we consider here as an example the socalled kuramoto shivashinsky equation to explain the methods but it applies also to a wide class of other problems like e g hydrodynamical problems or reaction diffusion equations too

maximum norm analysis of completely discrete finite element methods for parabolic problems we consider full discretizations of linear parabolic initial value problems in any space dimension finite elements are used for the discretization in space the resulting semidiscrete problem is integrated in time by means of a strongly a theta acceptable rational method stability estimates are derived and error bounds are established in the maximum norm for both smooth and rough initial data

global existence of solutions for a strongly coupled quasilinear parabolic system we are interested in existence uniqueness and a priori estimates of non negative solutions to the following problem begin gathered u t delta left left 1 alpha v right u right uf u v hfill v t delta v vg u v quad in 0 infty times omega hfill u v 0 quad in 0 infty times partial omega hfill u 0 u 0 v 0 v 0 quad in omega hfill end gathered 1 0

absolute continuity of heat kernel measure with pinned wiener measure on loop groups let t 0 k be a connected compact lie group equipped with an ad k invariant inner product on the lie algebra of k associated to this data are two measures 0 t and t 0 on l k the space of continuous loops based at e e k the measure 0 t is pinned wiener measure with variance t while the measure 0 t is a heat kernel measure on l k the measure 0 t is constructed using a k valued brownian motion while the measure 0 t is constructed using a l k valued brownian motion in this paper we show that 0 t is absolutely continuous with respect to 0 t and the radon nikodym derivative d 0 t d 0 t is bounded

integration by parts and quasi invariance for heat kernel measures on loop groups abstract integration by parts formulas are established both for wiener measure on the path space of a loop group and for the heat kernel measures on the loop group the wiener measure is defined to be the law of a certain loop group valued brownian motion and the heat kernel measures are time t t 0 distributions of this brownian motion a corollary of either of these integrations by parts formulas is the closability of the pre dirichlet form considered by b k driver and t lohrenz 1996 j functional anal 140 381 448 we also show that the heat kernel measures are quasi invariant under right under right and left translations by finite energy loops

cα regularity for non autonomous linear integrodifferential equations of parabolic type abstract linear integrodifferential equations in general banach space are studied and applications are given to linear integrodifferential partial differential equations

analyticity of the maximal solution of an abstract nonlinear parabolic equation where u t x 0 7 j x r r satisfies other suitable conditions problem p is studied under the assumption that af au generates an analytic semigroup in e and has domain f which corresponds to a parabolicity condition on equation p problems of this kind have been studied by monotonicity methods see brezis 111 crandall ligget 3 and crandall pazy 4 to obtain global but weak solutions of p here we find a maximally defined strict solution by means of a linearization method and by virtue of maximal regularity results for the linear case this method has been used by da prato grisvard in 5 to obtain strict solutions of p with arguments which seem to be more simple and suitable to our purpose we get also the analyticity of u with respect to 1 w when f is analytic in section 1 some notation and assumptions are given problem p is studied in section 2 where we prove existence and uniqueness of the strict solution of p which is defined on a maximal time interval contained in 0 t and it is continuous with respect to t 0 in section 3 we give conditions to get global existence of the solution in section 4 we study analyticity with respect to t ug of the maximally defined solution of p and in section 5 a simple example is given in a subsequent paper we will show how this theory applies to the study of the analyttctty of the solutions of nonlinear parabol ic partial differential equations in spaces of holder continuous functions

on linear volterra equations of parabolic type in banach spaces linear integrodifferential equations of volterra type in a banach space are studied in case the main part of the equation generates an analytic co semigroup under very general assumptions it is shown that a resolvent operator exists and that many of the solution properties of parabolic evolu tion equations are inherited the results are then applied to integro partial differential equations of parabolic type

analytic semigroups generated by ultraweak operators let be a regular open subset of r n we present an improved generation result for nonvariational operators in l 1 this result is obtained by studying ultraweak operators and by proving generation of analytic semigroups in l p l p and in we also characterise interpolation and extrapolation spaces

integration on loop group iii asymptotic peter weyl orthogonality using the wiener measure on the continuous loop group l g of a compact connected group g whose properties were studied in integration on loop group i we construct for g simply laced a stochastic central extension l g of l g despite the fact that this object has a multiplication defined almost everywhere we are able to define for it a convolution algebra

non explosion boundedness and ergodicity for stochastic semilinear equations abstract sufficient conditions for non explosion boundedness and ergodicity of solutions of a class of semilinear stochastic equations are given applications of the results are based on a new regularity theorem for solutions of the linear problem

on the continuity of ornstein uhlenbeck processes in infinite dimensions existence and continuity of ornstein uhlenbeck processes in banach and hilbert spaces are investigated under various assumptions

on the computation of switching surfaces in optimal control a grobner basis approach a number of problems in control can be reduced to finding suitable real solutions of algebraic equations in particular such a problem arises in the context of switching surfaces in optimal control a powerful new methodology for doing symbolic manipulations with polynomial data has been developed and tested namely the use of grobner bases we apply the grobner basis technique to find effective solutions to the classical problem of time optimal control

an identification problem for an ultraparabolic integrodifferential equation we prove the well posedness of an inverse problem related to an abstract integrodifferential problem in two time variables with applications to ultraparabolic partial integrodifferential equations of volterra type

on the evolution operator for a class of non autonomous abstract parabolic equations abstract we state existence and several properties of the evolution operator relevant to a family a t 0 t t of generators of analytic semigroups in a banach space the assumption that certain interpolation spaces do not depend on t allows us to weaken the usual regularity hypotheses on t a t requiring only continuity in suitable topologies