global solutions to semilinear parabolic systems for small data abstract we consider semilinear parabolic systems u t au f u g u 0 u 0 a being the generator of an analytic semigroup with spectrum a in the right half plane for appropriate small data g u 0 global solutions are obtained if 0 belongs to the resolvent set and local existence is proved if 0 a the inverse function theorem is used in a class of functions which are holder continuous with respect to time t the asymptotic behaviour of the solution is given too as a first application we prove the existence of global respectively local solutions to the navier stokes equations in l spaces in bounded respectively unbounded domains second we consider the case where a is an elliptic operator of order 2 m and f has bounded growth third a quasilinear example where the nonlinearity involves derivatives of the same order as a is studied then an example of a nonautonomous nonlinearity is given and finally the convergence of solutions of the time dependent system to solutions of the stationary semilinear elliptic one is derived for 0 a

deadbeat algorithms for output controllers in discrete time multivariable systems abstract this paper studies the problem of designing output deadbeat controllers to force the state of discrete time multivariable systems to zero in a finite number of samples two algorithms are considered the first is based on the fact that the closed loop eigenstructure assignable by output feedback is constrained by the requirement that the left and right eigenvectors must be in certain subspaces in the second algorithm the output gain matrix is computed through the optimization of certain parameters of the controller while maintaining its structural constraints computer programs have been developed to realize the two algorithms and examples are given to illustrate the feasibility of the techniques

monotonic decreasing and asymptotic behavior of the kinetic energy for weak solutions of the navier stokes equations in exterior domains existence of weak solutions of the three dimensional navier stokes problem was first proved by j leray in the case of the cauchy problem cf leray 1934 as is well known these solutions are important in that they are the only solutions which so far are known to exist for all times without restriction on the data unfortunately however the question of whether they are classical in an ordinary sense is still open even though partial conclusions regarding regularity are avilable leray 1934 scheffer 1976 1980 caffarelli kohn nirenberg 1982 subsequently e hopf 1951 using a different technique constructed weak solutions for a general initial boundary value problem however hopf s solution even for the cauchy problem has weaker properties than leray s solution among others we refer to the energy inequality for the velocity field u x t i e ntimits mega u 2 x t dx ntimits mega u 2 x s sx eqq 2ntimits s t ntimits mega abla u abla udxdau i for s 0 for almost all s 0 and for all t s

asymptotic stability of landau solutions to navier stokes system it is known that the three dimensional navier stokes system for an incompressible fluid in the whole space has a one parameter family of explicit stationary solutions that are axisymmetric and homogeneous of degree 1 we show that these solutions are asymptotically stable under any l2 perturbation

on energy inequality smoothness and large time behavior in l 2 for weak solutions of the navier stokes equations in exterior domains soit un domaine exterieur de r n n 3 4 a frontiere lisse sur le cylindre d espace temps 0 on considere le probleme de navier stokes non stationnaire u t u u u p f dans 0 u 0 dans 0 u 0 sur 0 u x 0 a x avec des donnees prescrites f et a

boundedness for large x of suitable weak solutions of the navier stokes equations with prescribed velocity at infinity we consider time dependent perturbationsu of r finn s stationary pr solution of the navier stokes equations which converges to a constant vectorv as x for a given time interval t we find a radiusk such thatu is essentially bounded on t x k

on stability of the spatially inhomogeneous navier stokes boussinesq system with general nonlinearity this paper considers l 2 asymptotic stability of the spatially inhomogeneous navier stokes boussinesq system with general nonlinearity including both power nonlinear terms and convective terms we construct a local in time strong solution of the system by applying semigroup theory on hilbert spaces and fractional powers of the stokes laplace operator it is also shown that under some assumptions on an energy inequality the system has a unique global in time strong solution when the initial datum is sufficiently small furthermore we investigate the asymptotic stability of the global in time strong solution by using an energy inequality maximal l p in time regularity for hilbert space valued functions and fractional powers of linear operators in a solenoidal l 2 space we introduce new methods for showing the asymptotic stability by applying an energy inequality and maximal l p in time regularity for hilbert space valued functions our approach in this paper can be applied to show the asymptotic stability of energy solutions for various incompressible viscous fluid systems and the stability of small stationary solutions whose structure is not clear

on the global existence and convergence to steady state of navier stokes flow past an obstacle that is started from rest a new approach to the existence theory for the navier stokes equations using a technique of kato 15 further developed in combination with estimates for oseen s equation by kobayashi shibata 17 and shibata 24 has made possible the solution of a long standing open problem often referred to as finn s starting problem this paper provides the solution

algebraic l 2 decay for weak solutions of a viscous boussinesq system in exterior domains abstract the algebraic decay rates for the total kinetic energy of weak solutions of the n dimensional viscous boussinesq system in exterior domains are established by means of the spectral decomposition method of fractional powers of the laplacian and stokes operators the results reveal that l 2 decay of the velocity field with zero divergence becomes slower than that of the temperature

linear regression with random projections we investigate a method for regression that makes use of a randomly generated subspace gp f of finite dimension p of a given large possibly infinite dimensional function space f for example l2 0 1 d r gp is defined as the span of p random features that are linear combinations of a basis functions of f weighted by random gaussian i i d coefficients we show practical motivation for the use of this approach detail the link that this random projections method share with rkhs and gaussian objects theory and prove both in deterministic and random design approximation error bounds when searching for the best regression function in gp rather than in f and derive excess risk bounds for a specific regression algorithm least squares regression in gp this paper stresses the motivation to study such methods thus the analysis developed is kept simple for explanations purpose and leaves room for future developments

weak solutions of the navier stokes equations with non zero boundary values in an exterior domain satisfying the strong energy inequality abstract in an exterior domain r 3 and a time interval 0 t 0 t consider the instationary navier stokes equations with initial value u 0 l 2 and external force f div f f l 2 0 t l 2 as is well known there exists at least one weak solution in the sense of j leray and e hopf with vanishing boundary values satisfying the strong energy inequality in this paper we extend the class of global in time leray hopf weak solutions to the case when u g with non zero time dependent boundary values g although uniqueness for these solutions cannot be proved we show the existence of at least one weak solution satisfying the strong energy inequality and a related energy estimate

asymptotic stability of large solutions with large perturbation to the navier stokes equations abstract consider weak solutions w of the navier stokes equations in serrin s class w l 0 l q for 2 3 q 1 with 3 q where is a general unbounded domain in r 3 we shall show that although the initial and external disturbances from w are large every perturbed flow v with the energy inequality converges asymptotically to w as v t w t l 2 0 v t w t l 2 o t 1 2 as t

output deadbeat control discrete time multivariate systems an algorithm is presented to compute output deadbeat controls for linear multivariable systems the algorithm based on quadratic optimization with no cost on control cheap control is numerically stable and covers the most general cases the resulting controls are internally stable and are shown to drive the outputs of the system to zero in minimum time using state feedback the central part of the algorithm is the construction of the nonunique state cost matrix two alternatives are presented having different complexities and resulting in solutions having different properties

an improved l2 decay for weak solutions of a boussinesq system in exterior domains abstract the algebraic decay for the total kinetic energy of weak solutions of the viscous boussinesq system in an exterior domain is given by means of the spectral analysis method of the stokes operator which gives a positive answer to the l 2 decay question in 11

solutions for semilinear parabolic equations in lp and regularity of weak solutions of the navier stokes system abstract we construct a unique local regular solution in l q 0 t l p for a class of semilinear parabolic equations which includes the semilinear heat equation u t u u u 0 and the navier stokes system here p and q are so chosen that the norm of l q 0 t l p is dimensionless or scaling invariant the main relation between p and q for the semilinear heat equation is 1 q 1 p 1 q n 2 provided that initial data are in l r with r n 2 1 where n is the space dimension applying our regular solutions to the navier stokes system we show that the k 2 dimensional hausdorff measure of possible time singularities of a turbulent solution is zero if the turbulent solution is in l q 0 t l p where k 2 q nq p p n 1 q we show moreover that a turbulent solution is regular if it is in c 0 t l n

on weighted norm estimates for nonstationary incompressible navier stokes flows in a 3d exterior domain abstract we study the time decay of weighted norms of weak and strong solutions to the navier stokes equations in a 3d exterior domain moment estimates for weak solutions and weighted l q estimates for strong solutions are deduced both of which seem to be optimal the relation is discussed between the space time decay and the vanishing of the total net force exerted by the fluid to the body a class of initial data is given so that the total net force associated to the corresponding fluid flows does not vanish

deadbeat error control of discrete multivariable systemsdagger abstract the main objective of this paper is to delineate conditions under which deadbeat error control can be achieved between a plant output and a discrete reference input signal all facets of this problem are considered in the linear multivariable case i e when the reference is any non diminishing signal whether the plant is truly discrete or a discretized continuous one and whether the initial conditions are zero or unknown the results presented can also be used to resolve the case of known non zero initial conditions on the state of the plant as well as the compensator

l2 decay for the navier stokes flow in halfspaces on etudie le comportement asymptotique dans l 2 des solutions faibles des equations de navier stokes dans des demi espaces

a direct computation of state deadbeat feedback gains a method for computing a feedback gain that achieves state deadbeat control is given from systems given in the staircase form this method derives the deadbeat gain in a numerically reliable way it is shown that the gain turns out to be lq optimal for some weightings

on the energy inequality for weak solutions to the navier stokes equations of compressible fluids on unbounded domains abstract we consider the navier stokes equations of compressible isentropic viscous fluids on an unbounded three dimensional domain with a compact lipschitz boundary under the condition that the total mass of the fluid is finite we show the existence of globally defined weak solutions satisfying the energy inequality in differential form

further properties of steady state solutions to the navier stokes problem past a three dimensional obstacle we show that the weak formulation of the steady state navier stokes problem in the exterior of a three dimensional compact set closure of a bounded domain corresponding to a nonzero velocity at infinity and subjected to a given body force is equivalent to a nonlinear equation in appropriate banach spaces we thus show that the relevant nonlinear operator enjoys a number of fundamental properties that allow us to derive many significant results for the original problem in particular we prove that the manifold constituted by the pairs u with the nondimensional speed at infinity reynolds number and u weak solution corresponding to and to a given body force f is for generic f a c one dimensional manifold and that for almost any the number of solutions is finite we also show that for any given f in the appropriate function space and any given 0 the corresponding solutions can be controlled by their specification only in a suitable neighborhood i of the boundary the size

on l 1 summability and asymptotic profiles for smooth solutions to navier stokes equations in a 3d exterior domain the exterior nonstationary problem is studied for the 3d navier stokes equations the l1 summability is proved for smooth solutions which correspond to initial data satisfying certain symmetry and moment conditions the result is then applied to show that such solutions decay in time more rapidly than observed in general furthermore an asymptotic expansion is deduced and a lower bound estimate is given for the rates of decay in time

abstract lp estimates for the cauchy problem with applications to the navier stokes equations in exterior domains abstract we apply an abstract perturbation theorem to derive global in time lq estimates for the cauchy problem and lq ls estimates for the nonstationary stokes equations in exterior domains this will be applied to obtain various new global in time estimates for weak solutions of the navier stokes equations

on l3 stability of the navier stokes system in exterior domains abstract this paper studies the stability of a stationary solution of the navier stokes system with a constant velocity at infinity in an exterior domain more precisely this paper considers the stability of the navier stokes system governing the stationary solution which belongs to the weak l 3 space l 3 under the condition that the initial datum belongs to a solenoidal l 3 space we prove that if both the l 3 norm of the initial datum and the l 3 norm of the stationary solution are sufficiently small then the system admits a unique global in time strong l 3 solution satisfying both l 3 asymptotic stability and l asymptotic stability moreover we investigate l 3 r asymptotic stability of the global in time solution using l p l q type estimates for the oseen semigroup and applying an equivalent norm on the lorentz space are key ideas to establish both the existence of a unique global in time strong or mild solution of our system and the stability of our solution

energy decay for a weak solution of the navier stokes equation with slowly varying external forces we consider the navier stokes system with slowly decaying external forces formula we show that the energy norm of a weak solution has non uniform decay u t 2 0 t under suitable conditions on the data f and a which make the energy of solution bounded in time also we show the exact rate of the decay uniform decay of the energy u t 2 c 1 t e for external forces with a given explicit decay rate

state deadbeat control of nonlinear systems construction via sets a geometric generalization of the discrete time linear deadbeat control problem is studied the proposed method to generate a deadbeat tracker for a given nonlinear system is constructive and makes use of sets that can be computed iteratively for demonstration derivations of the deadbeat feedback law and tracker dynamics are provided for an example system based on the method a simple algorithm that computes the deadbeat gain for a linear system with scalar input is given

domains of fractional powers of the stokes operator in lr spaces soit d un domaine borne de r n n 2 a frontiere lisse s soit 1 r et soit x r la fermeture dans l r d n de tous les champs vectoriels solenoidaux lisses a support compact dans d soit w r m d l espace de sobolev d ordre m et soit le laplacien on considere l operateur de stokes a r p r de domaine d a r w w r 2 d n w 5 0 mx 2 soit a un nombre reel positif pour chaque positif il y a une constante ce a telle que a r z ce a exp im z pour tout z satisfaisant a rez 0 ou a r z est la norme operateur de a r z

a specifically fully nonlinear perturbation of the heat equations in exterior domains there is extensive literature on solving nonlinear evolution equations with small initial data by means of energy estimates and l l estimates this approach has been applied to 1 1 in case g r cf l 3 in earlier work to solve such a fully nonlinear problem one established an a priori c2 estimate by using energy estimates which must be considered in a version of hilbert spaces hence there had to be attached the unpleasant requirement that f e ck and a is small in w g fl w g for k and i large enough we also refer to 4 where bandle and levine found that the problem

design of ripple free multivariable robust servomechanisms the application of standard robust servomechanism theory to sampled data systems guarantees asymptotic tracking at sampling instances and between sampling instants the output will normally contain ripple in this note the robust servomechanism theory is extended to sampled data systems and a technique is proposed for ripple free tracking of sinusoids and polynomials it is shown that a continuous internal model is necessary and sufficient to provide ripple free response

deadbeat control and tracking of discrete time systems a new approach for forcing the state of a linear discrete time system to zero in a minimum number of steps is discussed the problem is formulated as the solution to a steady state optimal control problem with no cost on the control this problem in turn is set up as the solution to an associated eigenvalue problem no special assumption on the open loop system matrix and or the ratio of the number of states to controls is required stable numerical techniques are presented for solving for the feedback gain robust deadbeat tracking is also discussed

technical communique a sparse and condensed qp formulation for predictive control of lti systems the computational burden that model predictive control mpc imposes depends to a large extent on the way the optimal control problem is formulated as an optimization problem we present a formulation where the input is expressed as an affine function of the state such that the closed loop dynamics matrix becomes nilpotent using this approach and removing the equality constraints leads to a compact and sparse optimization problem to be solved at each sampling instant the problem can be solved with a cost per interior point iteration that is linear with respect to the horizon length when this is bigger than the controllability index of the plant the computational complexity of existing condensed approaches grow cubically with the horizon length whereas existing non condensed and sparse approaches also grow linearly but with a greater proportionality constant than with the method presented here