on flows associated to sobolev vector fields in wiener spaces an approach a la diperna lions in this paper we extend the diperna lions theory of flows associated to sobolev vector fields to the case of cameron martin valued vector fields in wiener spaces e having a sobolev regularity the proof is based on the analysis of the continuity equation in e and on uniform gaussian commutator estimates in finite dimensional spaces

one dimensional transport equations with discontinuous coefficients we consider one dimensional linear transport equations with bounded but possibly discontinuous coefficient a the cauchy problem is studied from two different points of view in the first case we assume that a is piecewise continuous we give an existence result and a precise description of the solutions on the lines of discontinuity in the second case we assume that a satisfies a one sided lipschitz condition we give existence uniqueness and general stability results for backward lipschitz solutions and forward measure solutions by using a duality method we prove that the flux associated to these measure solutions is a product by some canonical representative of a key words linear transport equations discontinuous coefficients weak stability duality product of a measure by a discontinuous function nonnegative solutions 1991 mathematics subject classification primary 35f10 35b35 34a12 to appear in nonlinear analysis tma departement de mathematiques et applications umr cnrs 8553 ecole normale superieure et cnrs 45 rue d ulm 75230 paris cedex 05 france francois bouchut ens fr mathematiques applications et physique mathematique d orleans umr cnrs 6628 universite d orleans 45067 orleans cedex 2 france james math cnrs fr

convergence of numerical approximations to non linear continuity equations with rough force fields we prove quantitative regularity estimates for the solutions to non linear continuity equations and their discretized numerical approximations on cartesian grids when advected by a rough force field this allows us to not only recover the known optimal regularity for linear transport equations but also to obtain the convergence of a wide range of numerical schemes our proof is based on novel commutator estimates quantifying and extending to the non linear case the classical commutator approach of the theory of renormalized solutions

compactness for nonlinear continuity equations abstract we prove compactness and hence existence for solutions to a class of nonlinear transport equations the corresponding models combine the features of linear transport equations and scalar conservation laws we introduce a new method which gives quantitative compactness estimates compatible with both frameworks

on the uniqueness of solutions to continuity equations abstract we obtain sufficient conditions for the uniqueness of solutions to the cauchy problem for the continuity equation in classes of measures that need not be absolutely continuous

adaptive gpc based on laguerre filters modelling abstract a state space model based on laguerre filters is proposed to design an adaptive generalized predictive controller gpc for stable plants a sufficient stability condition is derived to study the convergence and stability of the proposed controller in case of a plant model match robustness conditions are also given for the case of plant model mismatch the results are applicable to gpc schemes based on moving average models in the presence of plant model mismatch illustrative examples show the robustness of the controller and compare it with other gpc versions

new restriction estimates for the 3 d paraboloid over finite fields abstract we improve the range of exponents for the restriction problem for the 3 d paraboloid over finite fields the key new ingredient is a variant of the bourgain katz tao finite field incidence theorem derived from sum product estimates in prime order fields we give an explicit relationship between the exponent in this incidence theorem and restriction estimates for the paraboloid

extension theorems for paraboloids in the finite field setting in this paper we study the l p l r boundedness of the extension operators associated with paraboloids in mathbb f q d where mathbb f q is a finite field of q elements in even dimensions d 4 we estimate the number of additive quadruples in the subset e of the paraboloids that is the number of quadruples x y z w in e 4 with x y z w as a result in higher even dimensions we obtain the sharp range of exponents p for which the extension operator is bounded independently of q from l p to l 4 in the case when 1 is a square number in mathbb f q using the sharp l p l 4 result we improve upon the range of exponents r for which the l 2 l r estimate holds obtained by mockenhaupt and tao duke math 121 35 74 2004 in even dimensions d 4 in addition assuming that 1 is not a square number in mathbb f q we extend their work done in three dimension to specific odd dimensions d 7 the discrete fourier analytic machinery and gauss sum estimates make an important role in the proof

stability and convergence analyses of an adaptive gpc based on state space modeling a generalized predictive controller gpc is derived based on a general state space model the equivalence of the predictive control problem to a perturbation problem is revealed in the case of a small perturbation the closed loop poles are calculated with high accuracy for the case of a general perturbation an upper bound on the permissible perturbation norm is derived assuming an open loop stable system both the plant model match and plant model mismatch cases are analysed the controller is proven to be robust and an adaptive implementation is motivated for open loop stable systems the convergence and stability of the control scheme are ensured by proper tuning of the control weight and prediction horizon the results are applicable to a wide range of predictive controllers the main contribution of this paper is not a new control algorithm but new techniques to analyse the gpc as well as new stability and convergence results

paper an input output view of robustness in adaptive control the stability and robustness properties of adaptive control systems are examined using input output stability theory i e passivity and small gain theory a generic adaptive error system is developed based on the concept of a tuned system an ideal converged nonadaptive closed loop system using this error system with passivity theory gives conditions for global stability where only boundedness in norm is required on the external inputs e g disturbance reference and initial conditions small gain theory is used to develop local stability results where the magnitudes of the external inputs are restricted in the global case a particular system operator not the plant is required to be strictly passive a condition which is unlikely to hold in actual use due to unmodeled dynamics the local results however are not so restricted and allow for unmodeled dynamics in this latter case an estimate of the stability margin is given under a persistent excitation condition

brief paper on the stabilizability condition in indirect adaptive control many recent studies on overall stability of adaptive control systems require among other conditions that the identified model be stabilizable at least asymptotically in fact the usual identification methods do not guarantee this property in the passive approach without exciting extra signals or if the model order is subject to over estimation in order to solve this problem it is shown in this paper that certain classical identification algorithms can be easily modified in order to ensure the above stabilizability condition without affecting the other properties

critical non sobolev regularity for continuity equations with rough velocity fields abstract we present a divergence free vector field in the sobolev space h 1 such that the flow associated to the field does not belong to any sobolev space the vector field is deterministic but constructed as the realization of a random field combining simple elements this construction illustrates the optimality of recent quantitative regularity estimates as it gives a straightforward example of a well posed flow which has nevertheless only very weak regularity

well posedness of the transport equation by stochastic perturbation we consider the linear transport equation with a globally holder continuous and bounded vector field with an integrability condition on the divergence while uniqueness may fail for the deterministic pde we prove that a multiplicative stochastic perturbation of brownian type is enough to render the equation well posed this seems to be the first explicit example of a pde of fluid dynamics that becomes well posed under the influence of a multiplicative noise the key tool is a differentiable stochastic flow constructed and analyzed by means of a special transformation of the drift of ito tanaka type

lagrangian flows and the one dimensional peano phenomenon for odes we consider the one dimensional ordinary differential equation with a vector field which is merely continuous and nonnegative and satisfies a condition on the amount of zeros although it is classically known that this problem lacks uniqueness of classical trajectories we show that there is uniqueness for the so called regular lagrangian flow by now usual notion of flow in nonsmooth situations as well as uniqueness of distributional solutions for the associated continuity equation the proof relies on a space reparametrization argument around the zeros of the vector field

conjecture and improved extension theorems for paraboloids in the finite field setting we study the extension estimates for paraboloids in d dimensional vector spaces over finite fields mathbb f q with q elements we use the connection between l 2 based restriction estimates and l p rightarrow l r extension estimates for paraboloids as a consequence we improve the l 2 rightarrow l r extension results obtained by lewko and lewko proc am math soc 140 2013 2028 2012 in even dimensions d ge 6 and odd dimensions d 4 ell 3 for ell in mathbb n our results extend the consequences for 3 d paraboloids due to lewko adv math 270 1 457 479 2015 to higher dimensions we also clarifies conjectures on finite field extension problems for paraboloids

deterministic adaptive control based on laguerre series representation the behaviour of adaptive controllers in the presence of unmodelled dynamics and the need for reduced a priori information have led us to abandon the usual arma transfer function representation for a representation by an orthonormal series the appeal of our new approach is that it eliminates the need for assumptions about the plant order and the time delay the plant is modelled by an orthonormal laguerre network put in state space form a simple predictive control law is proposed an explicit deterministic adaptive controller is then designed simulations show that it is easy to use able to handle non minimum phase plants and more robust than the conventional model based approach although we chose laguerre functions other orthonormal functions may be used we have already tested some with success

paper unification of discrete time explicit model reference adaptive control designs various discrete time explicit mrac designs based on stability considerations are presented from a unified point of view a general structure and a design procedure are given which contain as particular cases various possible designs some of these designs have been already given in the literature but some are new the general structure considered removes some of the restrictions encountered in previous designs and solves the important problem of independent specification of tracking and regulation objectives the proof of boundedness of the control variables is also given simulations are included in order to evaluate the various designs in a deterministic environment

ordinary differential equations transport theory and sobolev spaces we obtain some new existence uniqueness and stability results for ordinary differential equations with coefficients in sobolev spaces these results are deduced from corresponding results on linear transport equations which are analyzed by the method of renormalized solutions

weak version of restriction estimates for spheres and paraboloids in finite fields abstract we study l p l r restriction estimates for algebraic varieties in d dimensional vector spaces over finite fields unlike the euclidean case if the dimension d is even then it is conjectured that the l 2 d 2 d 3 l 2 stein tomas restriction result can be improved to the l 2 d 4 d 4 l 2 estimate for both spheres and paraboloids in finite fields in this paper we show that the conjectured l p l 2 restriction estimate holds in the specific case when test functions under consideration are restricted to d coordinate functions or homogeneous functions of degree zero to deduce our result we use the connection between the restriction phenomena for our varieties in d dimensions and those for homogeneous varieties in d 1 dimensions

stability result with application to adaptive control we prove a relatively simple stability result for a particular system of time varying difference equations we then display its usefulness in analysing the stability and convergence of discrete time adaptive control systems in particular we apply our result to demonstrate the stability and convergence of two disparate adaptive control schemes

design of a deadbeat adaptive tracking system using an orthogonal decomposition technique a matrix pseudo in version lemma is presented to derive recursive algorithms for computing the best approximate solution to the sequential estimation problem a deadbeat convergence property of the algorithms is shown and is applied to the design of a deadbeat adaptive tracking system simulations illustrate perfect tracking of the system output to the periodic reference trajectory in finite steps

restriction and kakeya phenomena for finite fields the restriction and kakeya problems in euclidean space have received much attention in the last few decades and they are related to many problems in harmonic analysis partial differential equations pdes and number theory in this paper we initiate the study of these problems on finite fields in many cases the euclidean arguments carry over easily to the finite setting and are in fact somewhat cleaner but there are some new phenomena in the finite case which deserve closer study

finite field restriction estimates for the paraboloid in high even dimensions abstract we prove that the finite field fourier extension operator for the paraboloid is bounded from l 2 l r for r 2 d 4 d in even dimensions d 8 which is the optimal l 2 estimate for d 6 we obtain the optimal range r 2 d 4 d 8 3 apart from the endpoint for d 4 we improve the prior range of r 16 5 3 2 to r 28 9 3 111 compared to the conjectured range of r 3 the key new ingredient is improved additive energy estimates for subsets of the paraboloid

sobolev estimates for solutions of the transport equation and ode flows associated to non lipschitz drifts it is known after jabin j differ equ 260 5 4739 4757 2016 and alberti et al ann pde 5 1 9 2019 that ode flows and solutions of the transport equation associated to sobolev vector fields do not propagate sobolev regularity even of fractional order in this paper we improve the result at clop and jylha j differ equ 266 8 4544 4567 2019 and show that some kind of propagation of sobolev regularity happens as soon as the gradient of the drift is exponentially integrable we provide sharp sobolev estimates and new examples as an application of our main theorem we generalize a regularity result for the 2d euler equation obtained by bahouri and chemin in bahouri and chemin arch ration mech anal 127 2 159 181 1994

stochastic lagrangian particle approach to fractal navier stokes equations in this article we study the fractal navier stokes equations by using the stochastic lagrangian particle path approach in constantin and iyer comm pure appl math lxi 330 345 2008 more precisely a stochastic representation for the fractal navier stokes equations is given in terms of stochastic differential equations driven by levy processes based on this representation a self contained proof for the existence of a local unique solution for the fractal navier stokes equation with initial data in mathbb w 1 p is provided and in the case of two dimensions or large viscosity the existence of global solutions is also obtained in order to obtain the global existence in any dimensions for large viscosity the gradient estimates for levy processes with time dependent and discontinuous drifts are proved

global existence of weak solutions to the fene dumbbell model of polymeric flows systems coupling fluids and polymers are of great interest in many branches of sciences one of the most classical models to describe them is the fene finite extensible nonlinear elastic dumbbell model we prove global existence of weak solutions to the fene dumbbell model of polymeric flows the main difficulty is the passage to the limit in a nonlinear term that has no obvious compactness properties the proof uses many weak convergence techniques in particular it is based on the control of the propagation of strong convergence of some well chosen quantity by studying a transport equation for its defect measure in addition this quantity controls a rescaled defect measure of the gradient of the velocity

an adaptive lq controller for non minimum phase systems abstract the paper studied the problem of the adaptive tracking of a reference signal by the output of any deterministic discrete siso linear system with unknown parameters an adaptive control scheme based on the lq control solution is proposed the stability of the resulting adaptively controlled system and conditions for a good tracking are studied several simulations illustrate the proposal

robust adaptive control conditions for global stability an input output approach is presented for analyzing the global stability and robustness properties of adaptive controllers to unmodeled dynamics the concept of a tuned system is introduced i e the control system that could be obtained if the plant were known comparing the adaptive system to the tuned system results in the development of a generic adaptive error system passivity theory is used to derive conditions which guarantee global stability of the error system associated with the adaptive controller and ensure boundedness of the adaptive gains specific bounds are presented for certain significant signals in the control systems limitations of these global results are discussed particularly the requirement that a certain operator be strictly positive real spr a condition that is unlikely to hold due to unmodeled dynamics

lagrangian solutions to the 2d euler system with l1 vorticity and infinite energy we consider solutions to the two dimensional incompressible euler system with only integrable vorticity thus with possibly locally infinite energy with such regularity we use the recently developed theory of lagrangian flows associated to vector fields with gradient given by a singular integral in order to define lagrangian solutions for which the vorticity is transported by the flow we prove strong stability of these solutions via strong convergence of the flow under the only assumption of l1 weak convergence of the initial vorticity the existence of lagrangian solutions to the euler system follows for arbitrary l1 vorticity relations with previously known notions of solutions are established

estimates and regularity results for the diperna lions flow in this paper we derive new simple estimates for ordinary differential equations with sobolev coefficients these estimates not only allow to recover some old and recent results in a simple direct way but they also have some new interesting corollaries

plane waves in nonlinear electromagnetism on enonce un theoreme d existence globale pour le probleme de cauchy des ondes planes les donnees initiales sont supposees mesurables et bornees mais pas necessairement petites