about the regularizing properties of the non cut off kac equation we prove in this work that under suitable assumptions the solution of the spatially homogeneous non cut off kac equation or of the spatially homogeneous non cut off 2d boltzmann equation with maxwellian molecules in the radial case becomes very regular with respect to the velocity variable as soon as the time is strictly positive

consistent linearization and finite element implementation of an incrementally objective canonical form return mapping algorithm for large deformation inelasticity the present paper focuses on the consistent linearization and finite element implementation of an incrementally objective canonical form return mapping algorithm a general and modular algorithmic setting suited for almost any rate constitutive equations is presented where the finite deformation consistent tangent modulus is obtained as a by product of the integration algorithm numerical examples illustrate the good performance of the proposed formulation especially for large deformation increments with noteworthy superimposed rotation where the consistent formulation converges quadratically in a reasonable number of iteration

robust exact differentiation via sliding mode technique abstract the main problem in differentiator design is to combine differentiation exactness with robustness in respect to possible measurement errors and input noises the proposed differentiator provides for proportionality of the maximal differentiation error to the square root of the maximal deviation of the measured input signal from the base signal such an order of the differentiation error is shown to be the best possible one when the only information known on the base signal is an upper bound for lipschitz s constant of the derivative

the bgk model with external confining potential existence long time behaviour and time periodic maxwellian equilibria we study global existence and long time behaviour for the inhomogeneous nonlinear bgk model for the boltzmann equation with an external confining potential for an initial datum f0 0 with bounded mass entropy and total energy we prove existence and strong convergence in l1 to a maxwellian equilibrium state by compactness arguments and multipliers techniques of particular interest is the case with an isotropic harmonic potential in which boltzmann himself found infinitely many time periodic maxwellian steady states this behaviour is shared with the boltzmann equation and other kinetic models for all these systems we study the multistability of the time periodic maxwellians and provide necessary conditions on f0 to identify the equilibrium state both in l1 and in lyapunov sense under further assumptions on f these conditions become also sufficient for the identification of the equilibrium in l1

mini max integral sliding mode control for multimodel linear uncertain systems an original linear time varying system with matched and unmatched disturbances and uncertainties is replaced by a finite set of dynamic models such that each one describes a particular uncertain case including exact realizations of possible dynamic equations as well as external unmatched bounded disturbances such a tradeoff between an original uncertain linear time varying dynamic system and a corresponding higher order multimodel system containing only matched uncertainties leads to a linear multi model system with known unmatched bounded disturbances and unknown matched disturbances as well each model from a given finite set is characterized by a quadratic performance index the developed minimax integral sliding mode control strategy gives an optimal minimax linear quadratic lq control with additional integral sliding mode term the design of this controller is reduced to a solution of an equivalent mini max lq problem that corresponds to the weighted performance indices with weights from a finite dimensional simplex the additional integral sliding mode controller part completely dismisses the influence of matched uncertainties from the initial time instant two numerical examples illustrate this study

chattering free full order sliding mode control in conventional sliding mode control systems the sliding mode motion is of reduced order two main problems hindering the application of the sliding mode control are the singularity in terminal sliding mode control systems and the chattering in both the conventional linear sliding mode and the terminal sliding mode control systems this paper proposes a chattering free full order terminal sliding mode control scheme since the derivatives of terms with fractional powers do not appear in the control law the control singularities are avoided a continuous control strategy is developed to achieve the chattering free sliding mode control during the ideal sliding mode motion the systems behave as a desirable full order dynamics rather than a desirable reduced order dynamics a systematic design method of full order sliding mode control for nonlinear systems is presented which allows both the chattering and singularity problems to be resolved simulations validate the proposed chattering free full order sliding mode control

on frequency domain criterion of finite time convergence of second order sliding mode control algorithms transient processes in the systems controlled by the twisting second order sliding mode sosm control algorithm and certain generic sosm given by the describing function are analyzed in the frequency domain the analysis is based on the approximate describing function method the relationship between the frequency response nyquist plot of the plant the shape of the negative reciprocal describing function of the controller and the transient process convergence rate is investigated a simple criterion of the existence of finite time convergence is proposed it is shown that the convergence rate in a system controlled by a sosm controller depends on the angle between the high frequency asymptote of the nyquist plot of the plant and the low amplitude asymptote of the negative reciprocal of the describing function of the controller which is named the phase deficit

on the finite time stabilization of uncertain nonlinear systems with relative degree three this technical note discusses a finite time stabilization problem in a nonlinear uncertain environment the main result of the present note is presented making reference to a triple integrator affected by bounded uncertainties and subject to ldquohardrdquo measurement nonlinearities such that the sign of the state variables is the only reliable information available for feedback we propose a discontinuous control scheme guaranteeing the practical finite time annihilation of the three state variables constructive proof and computer simulations as well as guidelines for practical implementations are provided throughout this technical note

on the trend to global equilibrium in spatially inhomogeneous entropy dissipating systems the linear fokker planck equation we study the long time behavior of kinetic equations in which transport and spatial confinement in an exterior potential or in a box are associated with a degenerate collision operator acting only in the velocity variable we expose a general method based on logarithmic sobolev inequalities and the entropy to overcome the well known problem due to the degeneracy in the position variable of the existence of infinitely many local equilibria this method requires that the solution be somewhat smooth in this paper we apply it to the linear fokker planck equation and prove decay to equilibrium faster than o t 1 for all 0 2001 john wiley sons inc

robust output tracking control of nonlinear mimo systems via sliding mode technique abstract the robust output tracking control problem of general nonlinear multi input multi output mimo systems is discussed the robustness against parameter uncertainties and unknown disturbances is considered a second order sliding mode control smc technique is used to establish the desired tracking input output i o linearization relative degree minimum phase and matching condition concepts are reviewed some earlier smc strategies which are restricted to the systems in canonical form are extended to a much broader class of nonlinear dynamics it is also shown that for unperturbed dynamics the sliding phase of the smc applications have a direct correspondence to the i o linearization operations interesting parametric flexibilities emanate within the formation of the second order smc designating the s dynamics and the error dynamics segments as frequency domain filters however a critical impasse is posed in the off line selections of the design parameters a set of example cases is presented for a spacecraft attitude control problem these examples manifest that the proposed control strategy is tunable to a desired response despite the disturbances and uncertainties

the boltzmann equation with large amplitude initial data in bounded domains abstract the paper is devoted to constructing the global solutions around global maxwellians to the initial boundary value problem on the boltzmann equation in general bounded domains with isothermal diffuse reflection boundaries we allow a class of non negative initial data which have arbitrary large amplitude and even contain vacuum the result shows that the oscillation of solutions away from global maxwellians becomes small after some positive time provided that they are initially close to each other in l 2 this yields the disappearance of any initial vacuum and the exponential convergence of large amplitude solutions to equilibrium in large time the isothermal diffuse reflection boundary condition plays a vital role in the analysis the most key ingredients in our strategy of the proof include i l x v 2 l x l v 1 l x v estimates along a bootstrap argument ii pointwise estimates on the upper bound of the gain term by the product of l norm and l 2 norm iii an iterative procedure on the nonlinear term

a novel higher order sliding mode control scheme a higher order sliding mode control algorithm is proposed for a class of uncertain multi input multi output nonlinear systems this problem can be viewed as the flnite time stabilization of a higher order input output dynamic system with bounded uncertainties the developed control scheme is based on geometric homogeneity and sliding mode control the proposed procedure provides explicit conditions on the controller parameters and guarantees robustness against uncertainties an illustrative example of a hovercraft vessel control demonstrates the advantages of the strategy

sliding order and sliding accuracy in sliding mode control the synthesis of a control algorithm that stirs a nonlinear system to a given manifold and keeps it within this constraint is considered usually what is called sliding mode is employed in such synthesis this sliding mode is characterized in practice by a high frequency switching of the control it turns out that the deviation of the system from its prescribed constraints sliding accuracy is proportional to the switching time delay a new class of sliding modes and algorithms is presented and the concept of sliding mode order is introduced these algorithms feature a bounded control continuously depending on time with discontinuities only in the control derivative it is also shown that the sliding accuracy is proportional to the square of the switching time delay

analysis of closed loop performance and frequency domain design of compensating filters for sliding mode control systems for a sliding mode sm system containing parasitic dynamics a model of chattering and a model of the averaged motions of the order equal to the original order of the system are obtained via the use of the locus of a perturbed relay system lprs method the latter model proves that the averaged motions in an sm system differ from those in the reduced order model in particular such phenomena as the nonideal disturbance rejection and the nonideal input tracking are observed it is shown that the non reduced order model transforms into the reduced one when the parasitic dynamics are removed yet the chattering and the averaged motion being the two components of the motion in the sm system belong to two different frequency ranges spectra as a result a frequency characteristic shaping aimed at achieving the desired frequency of chattering and enhancing the closed loop performance becomes possible this opportunity is explored in this paper examples that illustrate the proposed approach are provided

on multi input chattering free second order sliding mode control a solution to the problem of eliminating the chattering effect which is always associated with practical implementations of variable structure control is presented with reference to a class of uncertain multi input nonlinear systems the solution procedure relies on the application of an original control approach capable of enforcing a second order sliding mode i e a sliding regime on a surface s x t 0 in the system state space with s spl dot x t identically equal to zero a regime enforced by a control signal depending on s x t but directly acting only on s spl uml x t such an approach in its original formulation only applies to single input nonlinear systems with particular types of uncertainties in the present paper its validity is extended to multi input nonlinear systems characterized by uncertainties of more general nature covering a wide class of real processes

universal dynamics for the defocusing logarithmic schrodinger equation we consider the nonlinear schrodinger equation with a logarithmic nonlinearity in a dispersive regime we show that the presence of the nonlinearity affects the large time behavior of the solution the dispersion is faster than usual by a logarithmic factor in time and the positive sobolev norms of the solution grow logarithmically in time moreover after rescaling in space by the dispersion rate the modulus of the solution converges to a universal gaussian profile these properties are suggested by explicit computations in the case of gaussian initial data and remain when an extra power like nonlinearity is present in the equation one of the key steps of the proof consists in using the madelung transform to reduce the equation to a variant of the isothermal compressible euler equation whose large time behavior turns out to be governed by a parabolic equation involving a fokker planck operator

nonholonomic motion planning steering using sinusoids methods for steering systems with nonholonomic c onstraints between arbitrary configurations are investigated suboptimal trajectories are derived for systems that are not in canonical form systems in which it takes more than one level of bracketing to achieve controllability are considered the trajectories use sinusoids at integrally related frequencies to achieve motion at a given bracketing level a class of systems that can be steered using sinusoids claimed systems is defined conditions under which a class of two input systems can be converted into this form are given

on the trend to global equilibrium for spatially inhomogeneous kinetic systems the boltzmann equation as part of our study of convergence to equilibrium for spatially inhomogeneous kinetic equations started in 21 we derive estimates on the rate of convergence to equilibrium for solutions of the boltzmann equation like o t our results hold conditionally to some strong but natural estimates of smoothness decay at large velocities and strict positivity which at the moment have only been established in certain particular cases among the most important steps in our proof are 1 quantitative variants of boltzmann s h theorem as proven in 52 60 based on symmetry features hypercontractivity and information theoretical tools 2 a new quantitative version of the instability of the hydrodynamic description for non small knudsen number 3 some functional inequalities with geometrical content in particular the korn type inequality which we established in 22 and 4 the study of a system of coupled differential inequalities of second order by a treatment inspired from 21 we also briefly point out the particular role of conformal velocity fields when they are allowed by the geometry of the problem

technical communique analysis of a class of discrete time systems with power rule in this communique the properties of a class of discrete time dynamic systems with power rule are studied and comparisons with their continuous time counterparts as well as the linear discrete time systems are made it is shown that using power rule can be beneficial for improving dynamic behaviors of discrete time systems for example a power control law may stabilize a discrete time system which is not stabilizable by the linear control law

on weak and strong convergence to equilibrium for solutions to the linear boltzmann equation this paper considers the linear space inhomogeneous boltzmann equation for a distribution function in a bounded domain with general boundary conditions together with an external potential force the paper gives results on strong convergence to equilibrium whent for general initial data first in the cutoff case and then for infinite range collision forces the proofs are based on the properties of translation continuity and weak convergence to equilibrium to handle these problems generalh theorems concerning monotonicity in time of convex entropy functionals are presented furthermore the paper gives general results on collision invariants i e on functions satisfying detailed balance relations in a binary collision

the boltzmann equation with weakly inhomogeneous data in bounded domain this paper is concerned with the boltzmann equation with specular reflection boundary condition we construct a unique global solution and obtain its large time asymptotic behavior in the case that the initial data is close enough to a radially symmetric homogeneous datum the result extends the case of cauchy problem considered by arkeryd esposito pulvirenti 4 to the specular reflection boundary value problem in bounded domain

brief higher order sliding mode stabilization for a class of nonholonomic perturbed systems this paper deals with nonholonomic perturbed systems necessary and sufficient geometric conditions on the perturbation vector field are given in order to put the system into a perturbed one chained form two different sliding mode control strategies are then designed to robustly stabilize under some conditions this chained system one providing a practical stabilization the other performing a finite time convergence as a way of illustration simulations on the example of the unicycle type mobile robot are presented

existence and convergence to equilibrium of a kinetic model for cometary flows a kinetic equation with a relaxation time model for wave particle collisions is considered similarly to the bgk model of gas dynamics it involves a projection onto the set of equilibrium distributions nonlinearly dependent on moments of the distribution function an earlier existence result is extended to bounded domains with reflecting boundaries and to initial conditions permitting vacuum regions the long time behaviour is investigated convergence on compact time intervals shifted to infinity to the set of equilibrium solutions is proven the set of smooth equilibrium solutions is computed

analysis of second order sliding mode algorithms in the frequency domain a frequency domain analysis of the second order sliding mode algorithms particularly of the twisting algorithm is carried out in the frequency domain with the use of the describing function method and tsypkin s approach it is shown that in the presence of an actuator the transient process may converge to a periodic motion parameters of this periodic motion are analyzed a comparison of the periodic solutions in the systems with higher order sliding mode controllers and the oscillations that occur in classical sliding mode systems with actuators is done

universal single input single output siso sliding mode controllers with finite time convergence an universal controller is constructed formulated in input output terms only which causes the output of any uncertain smooth single input single output siso minimum phase dynamic system with known relative degree to vanish in finite time this allows exact tracking of arbitrary real time smooth signals only one parameter is to be adjusted since the approach is based on higher order finite time convergence sliding modes the control can be made arbitrarily smooth providing for the arbitrarily high tracking accuracy order with respect to the sampling step

asymptotic tracking of uncertain systems with continuous control using adaptive bounding this paper presents a robust adaptive control design method for a class of multiple input multiple output uncertain nonlinear systems in the presence of parametric and nonparametric uncertainties and bounded disturbances using the approximation properties of the unknown continuous nonlinearities and the adaptive bounding technique the developed controller achieves asymptotic convergence of the tracking error to zero while ensuring boundedness of parameter estimation errors the algorithm does not assume the knowledge of any bound on the unknown quantities in designing the controller it is based on an integral technique involving the filtered tracking error and produces a continuous control theoretical developments are illustrated via simulation results

guaranteed computation methods for compartmental in series models under uncertainty the pattern of some real phenomenon can be described by compartmental in series models nevertheless most of these processes are characterized by their variability which produces that the exact values of the model parameters are uncertain although they can be bounded by intervals the aim of this paper is to compute tight solution envelopes that guarantee the inclusion of all possible behaviors of such processes current methods such as monotonicity analysis enable us to obtain guaranteed solution envelopes however if the model includes non monotone compartments or parameters the computation of solution envelopes may produce a significant overestimation our proposal consists of performing a change of variables in which the output is unaltered and the model obtained is monotone with respect to the uncertain parameters the monotonicity of the new system allows us to compute the output bounds for the original system without overestimation these model transformations have been developed for linear and non linear systems furthermore if the conditions are not completely satisfied a novel method to compute tight solution envelopes is proposed the methods exposed in this paper have been applied to compute tight solution envelopes for two different models a linear system for glucose modeling and a non linear system for an epidemiological model

smooth solutions to the bgk equation and the es bgk equation with infinite energy abstract the bgk model and the es bgk model of the boltzmann equation are of great importance in the kinetic theory of rarefied gases for the cauchy problems with general initial data smooth solutions were obtained only in bounded domains for both models in this paper we establish the existence and uniqueness of smooth solutions in the whole space more important some smooth solutions can permit infiniteness of energy

on convergence time and disturbance rejection of super twisting control super twisting algorithm is one of the versions of high order sliding mode control the interest to this algorithm is explained by its attractive properties continuous control input finite convergence time disturbance rejection in this paper the upper bound of admissible unknown disturbances and low bound of the convergence time are found and shown that both the values can be achieved with any desired accuracy

higher order super twisting for perturbed chains of integrators in this paper we present a generalization of the super twisting algorithm for perturbed chains of integrators of arbitrary order this higher order super twisting host controller is homogeneous with respect to a family of dilations and is continuous it is built as a dynamic controller with respect to the state variable of the chain of integrators and the convergence analysis is performed by the use of a homogeneous strict lyapunov function which is explicitly constructed the effectiveness of the controller is finally illustrated with simulations for a chain of integrators of order four first pure then perturbed where we compare the performances of two host controllers

discontinuous integral action for arbitrary relative degree in sliding mode control abstract for uncertain minimum phase systems with arbitrary and well defined relative degree we propose the implementation of a sliding mode control through a homogeneous discontinuous integral action the output sliding variable will track exactly and in finite time any sufficiently smooth reference while rejecting smooth perturbations with an absolutely continuous control signal this controller generalizes the well known super twisting algorithm to arbitrary order continuous and homogeneous integral controllers are also proposed approximating the behaviour of the discontinuous controller smooth lyapunov functions are derived to show the convergence and robustness properties of all controllers the performance is illustrated by means of experimental results on a magnetic suspension system

the bang bang funnel controller for uncertain nonlinear systems with arbitrary relative degree the paper considers output tracking control of uncertain nonlinear systems with arbitrary known relative degree and known sign of the high frequency gain the tracking objective is formulated in terms of a time varying bound a funnel around a given reference signal the proposed controller is bang bang with two control values the controller switching logic handles arbitrarily high relative degree in an inductive manner with the help of auxiliary derivative funnels we formulate a set of feasibility assumptions under which the controller maintains the tracking error within the funnel furthermore we prove that under mild additional assumptions the considered system class satisfies these feasibility assumptions if the selected control values are sufficiently large in magnitude finally we study the effect of time delays in the feedback loop and we are able to show that also in this case the proposed bang bang funnel controller works under slightly adjusted feasibility assumptions

global existence of classical solutions to the fokker planck bgk equation a kinetic model of the fokker planck boltzmann equation is introduced by replacing the original boltzmann collision operator with the bhatnagar gross krook collision model bgk collision model this model equation which we call the fokker planck bgk equation has many physical features that the fokker planck boltzmann equation possesses we first establish an l existence result for this equation by which we construct the approximate solutions then by means of the regularizing effects of the linear fokker planck operator and lp estimates of local maxwellians we obtain some uniform estimates of the approximate solutions finally combining those estimates and regularizing effects we prove by a compactness argument that the equation has a global classical solution under rather general initial conditions

adaptive robust controller based on integral sliding mode concept abstractthis paper proposes for a class of uncertain nonlinear systems an adaptive controller based on adaptive second order sliding mode control and integral sliding mode control concepts the adaptation strategy solves the problem of gain tuning and has the advantage of chattering reduction moreover limited information about perturbation and uncertainties has to be known the control is composed of two parts an adaptive one whose objective is to reject the perturbation and system uncertainties whereas the second one is chosen such as the nominal part of the system is stabilised in zero to illustrate the effectiveness of the proposed approach an application on an academic example is shown with simulation results

on the cauchy problem for boltzmann equations global existence and weak stability we study the large data cauchy problem for boltzmann equations with general collision kernels we prove that sequences of solutions which satisfy only the physically natural a priori bounds converge weakly in l to a solution from this stability result we deduce global existence of a solution to the cauchy problem our method relies upon recent compactness results for velocity averages a new formulation of the boltzmann equation which involves nonlinear normalization and an analysis of subsolutions and supersolutions it allows us to overcome the lack of strong a priori estimates and define a meaningful collision operator for general configurations

some applications of the method of moments for the homogeneous boltzmann and kac equations using the method of moments we prove that any polynomial moment of the solution of the homogeneous boltzmann equation with hard potentials or hard spheres is bounded provided that a moment of order strictly higher than 2 exists initially we also give partial results of convergence towards the maxwellian equilibrium in the case of soft potentials finally exponential as well as maxwellian estimates are introduced for the kac equation

large time behavior of the broadwell model of a discrete velocity gas with specularly reflective boundary conditions this paper studies the behavior of the one dimensional broadwell model of a discrete three velocity gas on a bounded domain 0 x 1 with specularly reflective boundary condition at x 0 1 for smooth initial data we show that the initial boundary value problem possesses a unique smooth solution which tends as t to a free state consisting of traveling waves f 1e x ct f 2e x ct f 3e x where each f ie is 2 periodic the convergence is in the weak topology of an appropriate orlicz banach state space no smallness assumptions are made on the data

homogeneous high order sliding mode design a lyapunov approach abstract we provide a lyapunov based design of high order sliding mode hosm control of arbitrary order for a class of single input single output uncertain nonlinear systems first we provide a general method to design degree 0 homogeneous controllers based on a homogeneous control lyapunov function clf then for a perturbed chain of integrators we construct a smooth and homogeneous clf and use it to design a family of homogeneous and discontinuous control laws the controller drives in finite time the trajectories of the system to a sliding surface of arbitrary relative degree and keeps the trajectories on this manifold despite persistently acting matched perturbations so that a sliding mode is established we obtain a large and simply parametrized family of usual discontinuous and quasi continuous hosm controllers of nested and polynomial type that share the robustness and accuracy properties of the existing ones

trend to equilibrium of weak solutions of the boltzmann equation in a slab with diffusive boundary conditions recently r illner and the author proved that under a physically realistic truncation on the collision kernel the boltzmann equation in the one dimensional slab 0 1 with general diffusive boundary conditions at 0 and 1 has a global weak solution in the traditional sense here it is proved that when the maxwellians associated with the boundary conditions atx 0 andx 1 are the same maxwellianmw then the solution is uniformly bounded and tends tomw fort

a global existence theorem for the initial boundary value problem for the boltzmann equation when the boundaries are not isothermal we extend the existence theorem recently proved by hamdache for the initial boundary value problem for the nonlinear boltzmann equation in a vessel with isothermal boundaries to more general situations including the case when the boundaries are not isothermal in the latter case a cut off for large speeds is introduced in the collision term of the boltzmann equation

classification of equilibrium solutions of the cometary flow equation and explicit solutions of the euler equations for monatomic ideal gases the set of smooth equilibrium solutions of a kinetic model for cometary flows is split into equivalence classes according to similarity transformations for each equivalence class in the two and three dimensional cases a normal form is computed each such equilibrium solution gives rise to an explicit solution of the compressible euler equations for monatomic gases the set of these solutions is discussed with special emphasis on solutions containing vacuum regions

global weak solutions of the boltzmann equation in a slab with diffusive boundary conditions it is proved that under a physically realistic truncation of the collision kernel the boltzmann equation in the one dimensional slab 0 1 with general diffusive boundary conditions at 0 and 1 has a global weak solution in the traditional sense this solution satisfies the boundary conditions almost everywhere and has at worst exponentially growing total energy

exponential decay towards equilibrium for the inhomogeneous aizenman bak model the aizenman bak model for reacting polymers is considered for spatially inhomogeneous situations in which they diffuse in space with a non degenerate size dependent coefficient both the break up and the coalescence of polymers are taken into account with fragmentation and coagulation constant kernels we demonstrate that the entropy entropy dissipation method applies directly in this inhomogeneous setting giving not only the necessary basic a priori estimates to start the smoothness and size decay analysis in one dimension but also the exponential convergence towards global equilibria for constant diffusion coefficient in any spatial dimension or for non degenerate diffusion in dimension one we finally conclude by showing that solutions in the one dimensional case are immediately smooth in time and space while in size distribution solutions are decaying faster than any polynomial up to our knowledge this is the first result of explicit equilibration rates for spatially inhomogeneous coagulation fragmentation models

on the dynamical sliding mode control of nonlinear systems the consequences of the differential algebraic approach in the sliding mode control of nonlinear single input single output systems are reviewed in tutorial fashion input dependent sliding surfaces possibly including time derivatives of the input signal are shown to arise naturally from elementary differential algebraic results pertaining to the fliess s generalized controller canonical forms of nonlinear systems this class of switching surfaces generally leads to chattering free dynamically synthesized sliding regimes in which the highest time derivative of the input signal undergoes all the bang bang type discontinuities examples illustrating the obtained results are also included

on the initial boundary value problem for the boltzmann equation a theorem on the traces of the solutions of initial boundary value problems for the boltzmann equation is proved this result makes it possible to extend a recent theorem of existence proved by hamdache to more realistic situations

cheap suboptimal control of an integral sliding mode for uncertain systems with delays a cheap control problem for a system with state delays and matched uncertainties is studied first such a cheap control problem is considered for the nominal linear system associated with the original one a suboptimal state feedback composite control for this problem is constructed using a singular perturbation technique employing this composite control in the nominal system generates the suboptimal nominal trajectory secondly based on the composite control an integral sliding mode controller is designed for the original system providing its robust motion along the suboptimal nominal trajectory an illustrative numerical example is presented

lp solutions to the cauchy problem of the bgk equation the bgk model of the boltzmann equation plays an important role in the kinetic theory of rarefied gases some existence and uniqueness results of solutions to its cauchy problem were established for large initial data under various circumstances see for example perthame b global existence to the bgk model of boltzmann equation j differ equations 82 191 205 1989 in this paper by establishing some weighted lp estimates of the hydrodynamical quantities of a gas we prove the existence theorem of the lp solutions to the cauchy problem and establish the propagation properties of the lp moments for this kind of solutions

the quantum liouville bgk equation and the moment problem abstract this work is devoted to the analysis of the quantum liouville bgk equation this equation arises in the work of degond and ringhofer on the derivation of quantum hydrodynamical models from first principles their theory consists in transposing to the quantum setting the closure strategy by entropy minimization used for kinetic equations the starting point is the quantum liouville bgk equation where the collision term is defined via a so called quantum local equilibrium defined as a minimizer of the quantum free energy under a local density constraint we then address three related problems we prove new results about the regularity of these quantum equilibria we prove that the quantum liouville bgk equation admits a classical solution and we investigate the long time behavior of the solutions the core of the proofs is based on a fine analysis of the properties of the minimizers of the free energy

regularity of the moments of the solution of a transport equation let u u x v satisfy the transport equation u v xu f x rn v rnwhere f belongs to some space of type lp dx d v where is a positive bounded measure on rn we study the resulting regularity of the moment u x v d v in terms of fractional sobolev spaces for example counter examples are given in order to test the optimality of our results