on measure solutions of the boltzmann equation part i moment production and stability estimates the spatially homogeneous boltzmann equation with hard potentials is considered for measure valued initial data having finite mass and energy we prove the existence of emph weak measure solutions with and without angular cutoff on the collision kernel the proof in particular makes use of an approximation argument based on the mehler transform moment production estimates in the usual form and in the exponential form are obtained for these solutions finally for the grad angular cutoff we also establish uniqueness and strong stability estimate on these solutions

spectral lagrangian methods for collisional models of non equilibrium statistical states we propose a new spectral lagrangian based deterministic solver for the non linear boltzmann transport equation bte in d dimensions for variable hard sphere vhs collision kernels with conservative or non conservative binary interactions the method is based on symmetries of the fourier transform of the collision integral where the complexity in its computation is reduced to a separate integral over the unit sphere s d 1 the conservation of moments is enforced by lagrangian constraints the resulting scheme implemented in free space is very versatile and adjusts in a very simple manner to several cases that involve energy dissipation due to local micro reversibility inelastic interactions or elastic models of slowing down process our simulations are benchmarked with available exact self similar solutions exact moment equations and analytical estimates for the homogeneous boltzmann equation both for elastic and inelastic vhs interactions benchmarking of the simulations involves the selection of a time self similar rescaling of the numerical distribution function which is performed using the continuous spectrum of the equation for maxwell molecules as studied first in bobylev et al a v bobylev c cercignani g toscani proof of an asymptotic property of self similar solutions of the boltzmann equation for granular materials journal of statistical physics 111 2003 403 417 and generalized to a wide range of related models in bobylev et al a v bobylev c cercignani i m gamba on the self similar asymptotics for generalized non linear kinetic maxwell models communication in mathematical physics in press url the method also produces accurate results in the case of inelastic diffusive boltzmann equations for hard spheres inelastic collisions under thermal bath where overpopulated non gaussian exponential tails have been conjectured in computations by stochastic methods t v noije m ernst velocity distributions in homogeneously cooling and heated granular fluids granular matter 1 57 1998 m h ernst r brito scaling solutions of inelastic boltzmann equations with over populated high energy tails journal of statistical physics 109 2002 407 432 s j moon m d shattuck j swift velocity distributions and correlations in homogeneously heated granular media physical review e 64 2001 031303 i m gamba s rjasanow w wagner direct simulation of the uniformly heated granular boltzmann equation mathematical and computer modelling 42 2005 683 700 and rigorously proven in gamba et al i m gamba v panferov c villani on the boltzmann equation for diffusively excited granular media communications in mathematical physics 246 2004 503 541 39 and a v bobylev i m gamba v panferov moment inequalities and high energy tails for boltzmann equations with inelastic interactions journal of statistical physics 116 2004 1651 1682

global boundedness of moments of solutions of the boltzmann equation for forces of infinite range this paper proves the global boundedness in time of higher moments to a weak solution of the non linear space homogeneous boltzmann equation for inverse kth power forces with k 5 in the course of the proof a new collisional estimate of v s v s v s v s is obtained where v v are the velocities before and v v are the velocities after a binary collision

self similar asymptotic behavior for the solutions of a linear coagulation equation abstract in this paper we consider the long time asymptotics of a linear version of the smoluchowski equation which describes the evolution of a tagged particle moving in a random distribution of fixed particles the volumes v of these particles are independently distributed according to a probability distribution which decays asymptotically as a power law v the validity of the equation has been rigorously proved in 22 taking as a starting point a particle model and for values of the exponent 3 but the model can be expected to be valid on heuristic grounds for 5 3 the resulting equation is a non local linear degenerate parabolic equation the solutions of this equation display a rich structure of different asymptotic behaviors according to the different values of the exponent here we show that for 5 3 2 the linear smoluchowski equation is well posed and that there exists a unique self similar profile which is asymptotically stable

mathematics of granular materials this is a short and somewhat informal review on the most mathematical parts of the kinetic theory of granular media intended for physicists and for mathematicians outside the field

riccati equation solution of the minimum time output control problem by taking a geometrical approach and by using a type of orthogonality in constructing bases for the k th controllable subspaces a general solution is derived for the multivariable minimum time output control problem the solution is easier to implement than solutions derived by traditional approaches to output control and in fact it is shown that the optimal control sequence may be constructed from the solution to a specialized riccati equation this new approach thus relates geometrical approaches to the problem to the previously known result that output control is just a special case of the singular linear quadratic optimal control problem

the boltzmann equation for driven systems of inelastic soft spheres we study a generic class of inelastic soft sphere models with a binary collision rate g that depends on the relative velocity g this includes previously studied inelastic hard spheres 1 and inelastic maxwell molecules 0 we develop a new asymptotic method for analyzing large deviations from gaussian behavior for the velocity distribution function f c the framework is that of the spatially uniform nonlinear boltzmann equation and special emphasis is put on the situation where the system is driven by white noise depending on the value of exponent three different situations are reported for 2 it becomes an attractive fixed point with velocity distributions f c having stretched exponential behavior at large c the corresponding dominant behavior of f c is computed together with sub leading corrections in the marginally stable case 2 the high energy tail of f c is of power law type and the associated exponents are calculated our analytical predictions are confronted with monte carlo simulations with a remarkably good agreement

a new approach to inversion based feedforward control design for nonlinear systems the finite time transition between stationary setpoints of nonlinear siso systems is considered as a scenario for the presentation of a new design approach for inversion based feedforward control design techniques which are based on a stable system inversion result in input trajectories with pre and or post actuation intervals the presented approach treats the considered transition task as a two point boundary value problem bvp and yields causal feedforward trajectories which are constant outside the transition interval the main idea of this approach is to provide free parameters in the desired output trajectory to solve the bvp of the internal dynamics thereby a standard matlab function can be used for the numerical solution of the bvp feedforward control design techniques are illustrated by simulation results for a simple example

the structure of time optimal trajectories for single input systems in the plane the c nonsingular case for single input c infty systems in the plane in which the control enters linearly we prove if the system is suitably nondegenerate that the time optimal trajectories are finite concatenations of bang bang and singular arcs with local bounds on the number of switchings

on the inelastic enskog equation near vacuum the cauchy problem for the inelastic enskog equation is considered for near vacuum data the global existence of mild and classical solutions of the inelastic enskog equation is obtained then based on some new functionals we give the uniform l1 x v stability of solutions for the inelastic enskog equation

minimum time trajectory generation for constrained linear systems using flatness and b splines in this article we discuss minimum time trajectory generation for input and state constrained continuous time lti systems in the light of the notion of flatness and b splines parametrisation flat systems have the useful property that the input and the state trajectories can be completely characterised by the so called flat output we propose a splines parametrisation for the flat output and the corresponding parametrisations for the performance output the states and the inputs using this parametrisation the problem of minimum time constrained trajectory planning is cast into a feasibility search problem in the splines control point space in which the constraint region is characterised by a polytope a close approximation of the minimum time trajectory is obtained by systematically searching the end time that makes the constraint polytope to be minimally feasible

strict entropy production bounds and stability of the rate of convergence to equilibrium for the boltzmann equation we first consider the boltzmann equation with a collision kernel such that all kinematically possible collisions are run at equal rates this is the simplest boltzmann equation having the compressible euler equations as a scaling limit for it we prove a stability result for theh theorem which says that when the entropy production is small the solution of the spatially homogeneous boltzmann equation is necessarily close to equilibrium in the entropie sense and therefore strongl 1 sense we use this to prove that solutions to the spatially homogeneous boltzmann equation converge to equilibrium in the entropie sense with a rate of convergence which is uniform in the initial condition for all initial conditions belonging to certain natural regularity classes every initial condition with finite entropy andp th velocity moment for some p 2 belongs to such a class we then extend these results by a simple monotonicity argument to the case where the collision rate is uniformly bounded below which covers a wide class of slightly modified physical collision kernels these results are the basis of a study of the relation between scaling limits of solutions of the boltzmann equation and hydrodynamics which will be developed in subsequent papers the program is described here

generalized bang bang control for feedforward constrained regulation in the behavioral framework for continuous time linear scalar systems simple sufficient conditions for the solution of the minimum time rest to rest feedforward constrained control problem are provided the investigation of the time optimal input output pair reveals that the input or the output saturates on the assigned constraints at all times except for a set of zero measure the resulting optimal input is composed of sequences of bang bang functions and linear combinations of the modes associated to the zero dynamics this signal behavior constitutes a generalized bang bang control that can be fruitfully exploited for feedforward constrained regulation using discretization an arbitrarily good approximation of the optimal generalized bang bang control is found by solving a sequence of linear programming problems numerical examples are included

nonlinear minimum time control with pre and post actuation this article studies the time optimal output transition problem to change the system output from an initial value y t y for all time t 0 to a final value y t y for all time t t for invertible nonlinear systems the main contribution of the article is to show that the use of pre and post actuation input outside the transition interval i t 0 t can reduce the transition time t beyond the standard bang bang type inputs for optimal state transition the advantage of using pre and post actuation is demonstrated with an illustrative nonlinear example

noncausal inverses for linear systems suppose we are to guide a plant with a square transfer matrix through a series of way points this can be accomplished by output tracking of trajectories which are fitted together in a sufficiently smooth manner each segment of each output is prescribed in terms of a polynomial or sine function or the product thereof our technique is to compute 1 a noncausal particular solution based on the desired trajectory over each segment and the movement from one segment to the next and this particular solution consists of a particular control and a particular state trajectory and 2 a regulator to handle modeling errors disturbances etc we concentrate on the particular solution

a maxwellian lower bound for solutions to the boltzmann equation we prove that the solution of the spatially homogeneous boltzmann equation is bounded pointwise from below by a maxwellian i e a function of the formc1 exp c2v2 this holds for any initial data with bounded mass energy and entropy and for any positive timet t0 the constantsc1 andc2 depend on the mass energy and entropy of the initial data and ont0 0 only

l estimates for the space homogeneous boltzmann equation this paper studies the boundedness of solutionsf of the initial value problem for the space homogeneous boltzmann equation for inverse kth power forces whenk 5 and under angular cutoff the main result is that if the initial value isf0 0 with 1 2 0 el1 and 1 sf0e l for somes 2 then 1 s ftel fort 0 and ess tsup 1 s f t 5

the complexity of partial derivatives abstract let l denote the nonscalar complexity in k x 1 x n we prove l x 1 x n 3l using this we determine the complexity of single power sums single elementary symmetric functions the resultant and the discriminant as root functions up to order of magnitude also we linearly reduce matrix inversion to computing the determinant

lower bounds for special cases of syntactic multilinear abps abstract algebraic branching programs abps are standard models for computing polynomials syntactic multilinear abps smabps are restrictions of abps where every variable is allowed to occur at most once in every path from the source to the sink node proving super polynomial lower bounds for syntactic multilinear abps remains a challenging open question in algebraic complexity theory the current best lower bound for smabps is only quadratic in the number of variables 1 in this article we develop a new approach for proving syntactic multilinear branching program lower bounds convert the smabp into an equivalent multilinear formula with a super polynomial blow up in size and then exploit the structural limitations of the resulting formula to obtain an upper bound on the rank of partial derivative matrix of the polynomial computed by the smabp using this approach we prove exponential lower bounds for special cases of smabps namely sum of read once oblivious smabps multilinear r pass abps and set multilinear abps en route we also prove an exponential lower bound for a special class of syntactic multilinear arithmetic circuits using a similar approach

brief paper on the equivalence between stable inversion for nonminimum phase systems and reciprocal transfer functions defined by the two sided laplace transform in this paper the two sided laplace transform a classical but not very common mathematical tool is revived to express the stable inversion for linear nonminimum phase systems that was recently proposed from the viewpoint of state space representations it is demonstrated that those two different expressions for the stable inversion are mathematically equivalent simple examples are presented to illustrate the two sided laplace transform as a direct and intuitive approach to stable inversion the two sided laplace transform approach is also applied to the development of an iterative learning control for nonminimum phase systems that needs neither a precise inversion model nor fourier transform computations but instead requires only measuring the system response with time reversals

brief paper inversion of sampled data system approximates the continuous time counterpart in a noncausal framework although correspondence between the poles of a continuous time and sampled data system with a piecewise constant input is simple and desirable from the stability viewpoint the relationship between zeros is intricate inversion of a sampled data system is mostly unstable irrespective of the stability of the continuous time counterpart this makes it difficult to apply inversion based control techniques such as perfect tracking transient response shaping or iterative learning control to sampled data systems although recently developed noncausal inversion techniques help us to circumvent unboundedness of the inversion caused by unstable zeros whether the inversion of sampled data systems approximates the continuous time counterpart or not as the sample period is shortened is still to be determined this article gives a positive conclusion to this problem

moment inequalities for the boltzmann equation and applications to spatially homogeneous problems some inequalities for the boltzmann collision integral are proved these inequalities can be considered as a generalization of the well known povzner inequality the inequalities are used to obtain estimates of moments of the solution to the spatially homogeneous boltzmann equation for a wide class of intermolecular forces we obtain simple necessary and sufficient conditions on the potential for the uniform boundedness of all moments for potentials with compact support the following statement is proved if all moments of the initial distribution function are bounded by the corresponding moments of the maxwelliana exp bv2 then all moments of the solution are bounded by the corresponding moments of the other maxwelliana1 exp b1 t v2 for anyt 0 moreoverb t const for hard spheres an estimate for a collision frequency is also obtained

rate of convergence of the nanbu particle system for hard potentials and maxwell molecules we consider the numerically motivated nanbu stochastic particle system associated to the spatially homogeneous boltzmann equation for true hard potentials we establish a rate of propagation of chaos of the particle system to the unique solution of the boltzmann equation more precisely we estimate the expectation of the squared wasserstein distance with quadratic cost between the empirical measure of the particle system and the solution the rate we obtain is almost optimal as a function of the number of particles but is not uniform in time

contractive metrics for a boltzmann equation for granular gases diffusive equilibria we quantify the long time behavior of a system of partially inelastic particles in a stochastic thermostat by means of the contractivity of a suitable metric in the set of probability measures existence uniqueness boundedness of moments and regularity of a steady state are derived from this basic property the solutions of the kinetic model are proved to converge exponentially as t to this diffusive equilibrium in this distance metrizing the weak convergence of measures then we prove a uniform bound in time on sobolev norms of the solution provided the initial datum has a finite norm in the corresponding sobolev space these results are then combined using interpolation inequalities to obtain exponential convergence to the diffusive equilibrium in the strong l1 norm as well as various sobolev norms

boltzmann model for viscoelastic particles asymptotic behavior pointwise lower bounds and regularity we investigate the long time behavior of a system of viscoelastic particles modeled with the homogeneous boltzmann equation we prove the existence of a universal maxwellian intermediate asymptotic state with explicit rate of convergence towards it exponential lower pointwise bounds and propagation of regularity are also studied these results can be seen as a generalization of several classical results holding for the pseudo maxwellian and constant normal restitution models

multi linear formulas for permanent and determinant are of super polynomial size an arithmetic formula is multilinear if the polynomial computed by each of its subformulas is multilinear we prove that any multilinear arithmetic formula for the permanent or the determinant of an n n matrix is of size super polynomial in n previously super polynomial lower bounds were not known for any explicit function even for the special case of multilinear formulas of constant depth

stability regions of nonlinear autonomous dynamical systems a topological and dynamical characterization of the stability boundaries for a fairly large class of nonlinear autonomous dynamic systems is presented the stability boundary of a stable equilibrium point is shown to consist of the stable manifolds of all the equilibrium points and or closed orbits on the stability boundary several necessary and sufficient conditions are derived to determine whether a given equilibrium point or closed orbit is on the stability boundary a method for finding the stability region on the basis of these results is proposed the method when feasible will find the exact stability region rather than a subset of it as in the lyapunov theory approach several examples are given to illustrate the theoretical prediction

stable inversion for nonlinear nonminimum phase time varying systems we extend stable inversion to nonlinear time varying systems and study computational issues the technique is applicable to minimum phase as well as nonminimum phase systems the inversion technique is new even in the linear time varying case and relies on partitioning the dichotomic split of the linearized system dynamics into time varying stable and unstable submanifolds this dichotomic split is used to build time varying filters which are in turn the basis of a contraction used to find a bounded inverse input state trajectory finding the inverse input state trajectory allows the development of exact output tracking controllers the method is local to the time varying trajectory and requires that the internal dynamics vary slowly however the method represents a significant advance relative to presently available tracking controllers present techniques are restricted to time invariant nonlinear systems and in the general case track only asymptotically

inversion of nonlinear time varying systems a procedure for inverting nonlinear systems that presents some computational advantages when applied to flexible robot arms or more generally to mechanical structures is given the procedure is presented in a general setting for nonlinear time varying systems an iterative algorithm that computes a smooth controlled invariant time varying manifold the mathematical properties of which are illustrated and exploited is also proposed an application to a flexible two link manipulator illustrates the algorithm and its computational advantages

global existence of classical solutions to the inelastic vlasov poisson boltzmann system we study the global existence and uniqueness of classical solutions to the inelastic vlasov poisson boltzmann system for a soft potential in the near vacuum regime for the global existence of classical solutions we assume reasonable conditions on the restitution coefficient which represents the character of inelastic collisions we use the smallness of initial data and an algebraically decaying weight function in the spatial variable to control the self consistence force and collision operator

stable inversion for nonlinear systems abstract there have been two recent major developments in output tracking for nonlinear systems and our first main contribution is to relate these under appropriate assumptions we show that the bounded solution of the partial differential equation of isidori and byrnes for each trajectory of an exosystem must be given by an integral representation formula of devasia chen and paden under restrictive hypotheses devasia chen and paden develop a picard process that converges to the solution of the integral equation this solution to the integral equation is also a bounded solution to a dynamical equation driven by the desired outputs in aircraft applications our nonlinear systems are perturbations of pure feedback systems with outputs and we find a solution to the stable inversion problem i e finding bounded controls and bounded state trajectories in response to bounded output signals in two steps the first step ignores the perturbation error and computes the major part of the desired control and corresponding state trajectory the second step computes the remaining control and states by finding a noncausal and stable solution to an error driven dynamical equation in other words the method of devasia chen and paden is applied to an error system and not to the full system this two step procedure is the second main contribution of this paper throughout this paper it is assumed that our nonlinear systems have vector relative degree

stability convergence to self similarity and elastic limit for the boltzmann equation for inelastic hard spheres we consider the spatially homogeneous boltzmann equation for inelastic hard spheres in the framework of so called constant normal restitution coefficients alpha in 0 1 in the physical regime of a small inelasticity that is alpha in alpha 1 for some constructive alpha in 0 1 we prove uniqueness of the self similar profile for given values of the restitution coefficient alpha in alpha 1 the mass and the momentum therefore we deduce the uniqueness of the self similar solution up to a time translation

on the estimation of asymptotic stability regions state of the art and new proposals this paper deals with the problem of the estimation of regions of asymptotic stability for continuous autonomous nonlinear systems the first part of the work provides a comprehensive survey of the existing methods and of their applications in engineering fields in the second part certain topological considerations are first developed and the trajectory reversing method is then presented together with a theorem on which it is based in the final part several examples of application are reported showing the efficiency of the proposed technique for low order second and third systems

cooling process for inelastic boltzmann equations for hard spheres part i the cauchy problem we develop the cauchy theory of the spatially homogeneous inelastic boltzmann equation for hard spheres for a general form of collision rate which includes in particular variable restitution coefficients depending on the kinetic energy and the relative velocity as well as the sticky particles model we prove local in time non concentration estimates in orlicz spaces from which we deduce weak stability and existence theorem strong stability together with uniqueness and instantaneous appearance of exponential moments are proved under additional smoothness assumption on the initial datum for a restricted class of collision rates concerning the long time behaviour we give conditions for the cooling process to occur or not in finite time

optimal local well posedness theory for the kinetic wave equation abstract we prove local existence and uniqueness results for the space homogeneous 4 wave kinetic equation in wave turbulence theory we consider collision operators defined by radial but general dispersion relations satisfying suitable bounds and we prove two local well posedness theorems in nearly critical weighted spaces

conservation of energy entropy identity and local stability for the spatially homogeneous boltzmann equation for nonsoft potential collision kernels with angular cutoff we prove that under the initial condition f0 v 1 v 2 logf0 v l1 r3 the classical formal entropy identity holds for all nonnegative solutions of the spatially homogeneous boltzmann equation in the class l 0 l12 r3 c1 0 l1 r3 where l1s r3 f f v 1 v 2 s 2 l1 r3 and in this class the nonincrease of energy always implies the conservation of energy and therefore the solutions obtained all conserve energy moreover for hard potentials and the hard sphere model a local stability result for conservative solutions i e satisfying the conservation of mass momentum and energy is obtained as an application of the local stability a sufficient and necessary condition on the initial data f0 such that the conservative solutions f belong to l1loc 0 l12 r3 is also given

on the h function and convergence towards equilibrium for a space homogeneous molecular density this paper contains the following results about space homogeneous molecular densities 1 let f j in l 1 bf r 3 j in bf n be a sequence of velocity distributions with bounded mass energy and h function which converges weakly towards f in l 1 then by convexity it is proved that the h function for f exists and satisfies h leqq lim inf j to infty hf i 2 for a velocity distribution f evolving in time not necessarily governed by the boltzmann equation and conserving mass momentum and energy the convergence of the h function towards the h value of the corresponding maxwellian is proved to imply strong convergence in l 1 of f towards the maxwellian 3 certain solutions f to a weak space homogeneous boltzmann equation with forces of infinite range obtained as weak limits of solutions in the cut off case are considered by 1 these solutions have h functions bounded by the initial value hf 0 the solutions f are proved to converge weakly towards the maxwellian

a comparison principle for state constrained differential inequalities and its application to time optimal control in this paper we present a comparison principle that characterizes the maximal solutions of state constrained differential inequalities in terms of solutions of certain differential equations with discontinuous right hand sides for the sake of completeness we show through some set valued analysis that the differential equations determining the maximal solutions have the unique solutions in the carathe spl acute odory sense in spite of discontinuity of their right hand sides we apply our comparison principle to the explicit characterization of the solution to a time optimal control problem for a class of state constrained second order systems which includes the dynamic equations of robotic manipulators with geometric path constraints as well as single degree of freedom mechanical systems with friction specifically we show that the time optimal trajectory is uniquely determined by two curves that can be constructed by solving two scalar ordinary differential equations with continuous right hand sides hence the time optimal trajectory can be found in a computationally efficient way through the direct use of the well known euler or runge kutta methods another interesting feature is that our method to solve the time optimal control problem works even when there exist an infinite number of switching points finally some simulation results using a two degrees of freedom dof robotic manipulator are presented to demonstrate the practical use of our complete characterization of the time optimal solution

nonlinear inversion based output tracking an inversion procedure is introduced for nonlinear systems which constructs a bounded input trajectory in the preimage of a desired output trajectory in the case of minimum phase systems the trajectory produced agrees with that generated by hirschorn s inverse dynamic system however the preimage trajectory is noncausal rather than unstable in the nonminimum phase case in addition the analysis leads to a simple geometric connection between the unstable manifold of the system zero dynamics and noncausality in the nonminimum phase case with the addition of stabilizing feedback to the preimage trajectory asymptotically exact output tracking is achieved tracking is demonstrated with a numerical example and compared to the well known byrnes isidori regulator rather than solving a partial differential equation to construct a regulator the inverse is calculated using a picard like interaction when preactuation is not possible noncausal inverse trajectories can be truncated resulting in the tracking error transients found in other control schemes

entropy production estimates for boltzmann equations with physically realistic collision kernels we establish strict entropy production bounds for the boltzmann equation with the hard sphere collision kernel using these entropy production bounds we prove results asserting that the rate at which strongl 1 convergence to equilibrium occurs is uniform in wide classes of initial data this extends our previous results in this direction which applied only to a very special collision kernel moreover the present results provide computable lower bounds compactness arguments are entirely avoided the uniformity is an important ingredient in our study of scaling limits of solutions of the non spatially homogeneous boltzmann equation and is the main focus of this paper however the results obtained here provide the only framework known to us in which one can obtain computable estimates on the time it takes a solution of the spatially homogeneous boltzmann equation with initial data far from equilibrium to reach any given small strongl 1 neighborhood of equilibrium

continuous time linear predictive control and flatness a module theoretic setting with examples a general flatness based framework for linear continuous time predictive control is presented the mathematical setting which is valid for multivariable systems is provided by the algebraic theory of modules where a controllable system corresponds to a finitely generated free module over a principal ideal ring any basis of this free module is a flat output which yields an easy calculation of the predicted trajectory this formalism permits one to handle non minimum phase systems system constraints and to deal with additive perturbations three concrete case studies namely a dc motor a flexible system and a cement mill are analysed and simulations are given these examples are written in such a way that any reader who is not familiar with module theory may nevertheless grasp the proposed control strategy

on the transfer of energy towards infinity in the theory of weak turbulence for the nonlinear schrodinger equation we study the mathematical properties of a kinetic equation derived in escobedo and velazquez arxiv 1305 5746v1 math ph which describes the long time behaviour of solutions to the weak turbulence equation associated to the cubic nonlinear schrodinger equation in particular we give a precise definition of weak solutions and prove global existence of solutions for all initial data with finite mass we also prove that any nontrivial initial datum yields the instantaneous onset of a condensate by which we mean that for any nontrivial solution the mass of the origin is strictly positive for any positive time furthermore we show that the only stationary solutions with finite total measure are dirac masses at the origin we finally construct solutions with finite energy where the energy is transferred to infinity in a self similar manner

deadbeat control of linear multivariable generalized state space systems the classic idea of deadbeat control is extended to linear multivariable discrete time generalized state space systems using algebraic methods the asymptotic properties of the linear quadratic regulator theory are used to obtain the classes of deadbeat controllers using stabilizing full semistate feedback the solution is constructed from a cheap control problem both semistate and output deadbeat control laws are considered the main design criteria are to drive the semistate and or outputs of the system to zero in minimum time and that the closed loop system be internally stable unique properties of these types of control laws are discussed for semistate deadbeat control all the dynamic poles including the ones at infinity are moved to the origin whereas for output deadbeat some of the finite transmission zeros are canceled numerically reliable algorithms are developed to solve both problems

global existence of solutions of the boltzmann equation for bose einstein particles with anisotropic initial data abstract in this paper we prove the global in time existence and uniqueness of solutions of the spatially homogeneous boltzmann equation for bose einstein particles for the hard sphere model for bounded anisotropic initial data the main idea of our proof is as follows we first establish an intermediate equation which is closely related to the original equation and is relatively easily proven to have global in time and unique solutions then we use the multi step iterations of the collision gain operator to obtain a desired uniform l bound for solutions of the intermediate equation so that if an initial datum is sufficiently small relative to the inverse of the planck constant which belongs to the case of very high temperature then the corresponding solution of the intermediate equation becomes the solution of the original equation

a behavioral approach to inversion based control abstract a new simplified behavior theory is proposed to address inversion based control for linear nonminimum phase siso systems the chosen space of signals is the set of piecewise c functions and input output pairs as weak solutions satisfy a differential integral equation with additional smoothness requirements a related key result is the output input or inverse representation of the behavior set that leads to the solution of a general stable inversion problem where polynomially unbounded noncausal desired outputs are allowed it is shown that this problem has a solution if and only if the smoothness degree of the desired output is greater than or equal to the system relative degree minus one when this straightforward condition is satisfied a closed form expression provides the inverse input then an analysis on preaction and postaction control follows two examples are included showing the relevance of output signal design in control applications

optimal output transitions for linear systems this article addresses the optimal minimum input energy output transition problem for linear systems the goal is to transfer the output from an initial value y t email protected for all time t t f previous methods solve this output transition problem by transforming it into a state transition problem the initial and final states x t i x t f respectively are chosen and a minimum energy state to state transition problem is solved however the choice of the initial and final states can be ad hoc and the resulting output transition cost input energy may not be minimal the contribution of this article is the solution of the optimal output transition problem an example system with elastic dynamics is studied to illustrate the proposed method simulation results are presented that show substantial reduction of transition costs with the use of the proposed method when compared to the use of minimum energy state to state transitions

constrained reachability and trajectory generation for flat systems we consider the problem of trajectory generation for constrained differentially flat systems based on the topological properties of the set of admissible steady state values of a flat output we derive conditions which allow for an a priori verification of the feasibility of constrained set point changes we propose to utilize this relation to generate feasible reference trajectories to this end we suggest to split the trajectory generation problem into two stages a the planning of geometric reference paths in the flat output space combined with b an assignment of a dynamic reference motion to these paths this assignment is based on a reduced optimal control problem the unique feature of the approach is that due to the specific construction of the reference paths the optimal control problem to be solved is guaranteed to be feasible to illustrate our results we consider a van de vusse reactor as an example

kac s program in kinetic theory this paper is devoted to the study of propagation of chaos and mean field limits for systems of indistinguishable particles undergoing collision processes the prime examples we will consider are the many particle jump processes of kac and mckean kac in proceedings of the third berkeley symposium on mathematical statistics and probability 1954 1955 vol iii pp 171 197 1956 mckean in j comb theory 2 358 382 1967 giving rise to the boltzmann equation we solve the conjecture raised by kac proceedings of the third berkeley symposium on mathematical statistics and probability 1954 1955 vol iii pp 171 197 1956 motivating his program on the rigorous connection between the long time behavior of a collisional many particle system and the one of its mean field limit for bounded as well as unbounded collision rates

a quantum boltzmann equation for haldane statistics and hard forces the space homogeneous initial value problem the paper considers equations of boltzmann type for haldane exclusion statistics existence and some basic properties of the solutions are studied for the space homogeneous initial value problem with hard forces and angular cut off the approach uses strong l 1 compactness some of the technical estimates are based on l decay properties and the control of the filling factor on range estimates for the solutions

upper maxwellian bounds for the spatially homogeneous boltzmann equation for the spatially homogeneous boltzmann equation with cutoff hard potentials it is shown that solutions remain bounded from above uniformly in time by a maxwellian distribution provided the initial data have a maxwellian upper bound the main technique is based on a comparison principle that uses a certain dissipative property of the linear boltzmann equation implications of the technique to propagation of upper maxwellian bounds in the spatially inhomogeneous case are discussed

brief optimal noncausal set point regulation of scalar systems this paper presents a novel system inversion approach to the set point regulation of linear scalar minimum phase systems this approach uses closed form expressions of cause effect pairs that make possible an arbitrarily smooth transition between two given output set points the cause effect pairs are based on the introduced concept of transition polynomials simple optimization procedures are then proposed to solve the relevant optimal output synthesis and optimal input synthesis the potentiality of the method is highlighted by the examples presented namely a synthesis of elevator velocity profiles and the performance improving of a pid controller

universality of cluster dynamics we have studied the kinetics of cluster formation for dynamical systems of dimensions up to n 8 interacting through elastic collisions or coalescence these systems could serve as possible models for gas kinetics polymerization and self assembly in the case of elastic collisions we found that the cluster size probability distribution undergoes a phase transition at a critical time which can be predicted from the average time between collisions this enables forecasting of rare events based on limited statistical sampling of the collision dynamics over short time windows the analysis was extended to l p normed spaces p 1 to allow for some amount of interpenetration or volume exclusion the results for the elastic collisions are consistent with previously published low dimensional results in that a power law is observed for the empirical cluster size distribution at the critical time we found that the same power law also exists for all dimensions n 2 8 two dimensional l p norms and even for coalescing collisions in two dimensions this broad universality in behavior may be indicative of a more fundamental process governing the growth of clusters

optimal inversion based control for the set point regulation of nonminimum phase uncertain scalar systems we propose an optimal inversion based control for the set point constrained regulation of nonminimum phase uncertain scalar systems this approach is based on a new control architecture where the main purpose of the feedback is to reduce the sensitivity to parametric plant uncertainties permitting in such a way the effective use of a feedforward action determined via a stable dynamic inversion essential constituents of the architecture are a parameterized family of transition polynomials to shape ideal output transfers and a parameterized controller ensuring the internal model principle the methodology is then centered on the optimal combined design of the feedback controller and of the inversion based command signal in order to minimize the worst case settling time subject to an amplitude constraint on the control variable and to arbitrarily assigned overshoot and undershoot bounds finally an approximate or suboptimal solution to the resulting nonlinear optimization problem can be obtained by genetic algorithms a worked example highlights the effectiveness of the overall methodology

flatness and defect of non linear systems introductory theory and examples we introduce flat systems which are equivalent to linear ones via a special type of feedback called endogenous their physical properties are subsumed by a linearizing output and they might be regarded as providing another nonlinear extension of kalman s controllability the distance to flatness is measured by a non negative integer the defect we utilize differential algebra where flatness and defect are best defined without distinguishing between input state output and other variables many realistic classes of examples are flat we treat two popular ones the crane and the car with n trailers the motion planning of which is obtained via elementary properties of plane curves the three non flat examples the simple double and variable length pendulums are borrowed from non linear physics a high frequency control strategy is proposed such that the averaged systems become flat

using stable input output inversion for minimum time feedforward constrained regulation of scalar systems the paper approaches the feedforward minimum time smooth control of non minimum phase linear scalar systems for set point regulation the aim is to synthesize a bounded input subject to non saturating constraints on the input and its derivatives till a prespecified arbitrary order the provided solution relies on a stable input output dynamic inversion technique and an ad hoc parameterized family of output transition functions the synthesized minimum time input that exhibits pre and post actuation is determined by means of easily implementable closed form expressions examples are given to illustrate the methodology

sums of read once formulas how many summands are necessary abstract an arithmetic read once formula rof is a formula circuit of fan out 1 over where each variable labels at most one leaf every multilinear polynomial can be expressed as the sum of possibly exponentially many rofs in this work we prove for certain multilinear polynomials a tight lower bound on the number of summands in such an expression

time splitting error in dsmc schemes for the spatially homogeneous inelastic boltzmann equation the paper is concerned with the numerical treatment of the uniformly heated inelastic boltzmann equation by the direct simulation monte carlo dsmc method this technique is presently the most widely used numerical method in kinetic theory we consider three modifications of the dsmc method and study them with respect to their efficiency and convergence properties convergence is investigated with respect to both the number of particles and the time step the main issue of interest is the time step discretization error due to various splitting strategies a scheme based on the strang splitting strategy is shown to be of second order with respect to time step while there is only first order for the commonly used euler splitting scheme on the other hand a no splitting scheme based on appropriate markov jump processes does not produce any time step error it is established in numerical examples that the no splitting scheme is about two orders of magnitude more efficient than the euler splitting scheme the strang splitting scheme reaches almost the same level of efficiency as that of the no splitting scheme since the deterministic time step error vanishes sufficiently fast

estimates for the boltzmann collision operator via radial symmetry and fourier transform abstract we extend the l p theory of the boltzmann collision operator by using classical techniques based in the carleman representation and fourier analysis allied to new ideas that exploit the radial symmetry of this operator we are then able to greatly simplify existent technical proofs in this theory extend the range and obtain explicit sharp constants in some convolution like inequalities for the gain part of the boltzmann collision operator

dead beat controllability of polynomial systems symbolic computation approaches state and output dead beat controllability tests for a very large class of polynomial systems with rational coefficients may be based on the quantifier elimination by partial cylindrical algebraic decomposition qepcad symbolic computation program the method is unified for a very large class of systems and can handle one or two sided control constraints families of minimum time state output dead beat controllers are obtained the computational complexity of the test is prohibitive for general polynomial systems but by constraining the structure of the system we may beat the curse of complexity a computationally less expensive algebraic test for output dead beat controllability for a class of odd polynomial systems is presented necessary and sufficient conditions are given they are still very difficult to check therefore a number of easier to check sufficient conditions are also provided the latter are based on the grobner basis method and qepcad it is shown on a subclass of odd polynomial systems how it is possible to further reduce the computational complexity by exploiting the structure of the system

convolution inequalities for the boltzmann collision operator we study integrability properties of a general version of the boltzmann collision operator for hard and soft potentials in n dimensions a reformulation of the collisional integrals allows us to write the weak form of the collision operator as a weighted convolution where the weight is given by an operator invariant under rotations using a symmetrization technique in l p we prove a young s inequality for hard potentials which is sharp for maxwell molecules in the l 2 case further we find a new hardy littlewood sobolev type of inequality for boltzmann collision integrals with soft potentials the same method extends to radially symmetric non increasing potentials that lie in some l s weak or l s the method we use resembles a brascamp lieb and luttinger approach for multilinear weighted convolution inequalities and follows a weak formulation setting consequently it is closely connected to the classical analysis of young and hardy littlewood sobolev inequalities in all cases the inequality constants are explicitly given by formulas depending on integrability conditions of the angular cross section in the spirit of grad cut off as an additional application of the technique we also obtain estimates with exponential weights for hard potentials in both conservative and dissipative interactions

on the cauchy problem for the inelastic boltzmann equation with external force in this paper the cauchy problem for the inelastic boltzmann equation with external force is considered for near vacuum data under the assumptions on the bicharacteristic generated by external force which can be arbitrarily large we prove the global existence of mild solution for initial data small enough with respect to the sup norm with exponential weight by using the contraction mapping theorem furthermore we prove the uniform l1 stability of the mild solution following from the exponential decay estimate and the gronwall s inequality for the case of soft potentials

regularity and long time behavior of the boltzmann equation for granular gases abstract this paper investigates regularity of solutions of the boltzmann equation with dissipative collisions in a thermal bath in the case of pseudo maxwellian approximation we prove that for any initial datum f 0 in the set of probability density with zero bulk velocity and finite temperature the unique solution of the equation satisfies f t h r 3 for all t 0 furthermore for any t 0 0 and s 0 the h s norm of f t is bounded for t t 0 as a consequence the exponential convergence to the unique steady state is also established under the same initial condition

time optimal swing up of the planar pendulum this paper presents qualitative results on the global structure of the time optimal trajectories of the planar pendulum on a cart this mechanical system is a benchmark to test nonlinear control methods and various papers addressed the problem of computing time optimal open loop controls relying on the theory of optimal synthesis we provide a discontinuous feedback giving optimal solutions for any initial data the approach is that of geometric control theory

boltzmann equation for granular media with thermal force in a weakly inhomogeneous setting in this paper we consider the spatially inhomogeneous diffusively driven inelastic boltzmann equation in different cases the restitution coefficient can be constant or can depend on the impact velocity which is a more physically relevant case including in particular the case of viscoelastic hard spheres in the weak thermalization regime i e when the diffusion parameter is sufficiently small we prove existence of global solutions considering both the close to equilibrium and the weakly inhomogeneous regimes we also study the long time behavior of these solutions and prove a convergence to equilibrium with an exponential rate the basis of the proof is the study of the linearized equation we obtain a new result on it we prove existence of a spectral gap in weighted stretched exponential and polynomial sobolev spaces more precisely there is a one dimensional eigenvalue which is negative and which tends to 0 when the diffusion parameter tends to 0 to do that we take advantage of the recent paper cite gmm where the spatially inhomogeneous equation for elastic hard spheres has been considered as far as the case of a constant coefficient is concerned the present paper improves similar results obtained in cite mm2 in a spatially homogeneous framework concerning the case of a non constant coefficient this kind of results is new and we use results on steady states of the linearized equation from cite al3

on the boltzmann equation for diffusively excited granular media we study the boltzmann equation for a space homogeneous gas of inelastic hard spheres with a diffusive term representing a random background forcing under the assumption that the initial datum is a nonnegative l 2 n function with bounded mass and kinetic energy second moment we prove the existence of a solution to this model which instantaneously becomes smooth and rapidly decaying under a weak additional assumption of bounded third moment the solution is shown to be unique we also establish the existence but not uniqueness of a stationary solution in addition we show that the high velocity tails of both the stationary and time dependent particle distribution functions are overpopulated with respect to the maxwellian distribution as conjectured by previous authors and we prove pointwise lower estimates for the solutions

a b spline based pseudo inversion approach for constrained optimal output transition the contribution of this paper is to propose a new method for the continuous time optimal output transition problem under constraints on the control effort the new method situates in the recently

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

accurate numerical methods for the collisional motion of heated granular flows in this paper we extend the spectral method developed in l pareschi b perthame a fourier spectral method for homogeneous boltzmann equations trans theo stat phys 25 1996 369 383 l pareschi g russo numerical solution of the boltzmann equation i spectrally accurate approximation of the collision operator siam j numer anal 37 2000 1217 1245 to the case of the inelastic boltzmann equation describing the collisional motion of a granular gas with and without a heating source the schemes are based on a fourier representation of the equation in the velocity space and provide a very accurate description of the time evolution of the distribution function several numerical results in dimension one to three show the efficiency and accuracy of the proposed algorithms some mathematical and physical conjectures are also addressed with the aid of the numerical simulations

nontrivial velocity distributions in inelastic gases we study spatially homogeneous inelastic gases using the boltzmann equation we consider uniform collision rates and obtain analytical results valid for arbitrary spatial dimension d and arbitrary dissipation coefficient in the unforced case we find that the velocity distribution decays algebraically p v t v for sufficiently large velocities the exponent d exhibits nontrivial dependence on the spatial dimension and the dissipation coefficient

entropy dissipation and moment production for the boltzmann equation leth f m flog f m dv be the relative entropy off and the maxwellian with the same mass momentum and energy and denote the corresponding entropy dissipation term in the boltzmann equation byd f q f f logf dv an example is presented which shows that d f h f m can be arbitrarily small this example is a sequence of isotropic functions and the estimates are very explicitly given by a simple formula ford which holds for such functions the paper also gives a simplified proof of the so called povzner inequality which is a geometric inequality for the magnitudes of the velocities before and after an elastic collision that inequality is then used to prove that f v v s dt c t wheref is the solution of the spatially homogeneous boltzmann equation herec t is an explicitly given function dependings and the mass energy and entropy of the initial data

multi objective optimal trajectory design and tracking with non periodic tracking transition switching for non minimum phase linear systems abstractin this paper the problem of trajectory design and tracking of non periodic tracking transition switching for non minimum phase linear systems is considered such a problem exists in various applications where the output trajectory consists of application dependent tracking sessions and to be designed transition sessions the challenge arises when multiple control objectives are considered including the smooth transition from one output tracking session to the next one without large oscillations during the transition smooth tracking transition switching without inducing pre and or post switching oscillations input energy minimisation without saturation under amplitude constraint and furthermore minimisation of the overall transition time the proposed approach extends the previous work that attained smooth output transition and smooth tracking transition switching to further achieve amplitude constrained input energy minimisation and transition time minimisation first the constrained inp

regularity theory for the spatially homogeneous boltzmann equation with cut off we develop the regularity theory of the spatially homogeneous boltzmann equation with cut off and hard potentials for instance hard spheres by i revisiting the lp theory to obtain constructive bounds ii establishing propagation of smoothness and singularities iii obtaining estimates on the decay of the singularities of the initial datum our proofs are based on a detailed study of the regularity of the gain operator an application to the long time behavior is presented

on spatially homogeneous solutions of a modified boltzmann equation for fermi dirac particles the paper considers a modified spatially homogeneous boltzmann equation for fermi dirac particles bfd we prove that for the bfd equation there are only two classes of equilibria the first ones are fermi dirac distributions the second ones are characteristic functions of the euclidean balls and they can be simply classified in terms of temperatures t 2 5tf and t 2 5tf where tf denotes the fermi temperature in general we show that the l bound 0 f 1 e derived from the equation for solutions implies the temperature inequality t 2 5tf and if t 2 5tf then f trend towards fermi dirac distributions if t 2 5tf then f are the second equilibria in order to study the long time behavior we also prove the conservation of energy and the entropy identity and establish the moment production estimates for hard potentials

the inelastic enskog equation with external force in this paper the cauchy problem for the inelastic enskog equation with external force is considered for near vacuum data under the assumptions on the bi characteristic we proved the global existence of mild solution

global l p properties for the spatially homogeneous boltzmann equation this paper studies the lp behavior for 1 p of solutions of the nonlinear spatially homogeneous boltzmann equation for a class of collision kernels including inverse kth power forces with k 5 and angular cut off the following topics are treated differentiability in lp together with global boundedness in time for lp moments that exist initially translation continuity in lp uniformly in time and strong convergence to equilibrium