a boundary value problem for the minimum time function a natural boundary value problem for the dynamic programming partial differential equation associated with the minimum time problem is proposed the minimum time function is shown to be the unique viscosity solution of this boundary value problem

nilpotent and high order approximations of vector field systems this paper is concerned with the study of structurally stable properties associated with systems described by one or several real analytic vector fields via approximating systems which retain inf

a general theorem on local controllability we prove a general sufficient condition for local controllability of a nonlinear system at an equilibrium point earlier results of brunovsky hermes jurdjevic crouch and byrnes sussmann and grossmann are shown to be particular cases of this result also a number of new sufficient conditions are obtained all these results follow from one simple general principle namely that local controllability follows whenever brackets with certain symmetries can be neutralized in a suitable way by writing them as linear combinations of brackets of a lower degree both the class of symmetries and the definition of degree can be chosen to suit the problem

the state constrained bilateral minimal time function abstract we generalize to the bilateral case that is with variable initial and end points the main results of nour and stern c nour r j stern regularity of the state constrained minimal time function nonlinear anal 66 1 2007 62 72 and stern r j stern characterization of the state constrained minimal time function siam j control optim 43 2004 697 707 where the regularity and hamilton jacobi characterization of the state constrained unilateral minimal time function were studied

nonlinear elliptic equations with singular boundary conditions and stochastic control with state constraints hamiltoniens sousquadratiques conditions aux limites infinies et donnees d explosion hamiltoniens superquadratiques formulation viscosite des conditions aux limites le probleme ergodique commande stochastique optimale avec contraintes d etat

on the large time behavior of solutions of hamilton jacobi equations associated with nonlinear boundary conditions in this article we study the large time behavior of solutions of first order hamilton jacobi equations set in a bounded domain with nonlinear neumann boundary conditions including the case of dynamical boundary conditions we establish general convergence results for viscosity solutions of these cauchy neumann problems by using two fairly different methods the first one relies only on partial differential equations methods which provides results even when the hamiltonians are not convex and the second one is an optimal control dynamical system approach named the weak kam approach which requires the convexity of hamiltonians and gives formulas for asymptotic solutions based on aubry mather sets

perron s method for hamilton jacobi equations on considere l existence des solutions d equations aux derivees partielles non lineaires scalaires d ordre 1 f x u du 0 dans ou est un sous ensemble ouvert de r n f r r n r est continue u r est l inconnue

asymptotic analysis for a closed processor sharing system with switching times normal usage in a closed processor sharing system with cpu switching times the probability distribution of the waiting time w p varepsilon t mathbb p w t varepsilon geqq 0 where varepsi

a processor shared queue that models switching times heavy usage asymptotics a closed processor sharing queueing model that corresponds to m terminals in series with a central processing unit cpu is considered under heavy usage conditions the effects of switching times between the cpu and the individual jobs are included by assuming that the effective service rate that the cpu provides is a decreasing function of the instantaneous number of jobs being processed using perturbation methods asymptotic expansions are constructed for the first two moments of a customer s response sojourn time conditioned on that customer s total required service from the cpu the asymptotic limit assumes that the number of terminals m is large m gg 1 that the rate at which the cpu provides service is fast o m and that the duration of a typical switching time is small compared to the time the cpu actually spends servicing the individual jobs extensive numerical comparisons show the quality of our asymptotic approximations

continuum theory of epitaxial crystal growth i we present various continuum limits to describe epitaxial thin film growth we consider a hierarchy of models which can take into account the diffusion of terrace adatoms attachment and detachment of edge adatoms vapor phase diffusion and the effect of multiple species the starting point is the burton cabrera frank type step flow model we have obtained partial differential equations in the form of a coupled system of diffusion equation for the adatom density and a hamilton jacobi equation for the film height function this is supplemented with appropriate boundary conditions at the continuum level to describe the growth at the peaks and valleys on the film the results here can be used in a macroscopic description of thin film growth

existence of c 1 critical subsolutions of the hamilton jacobi equation let m be a c second countable manifold without boundary we denote by tm the tangent bundle and by tm m the canonical projection a point in tm will be denoted by x v with x m and v tx m 1 x in the same way a point of the cotangent space t m will be denoted by x p with x m and p t x m a linear form on the vector space tx m we will suppose that g is a complete riemannian metric on m for v tx m the norm v is g v v 1 2 we will denote by the dual norm on t x m we will also use the notations v x for v tx m and p x for v t x m we will assume in the whole paper that h t m r is a function of class at least c2 which satisfies the following three conditions 1 uniform superlinearity for every k 0 there exists c k r such that

renormalisation on sierpinski type fractals presents a family of deterministic fractals which generalise the d dimensional sierpinski gaskets and establishes their order of ramification and their fractal and spectral dimensions random walks on these fractals are renormalisable and lead to rational not necessarily polynomial mapping

uniqueness of lower semicontinuous viscosity solutions for the minimum time problem we obtain the uniqueness of lower semicontinuous lsc viscosity solutions of the transformed minimum time problem assuming that they converge to zero on a reachable part of the target in appropriate directions we present a counter example which shows that the uniqueness does not hold without this convergence assumption it was shown by soravia that the uniqueness of lsc viscosity solutions having a subsolution property on the target holds in order to verify this subsolution property we show that the dynamic programming principle dpp holds inside for any lsc viscosity solutions in order to obtain the dpp we prepare appropriate approximate pdes derived through barles inf convolution and its variant

conditioned response time distribution for a large closed processor sharing system in very heavy usage the service of an interactive computer system is characterized by the random response waiting time perceived by users the system consists of a bank of user terminals each of which asynchronous

fractal sequences derived from the self similar extensions of the sierpinski gasket the well known sierpinski gasket has been generalised by the authors into the generalised pascal sierpinski gaskets gpsg of orders k l where both k and l are or 2 it has been shown here that families of self similar sequences can be derived from these extensions of the sierpinski gasket when k 2 and l is a prime number

aubry mather theory and periodic solutions of the forced burgers equation consider a hamiltonian system with hamiltonian of the form h x t p where h is convex in p and periodic in x a ndt and x2 r 1 it is well known that its smooth invariant curves correspond to smoothz 2 periodic solutions of the pde ut h x t u x 0 in this paper we establish a connection between the aubry mather theory of invariant sets of the hamiltonian system andz 2 periodic weak solutions of this pde by realizing the aubry mather sets as closed subsets of the graphs of these weak solutions we show that the complement of the aubry mather set on the graph can be viewed as a subset of the generalized unstable manifold of the aubry mather set defined in 2 24 the graph itself is a backward invariant set of the hamiltonian system the basic idea is to embed the globally minimizing orbits used in the aubry mather theory into the characteristic fields of the above pde this is done by making use of one and two sided minimizers a notion introduced in 12 and inspired by the work of morse on geodesics of type a 26 the asymptotic slope of the minimizers also known as the rotation number is given by the derivative of the homogenized hamiltonian defined in 21 as an application we prove that thez 2 periodic weak solution of the above pde with given irrational asymptotic slope is unique a similar connection also exists in multidimensional problems with the convex hamiltonian except that in higher dimensions two sided minimizers with a specified asymptotic slope may not exist c 1999 john wiley sons inc

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

large time asymptotics for one dimensional dirichlet problems for hamilton jacobi equations with noncoercive hamiltonians we study large time asymptotics for a class of noncoercive hamilton jacobi equations with dirichlet boundary condition in one space dimension we prove that the average growth rate of a solution is constant only in a subset of the whole domain and give the asymptotic profile in the subset we show that the large time behavior for noncoercive problems may depend on the space variable in general which is different from the usual results under the coercivity condition this work is an extension with more rigorous analysis of a recent paper by e yokoyama y giga and p rybka in which a growing crystal model is established and the asymptotic behavior described above is first discovered

asymptotic analysis of the waiting time distribution for a large closed processor sharing system the service of an interactive computer system is characterized by the random waiting or response time perceived by users the system consists of a bank of terminals each of which asynchronously alternates between thinking and waiting for service from a central processing unit which operates under the processor sharing discipline it is assumed that there are n 1 terminals where n is large and that rho np o 1 where p is the ratio of the mean required service time to the mean think time asymptotic approximations to the equilibrium distribution of the waiting time are derived for normal usage rho 1 singular perturbation techniques are used and the matching of the heavy usage approximation with both the normal and very heavy usage approximations is investigated numerical results are presented which indicate that the approximations are useful for n geqq 100

response time distribution for a processor sharing system we consider the response time for jobs in a processor sharing system with a poisson arrival process and exponentially distributed required service time i e an m m 1 ps queue the response time tilde w is the sum of the delay and the required service time we derive an integral representation for the equilibrium response time distribution pr tilde w t and evaluate this integral numerically for several values of the traffic intensity rho t in the heavy traffic case when rho is close to 1 for rho 1 alpha 0 t for 0 75 leqq rho leqq 0 99 the relative error of this approximation is less than 0 62 alpha 2 in the range for which 1 geqq pr tilde w t geqq 0 01

on the spatial fourier transforms of the pascal sierpinski gaskets it has been shown here that the fourier transform which can be obtained experimentally of an evolving fractal can be manipulated to yield the fractal dimension although only the pascal sierpinski gaskets have been considered here because of their generality it is expected that this technique can be utilized to identify other planar fractals as well

on cell problems for hamilton jacobi equations with non coercive hamiltonians and their application to homogenization problems abstract we study a cell problem arising in homogenization for a hamilton jacobi equation whose hamiltonian is not coercive we introduce a generalized notion of effective hamiltonians by approximating the equation and characterize the solvability of the cell problem in terms of the generalized effective hamiltonian under some sufficient conditions the result is applied to the associated homogenization problem we also show that homogenization for non coercive equations fails in general

phase transitions on fractals ii sierpinski gaskets for pt i see ibid vol 16 p 1267 1983 the authors construct and investigate a family of fractals which are generalisations of the sierpinski gaskets sgs to all euclidean dimensionalities these fractal lattices have a finite order of ramification and can be considered marginal between one dimensional and higher dimensional geometries physical models defined on them are exactly solvable the authors argue that short range spin models on the sg show no finite temperature phase transitions as examples they solve a few spin models and study the resistor network and percolation problems on these lattices

computing the effective hamiltonian using a variational approach a numerical method for homogenization of hamilton jacobi equations is presented and implemented as an l infty calculus of variations problem solutions are found by solving a nonlinear convex optimization problem the numerical method is shown to be convergent and error estimates are provided one and two dimensional examples are worked in detail comparing known results with the numerical ones and computing new examples the cases of nonstrictly convex hamiltonians and hamiltonians for which the cell problem has no solution are treated

hamilton jacobi equations with state constraints in the present paper we consider hamilton jacobi equations of the form h x u vu 0 x e q where q is a bounded open subset of rn h is a given continuous real valued function of x s p e q x r x rn and vu is the gradient of the unknown function u we are interested in particular solutions of the above equation which are required to be supersolutions in a suitable weak sense of the same equation up to the boundary of q this requirement plays the role of a boundary condition the main motivation for this kind of solution comes from deterministic optimal control and differential games problems with constraints on the state of the system as well from related questions in constrained geodesics

the minimum time function for semilinear evolutions we study the bellman equation for the minimum time problem associated with a semilinear evolution system from the point of view of contingent solutions for this purpose new contingent derivatives are defined which involve functions instead of vectors as directions our analysis is based on flow invariance results for appropriate differential inclusions

on nonlinear optimal control problems with state constraints this paper is concerned with an optimal control problem where the state is constrained to stay either in a smooth open set omega or in its closure overline omega under a higher order sufficient condition for the viability of omega and overline omega we prove that the optimal cost function v omega is the unique continuous constrained solution of the hamilton jacobi bellman equation furthermore we show that v omega coincides with the optimal cost function v overline omega on omega

large time behavior of solutions of hamilton jacobi equations with periodic boundary data abstract we study the large time behavior of viscosity solutions of hamilton jacobi equations with periodic boundary data on bounded domains we establish a result on convergence of viscosity solutions to state constraint asymptotic solutions or periodic asymptotic solutions depending on the sign of critical value as time goes to infinity

asymptotic expansion of the waiting time distribution of two models of a closed processor sharing system heavy usage the authors consider the probability distribution pr w t of the waiting time w of the waiting time win a closed processor sharing system with cpu switching times the waiting time is the time needed for a particular job called tagged job to be served by the cpu the switching time epsilon is the time spent by the cpu to transfer the control from one job to the next one in this paper the authors consider two different models of this processor sharing system in modeli a cpu discipline with switching time epsilon 0 and one class of users is considered whereas in model2 a cpu discipline with switching time epsilon 0 and two classes of users are considered for these models the authors derive an asymptotic formula for pr w t using matched asymptotic expansions that hold in the heavy usage region

neumann type boundary conditions for hamilton jacobi equations abstract in this paper we present a notion of viscosity solutions of hamilton jacobi equations for neumann type boundary conditions or more generally oblique derivative in particular we prove the existence uniqueness stability of such solutions and we show that the vanishing viscosity method yields such solutions next we check that value functions of control problems or differential games problems for reflected dynamical processes are solutions in that sense of the associated bellman or isaacs equations finally we consider the ergodic problems

the hamilton jacobi bellman equation for time optimal control in this paper several assertions concerning viscosity solutions of the hamilton jacobi bellman equation for the optimal control problem of steering a system to zero in minimal time are proved first two rather general uniqueness theorems are established asserting that any positive viscosity solution of the hjb equation must in fact agree with the minimal time function near zero if also a boundary condition introduced by bardi siam j control optim 27 1988 pp 776 785 is satisfied then the agreement is global additionally the holder continuity of any subsolution of the hjb equation is proved in the case where the related dynamics satisfy a hormander type hypothesis this last assertion amounts to a half derivative analogue of a theorem of crandall and lions traps amer math soc 277 1 983 pp 1 42 concerning lipschitz viscosity solutions

proximal analysis and the minimal time function under general hypotheses on the target set s and the dynamics of the system we show that the minimal time function ts cdot is a proximal solution to the hamilton jacobi equation uniqueness results are obtained with two different kinds of boundary conditions a new propagation result is proven and as an application we give necessary and sufficient conditions for ts cdot to be lipschitz continuous near s a petrov type modulus condition is also shown to be sufficient for continuity of ts cdot near s

optimal control with state space constraint i optimal control of piecewise deterministic processes with state space constraint is studied under appropriate assumptions it is shown that the optimal value function is the only viscosity solution on the open domain which is also a supersolution on the closed domain finally the uniform continuity of the value function is obtained under a condition on the deterministic drift

studies of the spectral dimension of the vicsek snowflake fractal by an exact real space renormalization group method the authors obtain the spectral dimension of the vicsek snowflake fractal the result seems to cast some doubt on the range of validity of the einstein relation on the deterministic fractal

on the sojourn time distribution in a finite capacity processor shared queue we consider a processor shared m m 1 queue that can accommodate atmost a finite number k of customers using singular perturbation techniques we construct asymptotic approximations to the distribution of acustomer s sojourn time we assume that k is large and treat several different cases of themodel parameters and also treat different time scales extensivenumerical comparisons are used to back up our asymptotic formulas author s abstract

regularity of the state constrained minimal time function abstract the regularity of the state constrained minimal time function is studied we generalize p wolenski y zhuang proximal analysis and the minimal time function siam j control optim 36 1998 1048 1072 theorem 6 1 in which wolenski and zhuang give necessary and sufficient conditions for lipschitz continuity of the unconstrained minimal time function and discuss certain ramifications

asymptotic expansions of moments of the waiting time in a shared processor of an interactive system an interactive computer system service is characterized by the random waiting or response time perceived by users this paper presents a novel solution to the problem of efficiently computing th

regularity results for the minimum time function with hormander vector fields abstract in a bounded domain of r n with boundary given by a smooth n 1 dimensional manifold we consider the homogeneous dirichlet problem for the eikonal equation associated with a family of smooth vector fields x 1 x n subject to hormander s bracket generating condition we investigate the regularity of the viscosity solution t of such problem due to the presence of characteristic boundary points singular trajectories may occur first we characterize these trajectories as the closed set of all points at which the solution loses point wise lipschitz continuity then we prove that the local lipschitz continuity of t the local semiconcavity of t and the absence of singular trajectories are equivalent properties finally we show that the last condition is satisfied whenever the characteristic set of x 1 x n is a symplectic manifold we apply our results to several examples

heavy traffic analysis of the sojourn time in tandem queues with overtaking a tandem jackson network with a processor sharing ps node followed by either another ps node or a first in first out fifo node is considered it is assumed that both nodes are in heavy traffic three term asymptotic approximations to the distribution of the total sojourn response time through the two nodes are computed the leading term in the asymptotic approximation corresponds to the heavy traffic limit in which the effects of overtaking disappear the correction terms in the approximations give a quantitative measure of the erects of overtaking which occurs due to the presence of the ps node s

critical spin wave dynamics for two dimensional sierpinski type fractal the dynamic critical exponent z for a system of nearest neighbour interacting heisenberg spins on a two dimensional sierpinski type fractal is obtained by applying a real space renormalisation group transformation of the equations of motion

long time behavior of solutions of hamilton jacobi equations with convex and coercive hamiltonians we investigate the long time behavior of viscosity solutions of hamilton jacobi equations in mathbb r n with convex and coercive hamiltonians and give three general criteria for the convergence of solutions to asymptotic solutions as time goes to infinity we apply the criteria to obtain more specific sufficient conditions for the convergence to asymptotic solutions and then examine them with examples we take a dynamical approach based on tools from weak kam theory such as extremal curves aubry sets and representation formulas for solutions for these investigations

a microscopic time scale approximation to the behavior of the local slope on the faceted surface under a nonuniformity in supersaturation abstract the morphological stability of a growing faceted crystal is discussed we argue that the interplay between nonuniformity in supersaturation on a growing facet and anisotropy of surface kinetics derived from the lateral motion of steps leads to a faceted instability qualitatively speaking as long as the nonuniformity in supersaturation on the facet is not too large it can be compensated for by a variation of step density along the facet and the faceted crystal can grow in a stable manner the problem can be modeled as a hamilton jacobi equation for height of the crystal surface the notion of a maximal stable region of a growing facet is introduced for microscopic time scale approximation of the original hamilton jacobi equation it is shown that the maximal stable region keeps its shape determined by profile of the surface supersaturation with constant growth rate by studying large time behavior of solution of macroscopic time scale approximation a quantitative criterion for the facet stability is given