global attractors for multivalued random dynamical systems generated by random differential inclusions with multiplicative noise in this paper we consider a stochastic differential inclusion with multiplicative noise it is shown that it generates a multivalued random dynamical system for which there also exists a global random attractor

kernel sections and uniform attractors of multi valued semiprocesses abstract we present the existence of kernel sections which are all compact invariant and pullback attracting of an infinite dimensional general multi valued process constructed by the set valued backward extension of multi valued semiprocesses moreover the structure of the uniform attractors of a family of multi valued semiprocesses and the uniform forward attraction of kernel sections of a family of general multi valued processes are investigated finally we explain our abstract results by considering the mixed wave systems with supercritical exponent and ordinary differential equations

typical performance of regular low density parity check codes over general symmetric channels typical performance of low density parity check ldpc codes over a general binary input output symmetric memoryless channel is investigated using methods of statistical mechanics relationship between the free energy in statistical mechanics approach and the mutual information used in the information theory literature is established within a general framework gallager and mackay neal codes are studied as specific examples of ldpc codes it is shown that basic properties of these codes known for particular channels including their potential to saturate shannon s bound hold for general symmetric channels the binary input additive white gaussian noise channel and the binary input laplace channel are considered as specific channel models

semiconcave solutions of partial differential inclusions abstract we obtain new results on the propagation of singularities for semiconcave solutions of partial differential inclusions these results will be used to study the behavior of singularities of the value function for a reflected control problem

on the multiplicity of equilibrium solutions to a nonlinear diffusion equation on a manifold arising in climatology abstract we analyze the sensitivity of a climatological model with respect to small changes in one of the distinguished parameters the solar constant we start by proving the stabilization of solutions of the evolution model when time tends to infinity later we study the stationary problem and obtain uniqueness or a multiplicity of solutions for different values of the solar constant

the application of dynamic programming to optimal inventory control this paper concerns a class of deterministic impulse control problems arising in inventory control a notable feature of the problem formulation is the presence of an end point constraint in consequence the value function may be discontinuous viability theory provides a characterization of the value function as the unique lower semicontinuous solution to a bensoussan lions type quasi variational inequality qvi suitably interpreted for nondifferentiable extended valued functions yet there are few examples in the literature of the use of this analytical machinery this paper provides such an example the example which concerns a problem for which the value function is neither everywhere finite valued nor continuously differentiable on the interior of its effective domain illustrates what is involved in calculating subdifferentials and checking satisfaction of qvi in a generalized sense this paper also provides a summary of the underlying theory and gathers in the appendix proofs of key results

on the dynamics of nonautonomous periodic general dynamical systems and differential inclusions abstract we establish the existence and stability results for periodic nonautonomous uniform forward attractors of periodic general dynamical systems set valued dynamical systems we also investigate the dynamical behavior of nonautonomous periodic differential inclusion x t f t x t on r m with only upper semi continuous right hand side by applying the abstract results firstly we show that if the system has a compact uniformly attracting set then it has a periodic nonautonomous uniform forward attractor a secondly we prove that a is robust with respect to both internal and external perturbations finally we apply the robustness result to discuss the effects of small time delays to asymptotic stability of the system

differential games with ergodic payoff we address a zero sum differential game with ergodic payoff we study this problem via the viscosity solutions of an associated hamilton jacobi isaacs equation under certain condition we establish the existence of a value and prove certain representation formulae

viability solutions to structured hamilton jacobi equations under constraints structured hamilton jacobi partial differential equations are hamilton jacobi equations where the time variable is replaced by a vector valued variable structuring the system it could be the time age pair hamilton jacobi mckendrick equations or candidates for initial or terminal conditions hamilton jacobi cournot equations among a manifold of examples here we define the concept of viability solution which always exists and can be computed by viability algorithms this is a constructive approach allowing us to derive from the tools of viability theory dealing with sets instead of functions some known as well as new properties of its solutions regarding functions as their epigraphs we bypass the regularity issues to arrive directly at the concept of barron jensen frankowska viscosity solutions above all we take into account viability constraints and extend classical boundary conditions to other internal conditions beyond that we use the fenchel legendre transform to associate a lagrangian with the hamiltonian and uncover an underlying variational problem it is proved that the viability solution to the structured hamilton jacobi equation is the valuation function of this variational principle the optimal evolutions of which are regulated by a regulation map which is constructed from the viability solution and can be computed by viability algorithms in other words the viability solution solves all at once a class of first order partial differential equations of intertemporal optimization problems and the regulation of their optimal evolutions

convexity in hamilton jacobi theory ii envelope representations upper and lower envelope representations are developed for value functions associated with problems of optimal control and the calculus of variations that are fully convex in the sense of exhibiting convexity in both the state and the velocity such convexity is used in dualizing the upper envelope representations to get the lower ones which have advantages not previously perceived in such generality and in some situations can be regarded as furnishing at least for value functions extended hopf lax formulas that operate beyond the case of state independent hamiltonians the derivation of the lower envelope representations centers on a new function called the dualizing kernel which propagates the legendre fenchel envelope formula of convex analysis through the underlying dynamics this kernel is shown to be characterized by a kind of double hamilton jacobi equation and despite overall nonsmoothness to be smooth with respect to time and concave convex in the primal and dual states it furnishes a means whereby in principle value functions and their subgradients can be determined through optimization without having to deal with a separate and typically much less favorable hamilton jacobi equation for each choice of the initial or terminal cost data

target problems under state constraints for nonlinear controlled impulsive systems we study the problem of reaching a target without leaving a prescribed constraint set for a dynamical system described by a controlled differential equation and a controlled instantaneous reset function we characterize all initial conditions from which the objective can be reached then we characterize the value function associated with the optimal reaching time problem

asymptotic perron s method and simple markov strategies in stochastic games and control we introduce a modification of perron s method where semisolutions are considered in a carefully defined asymptotic sense with this definition we can show in a rather elementary way that in a zero sum game or a control problem with or without model uncertainty the value function over all strategies coincides with the value function over markov strategies discretized in time therefore there are always discretized markov varepsilon optimal strategies uniform with respect to the bounded initial condition with a minor modification the method produces a value and approximate saddle points for an asymmetric game of feedback strategies versus counterstrategies

on dynamical properties of general dynamical systems and differential inclusions abstract let f be a general dynamical system defined on a complete locally compact metric space x we give a slightly improved version of the lyapunov characterization of asymptotic stability in one of our previous works and provide a short self contained proof for the existence of arbitrarily small positively invariant neighborhoods of compact asymptotically stable sets in the present context based on these results we prove that the uniform attractors of f are connected and are stable with respect to perturbations under appropriate conditions we are also interested in the dynamical properties of differential inclusion x t f x t on r m first we show that if no solutions of the system blow up in finite future time then its reachable mapping f is a general dynamical system then we discuss some asymptotic stability properties of the system in particular we prove that if there exists a nonempty compact connected subset m of r m such that m attracts a neighborhood of itself then the system has a connected uniform attractor a we also prove that a is stable with respect to both internal and external perturbations more precisely we prove that when 0 is sufficiently small the perturbed system x t con f x t b 1 b 1 has a connected uniform attractor a moreover h a a 0 as 0 where h is the hausdorff distance in r m

an lp approach to dynamic programming principles for stochastic control problems with state constraints abstract we study a class of nonlinear stochastic control problems with semicontinuous cost and state constraints using a linear programming lp approach first we provide a primal linearized problem stated on an appropriate space of probability measures with support contained in the set of constraints this space is completely characterized by the coefficients of the control system second we prove a semigroup property for this set of probability measures appearing in the definition of the primal value function this leads to dynamic programming principles for control problems under state constraints with general bounded costs a further linearized dpp is obtained for lower semicontinuous costs

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

cal h _ infty control of nonlinear systems differential games and viscosity solutions dealing with disturbances is one of the most important questions for controlled systems cal h infty optimal control theory is a deterministic way to tackle the problem in the presence of unfavorable disturbances the theory of differential games and the study of the associated hamilton jacobi isaacs equation appear to be basic tools of the theory we consider a general nonlinear system and prove that the existence of a continuous local viscosity supersolution of the isaacs equation corresponding to the cal h infty control problem is sufficient for its solvability we also show that the existence of a lower semicontinuous viscosity supersolution is necessary

attractors for random dynamical systems a criterion for existence of global random attractors for rds is established existence of invariant markov measures supported by the random attractor is proved for spde this yields invariant measures for the associated markov semigroup the results are applied to reation diffusion equations with additive white noise and to navier stokes equations with multiplicative and with additive white noise

discontinuous differential games and control systems with supremum cost abstract this paper concerns with the existence of a value for a zero sum two player differential game with supremum cost of the form c t 0 x 0 u v sup t 0 t h x t 0 x 0 u v under isaacs condition we characterize the value function as the unique solution in a suitable sense to a pde namely the hamilton jacobi isaacs equation as a byproduct we obtain a pde characterization of the value function for control system

on the asymptotic behaviour of solutions of a stochastic energy balance climate model abstract we prove the existence of a random global attractor for the multivalued random dynamical system associated to a nonlinear multivalued parabolic equation with a stochastic term of amplitude of the order of e the equation was initially suggested by north and cahalan following a previous deterministic model proposed by m i budyko for the modeling of some non deterministic variability as for instance the cyclones which can be treated as a fast varying component and are represented as a white noise process in the context of energy balance climate models we also prove the convergence in some sense of the global attractors when e 0 i e the convergence to the global attractor for the associated deterministic case e 0

mayer and optimal stopping stochastic control problems with discontinuous cost we study two classes of stochastic control problems with semicontinuous cost the mayer problem and optimal stopping for controlled diffusions the value functions are introduced via linear optimization problems on appropriate sets of probability measures these sets of constraints are described deterministically with respect to the coefficient functions both the lower and upper semicontinuous cases are considered the value function is shown to be a generalized viscosity solution of the associated hjb system respectively of some variational inequality dual formulations are given as well as the relations between the primal and dual value functions under classical convexity assumptions we prove the equivalence between the linearized mayer problem and the standard weak control formulation counter examples are given for the general framework

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

chaperoning state evolutions by variable durations mckendrick partial differential equations in demography provide the state of a population depending on chronological time t and age d and involve two time evolutions the chronological one t in 0 t mapsto t geq 0 and the calendar age t in t d t mapsto t t d geq 0 both with constant velocity equal to 1 the calendar age evolution chaperons so to speak the evolution t in t d t mapsto x t of the state of a system some physical biological and economic problems motivate the introduction of variable durations with variable velocities representing the fluidity of time offering mathematical metaphors of a subjective fleeting specious time passing more or less slowly variable durations are no longer prescribed but chosen among those available and regulated the joint evolution of the variable evolution and the state is assumed to be governed by a differential inclusion or control system and provides as a byproduct the unknown temporal windows on which they evolve toge

autonomous and non autonomous attractors for differential equations with delays abstract the asymptotic behaviour of some types of retarded differential equations with both variable and distributed delays is analysed in fact the existence of global attractors is established for different situations with and without uniqueness and for both autonomous and non autonomous cases using the classical notion of attractor and the recently new concept of pullback one respectively

extended hamilton jacobi characterization of value functions in optimal control this paper examines generalized solutions to the hamilton jacobi equation the hamiltonian h t x p is assumed convex in p but is not constrained to have linear growth in this variable this corresponds to a certain class of generalized bolza problems related to optimal control lower semicontinuous solutions are considered and it is shown that there is a unique solution the so called value function of the underlying bolza problem in proving the main result we use recent improved necessary optimality conditions viability is also used in a new way in connection to differential inclusions with unbounded images

impulse capture basins of sets under impulse control systems given a target contained in a constrained set and an impulse control system governing the evolutions of runs or executions that are hybrids of continuous and discrete evolutions this paper studies and provides several characterizations of the capture basin of the target viable in the constrained set it is the subset of initial runs from which start at least one run viable in the constrained set until it reaches the target in finite time it also provides algorithms and regulation rules governing the runs that reach the targets while obeying state constraints

optimal control problems on manifolds a dynamic programming approach abstract dynamic programming identifies the value function of continuous time optimal control with a solution to the hamilton jacobi equation appropriately defined this relationship in turn leads to sufficient conditions of global optimality which have been widely used to confirm the optimality of putative minimisers in continuous time optimal control the dynamic programming methodology has been used for problems with state space a vector space however there are many problems of interest in which it is necessary to regard the state space as a manifold this paper extends dynamic programming to cover problems in which the state space is a general finite dimensional c manifold it shows that also in a manifold setting we can characterise the value function of a free time optimal control problem as a unique lower semicontinuous lower bounded generalised solution of the hamilton jacobi equation the application of these results is illustrated by the investigation of minimum time controllers for a rigid pendulum

differential games optimal control and directional derivatives of viscosity solutions of bellman s and isaacs equations recent work by the authors and others has demonstrated the connections between the dynamic programming approach to optimal control theory and to two person zero sum differential games problems and the new notion of viscosity solutions of hamilton jacobi pde s introduced by m g crandall and p l lions in particular it has been proved that the dynamic programming principle implies that the value function is the viscosity solution of the associated hamilton jacobi bellman and isaacs equations in the present work it is shown that viscosity super and subsolutions of these equations must satisfy some inequalities called super and subdynamic programming principle respectively this is then used to prove the equivalence between the notion of viscosity solutions and the conditions introduced by a subbotin concerning the sign of certain generalized directional derivatives

chain differentials with an application to the mathematical fear operator abstract we introduce a concept of derivative in a topological vector space that yields the chain rule of differentiation for composition of functions akin to but simpler than the epiderivative of aubin frankowska we show that if a function has a g teaux differential with suitable continuity properties it is a chain differential as an example we show that the mathematical fear operator is chain differentiable with respect to the cost density distribution in the space of continuous functions endowed with the topology of pointwise convergence uniform on every compact subsets

characterization of the value function of final state constrained control problems with bv trajectories this paper aims to investigate a control problem governed by differential equations with radon measure as data and with final state constraints by using a known reparametrization method by dal maso and rampazzo 18 we obtain that the value function can be characterized by means of an auxiliary control problem of absolutely continuous trajectories involving time measurable hamiltonian we study the characterization of the value function of this auxiliary problem and discuss its numerical approximations

singular perturbation of optimal control problems on multidomains the goal of this paper is to study a singular perturbation problem in the framework of optimal control on multi domains we consider an optimal control problem in which the controlled system contains a fast and a slow variables this problem is reformulated as an hamilton jacobi bellman hjb equation the main difficulty comes from the fact that the fast variable lives in a multi domain the geometric singularity of the multi domains leads to the discontinuity of the hamiltonian under a controllability assumption on the fast variables the limit equation as the velocity of the fast variable goes to infinity is obtained via a pde approache and by means of the tools of the control theory

convexity in hamilton jacobi theory i dynamics and duality value functions propagated from initial or terminal costs and constraints by way of a differential inclusion or more broadly through a lagrangian that may take on infty are studied in the case where convexity persists in the state argument such value functions themselves taking on infty are shown to satisfy a subgradient form of the hamilton jacobi equation which strongly supports properties of local lipschitz continuity semidifferentiability and clarke regularity an extended method of characteristics is developed which determines them from the hamiltonian dynamics underlying the given lagrangian close relations with a dual value function are revealed

lax hopf based incorporation of internal boundary conditions into hamilton jacobi equation part ii computational methods this article presents a new method for explicitly computing solutions to a hamilton jacobi partial differential equation for which initial boundary and internal conditions are prescribed as piecewise affine functions based on viability theory a lax hopf formula is used to construct analytical solutions for the individual contribution of each affine condition to the solution of the problem the results are assembled into a lax hopf algorithm which can be used to compute the solution to the partial differential equation at any arbitrary time at no other cost than evaluating a semi analytical expression numerically the method being semi analytical it performs at machine accuracy compared to the discretization error inherent to finite difference schemes the performance of the method is assessed with benchmark analytical examples the running time of the algorithm is compared with the running time of a godunov scheme

regularity of trajectory attractor and upper semicontinuity of global attractor for a 2d non newtonian fluid abstract there are two results within this paper the one is the regularity of trajectory attractor and the trajectory asymptotic smoothing effect of the incompressible non newtonian fluid on 2d bounded domains for which the solution to each initial value could be non unique the other is the upper semicontinuity of global attractors of the addressed fluid when the spatial domains vary from m to r l l where m m 1 is an expanding sequence of simply connected bounded and smooth subdomains of such that m as m that is let a and a m be the global attractors of the fluid corresponding to and m respectively we establish that for any neighborhood o a of a the global attractor a m enters o a if m is large enough

minimum time control problems for non autonomous differential equations in this paper we investigate a minimum time problem for controlled non autonomous differential systems with dynamics depending on the final time the minimal time function associated to this problem does not satisfy the dynamic programming principle however we will prove that it is related to a standard front propagation problem via the reachability function two simple numerical examples are given to illustrate our approach

statistical mechanics of low density parity check codes we review recent theoretical progress on the statistical mechanics of error correcting codes focusing on low density parity check ldpc codes in general and on gallager and mackay neal codes in particular by exploiting the relation between ldpc codes and ising spin systems with multispin interactions one can carry out a statistical mechanics based analysis that determines the practical and theoretical limitations of various code constructions corresponding to dynamical and thermodynamical transitions respectively as well as the behaviour of error exponents averaged over the corresponding code ensemble as a function of channel noise we also contrast the results obtained using methods of statistical mechanics with those derived in the information theory literature and show how these methods can be generalized to include other channel types and related communication problems

stable representation of convex hamiltonians abstract existence and uniqueness of solutions to a hamilton jacobi equation v t h t x v x 0 v 0 with h convex with respect to the last variable can be proved by associating to h either a calculus of variations or an optimal control problem the data of the new problem should be so that its hamiltonian coincides with h and should also inherit appropriate continuity local lipschitz continuity properties of h in other words h can be represented by functions describing an optimization problem in this paper we provide further developments of representation theorems from rampazzo 2005 in particular our results imply continuous dependence of representations on the mapping h we apply them to study existence of solutions to the hamilton jacobi equation with h possibly discontinuous in t

on reflecting boundary problem for optimal control this paper deals with mayer s problem for controlled systems with reflection on the boundary of a closed subset k the main result is the characterization of the possibly discontinuous value function in terms of a unique solution in a suitable sense to a partial differential equation of hamilton jacobi bellman type

attractors for differential equations with unbounded delays we prove the existence of attractors for some types of dierential problems contain ing in nite delays applications and examples are provided to illustrate the theory which is valid for both cases with and without explicit dependence of time and with or without uniqueness of solutions as well

invariant solutions of differential games and hamilton jacobi isaacs equations for time measurable hamiltonians we characterize invariance of time varying domains with respect to differential games with time measurable dynamics we deduce from this result a new definition of viscosity solutions to some first order hamilton jacobi equations with time measurable hamiltonians

pullback attractor for differential evolution inclusions with infinite delays we analyze the existence of pullback attractor for non autonomous differential inclusions with infinite delays in banach spaces by using measures of noncompactness the obtained results can be applied to control systems driven by semilinear partial differential equations and multivalued feedbacks

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

an initial condition reconstruction in hamilton jacobi equations abstract we describe the family of initial conditions for hamilton jacobi equations hje corresponding to optimal control problems that can be retrieved by solving hje backward in time

global attractor for a non autonomous integro differential equation in materials with memory abstract the long time behavior of an integro differential parabolic equation of diffusion type with memory terms expressed by convolution integrals involving infinite delays and by a forcing term with bounded delay is investigated in this paper the assumptions imposed on the coefficients are weak in the sense that uniqueness of solutions of the corresponding initial value problems cannot be guaranteed then it is proved that the model generates a multivalued non autonomous dynamical system which possesses a pullback attractor first the analysis is carried out with an abstract parabolic equation then the theory is applied to the particular integro differential equation which is the objective of this paper the general results obtained in the paper are also valid for other types of parabolic equations with memory

lower semicontinuous solutions of hamilton jacobi bellman equations the value function of mayer s problem arising in optimal control is investigated and lower semicontinuous solutions of the associated hamilton jacobi bellman equation are defined in three equivalent ways under quite weak assumptions about the control system the value function is the unique solution moreover it is stable with respect to perturbations of the control system and the cost it coincides with the viscosity solution whenever it is continuous

morse decompositions for general dynamical systems and differential inclusions with applications to control systems the morse decomposition theory for single valued dynamical systems is extended to general dynamical systems and differential inclusions relations between chain recurrent sets chain recurrent components chain control sets and morse decompositions for open loop systems are discussed the robustness of feedback laws of closed loop systems with respect to small time delays and sample hold controls is also addressed

faithful representations for convex hamilton jacobi equations when a hamiltonian h h t x p is convex in the adjoint variable p the corresponding hamilton jacobi equation 0 1 ut h t x ux 0 is known to be the bellman equation of a suitable optimal control problem of course the latter is not unique so it is interesting to select a good optimal control problem among those representing 0 1 we call such a representation faithful if i it involves a dynamics which is locally lipschitz continuous in the state variable so that a unique trajectory corresponds to any given control and initial point and ii the lagrangian displays the same regularity as h in the x variable the main result of the present paper establishes the existence of faithful representations for a large class of hamiltonians including those for which the standard comparison theorems of viscosity solution theory are valid moreover our investigation includes t measurable hamiltonians as well if a faithful control theoretical representation does exist and 0 1 enjoys uniqueness properties one can infer sharp regularity results for the solution of 0 1 just by studying the regularity of the value function of the associated optimal control problem a further application consists of a simple interpretation of the front propagation phenomenon in terms of optimal trajectories of the underlying minimum problem

viscosity solutions for partial differential equations with neumann type boundary conditions and some aspects of aubry mather theory we study partial differential inequalities pdi of the type c h x v x nk x 0 where nk is the normal cone to the set k we prove existence of a constant c c such that the pdi of hamilton jacobi type has a unique global lipschitz viscosity solution we provide a formula to calculate this constant moreover we define a subset k of k such that any two solutions of the previous pdi which coincide on k will coincide on k our paper generalizes results of the case without boundary conditions for convex hamiltonians obtained by l c evans and a fathi

stochastic perron s method and elementary strategies for zero sum differential games we develop here the stochastic perron method in the framework of two player zero sum differential games we consider the formulation of the game where both players play symmetrically feedback strategies as opposed to the elliott kalton formulation prevalent in the literature the class of feedback strategies we use is carefully chosen so that the state equation admits strong solutions and the technicalities involved in the stochastic perron method carry through in a rather simple way more precisely we define the game over elementary strategies which are well motivated by intuition within this framework the stochastic perron method produces a viscosity subsolution of the upper isaacs equation dominating the upper value of the game and a viscosity supersolution of the upper isaacs equation lying below the upper value of the game using a viscosity comparison result we obtain that the upper value is the unique and continuous viscosity solution of the upper isaacs equation an identical statement hol

on the regularization effect of space time white noise on quasi linear parabolic partial differential equations we show that an existence and uniqueness and a comparison theorem hold if we add a space time white noise to a quasi linear parabolic equation in one space dimension even if the nonlinearity is only measurable and not even locally bounded

value functions for differential games and control systems with discontinuous terminal cost this paper deals with mayer s problem for control systems and differential games with discontinuous terminal cost there are two main results in the paper the first one says that the value function for control systems can be characterized as the unique solution in suitable sense to the hamilton jacobi bellman equation without any regularity assumptions on the terminal cost for differential games satisfying isaacs s minmax condition the second main result says that the value function is the unique solution to the hamilton jacobi isaacs equation when the terminal cost is semicontinuous this allows to prove the existence of the value under isaacs s condition this paper extends some results already well known in the continuous case

uniform attractors for non autonomous random dynamical systems abstract this paper is devoted to establishing a random uniform attractor theory for non autonomous random dynamical systems nrds the uniform attractor is defined as the minimal compact uniformly pullback attracting random set nevertheless the uniform pullback attraction in fact implies a uniform forward attraction in probability and implies also an almost uniform pullback attraction for discrete time sequences though no invariance is required by definition the uniform attractor can have a negative semi invariance under certain conditions several existence criteria for uniform attractors are given and the relationship between uniform and cocycle attractors is carefully studied to overcome the measurability difficulty the symbol space is required to be polish which is shown fulfilled by the hulls of l l o c p r l r functions p r 1 moreover uniform attractors for continuous nrds are shown determined by uniformly attracting deterministic compact sets finally the uniform attractor for a stochastic reaction diffusion equation with translation bounded external forcing are studied as applications

magnetization enumerator of real valued symmetric channels in gallager error correcting codes using the magnetization enumerator method we evaluate the practical and theoretical limitations of symmetric channels with real outputs results are presented for several regular gallager code constructions

semicontinuous solutions of hamilton jacobi bellman equations with degenerate state constraints abstract in this paper the value function of a bolza optimal control problem with state constraints is characterized as the unique lower semicontinuous solution of a hamilton jacobi equation the state constraints are given by an arbitrary closed set with possibly empty interior in particular soner s inward pointing condition is extended here to the case of degenerate state constraints

reflected differential games this paper concerns the existence of a possible discontinuous value for a zero sum two player reflected differential game under isaacs condition we characterize the value function as the unique solution in a suitable sense to a variational inequality namely the hamilton jacobi isaacs differential inclusion

stochastic verification theorems within the framework of viscosity solutions this paper studies controlled systems governed by ito s stochastic differential equations in which control variables are allowed to enter both drift and diffusion terms a new verification theorem is derived within the framework of viscosity solutions without involving any derivatives of the value functions this theorem is shown to have wider applicability than the restrictive classical verification theorems which require the associated dynamic programming equations to have smooth solutions based on the new verification result optimal stochastic feedback controls are obtained by maximizing the generalized hamiltonians over both the control regions and the superdifferentials of the value functions

lyapunov stability theory of nonsmooth systems this paper develops nonsmooth lyapunov stability theory and lasalle s invariance principle for a class of nonsmooth lipschitz continuous lyapunov functions and absolutely continuous state trajectories computable tests based on filipov s differential inclusion and clarke s generalized gradient are derived the primary use of these results is in analyzing the stability of equilibria of differential equations with discontinuous right hand side such as in nonsmooth dynamic systems or variable structure control

boundary value problems for systems of hamilton jacobi bellman inclusions with constraints we study in this paper boundary value problems for systems of hamilton jacobi bellman first order partial differential equations and variational inequalities the solutions of which are constrained to obey viability constraints they are motivated by some control problems such as impulse control and financial mathematics we shall prove the existence and uniqueness of such solutions in the class of closed set valued maps by giving a precise meaning to what a solution means in this case we shall also provide explicit formulas for this problem when we deal with hamilton jacobi bellman equations we obtain the existence and uniqueness of frankowska contingent episolutions we shall deduce these results from the fact that the graph of the solution is the viable capture basin of the graph of the boundary conditions under an auxiliary system and then from their properties and their characterizations proved in j p aubin siam j control optim 40 2001 pp 853 881

s shaped bifurcation branch in a quasilinear multivalued model arising in climatology abstract in this paper we show the existence of a continuous and unbounded connected s shaped set q u where q is the solar constant and u satisfies a quasilinear eventually multivalued stationary equation on a riemannian manifold without boundary arising as a stationary energy balance model for the earth surface temperature