on differential games for infinite dimensional systems with nonlinear unbounded operators abstract we consider a two player zero sum differential game governed by an abstract nonlinear differential equation of accretive type in an infinite dimensional space we prove that the value function of the game is the unique viscosity solution of the corresponding hamilton jacobi isaacs equation in the sense of 12 we also discuss some properties of this notion of solution

nonsmooth lyapunov pairs for infinite dimensional first order differential inclusions the main objective of this paper is to provide new explicit criteria to characterize weak lower semicontinuous lyapunov pairs or functions associated to first order differential inclusions in hilbert spaces these inclusions are governed by a lipschitzian perturbation of a maximally monotone operator the dual criteria we give are expressed by means of the proximal and basic subdifferentials of the nominal functions while primal conditions are described in terms of the contingent directional derivative we also propose a unifying review of many other criteria given in the literature our approach is based on advanced tools of variational analysis and generalized differentiation

motion of a set by the curvature of its boundary abstract the study of a crystal shrinking or growing in a melt gives rise to equations relating the normal velocity of the motion to the curvature of the crystal boundary often these equations are anisotropic indicating the preferred directions of the crystal structure in the isotropic case this equation is called the curve shortening or the mean curvature flow equation and has been studied by differential geometric tools in general there are no classical solutions to these equations in this paper we develop a weak theory for the generalized mean curvature equation using the newly developed theory of viscosity solutions our approach is closely related to that of osher and sethian chen giga and goto and evans and spruck who view the boundary of the crystal as the level set of a solution to a nonlinear parabolic equation although we use their results in an essential way we give an intrinsic definition our main results are the existence of a solution large time asymptotics of this solution and its connection to the level set solution of osher and sethian chen giga and goto and evans and spruck in general there is no uniqueness even for classical solutions but we prove a uniqueness result under restrictive assumptions we also construct a class of explicit solutions which are dilations of wulff crystals

nonsmooth lyapunov pairs for differential inclusions governed by operators with nonempty interior domain the general theory of lyapunov stability of first order differential inclusions in hilbert spaces has been studied by the authors in the previous paper adly et al in nonlinear anal 75 3 985 1008 2012 this new contribution focuses on the case when the interior of the domain of the maximally monotone operator governing the given differential inclusion is nonempty this includes in a natural way the finite dimensional case the current setting leads to simplified more explicit criteria and permits some flexibility in the choice of the generalized subdifferentials some consequences of the viability of closed sets are given our analysis makes use of standard tools from convex and variational analysis

existence of lipschitz and semiconcave control lyapunov functions given a locally lipschitz control system which is globally asymptotically controllable at the origin we construct a control lyapunov function for the system which is lipschitz on bounded sets and we deduce the existence of another one which is semiconcave and so locally lipschitz outside the origin the proof relies on value functions and nonsmooth calculus

level set approach to mean curvature flow in arbitrary codimension we develop a level set theory for the mean curvature evolution of surfaces with arbitrary co dimension thus generalizing the previous work 8 15 on hypersurfaces the main idea is to surround the evolving surface of co dimension k in r by a family of hypersurfaces the level sets of a function evolving with normal velocity equal to the sum of the d k smallest principal curvatures the existence and the uniqueness of a weak level set solution is easily established using mainly the results of 8 and the theory of viscosity solutions for second order nonlinear parabolic equations the level set solutions coincide with the classical solutions whenever the latter exist the proof of this connection uses a careful analysis of the squared distance from the surfaces it is also shown that varifold solutions constructed by brakke 7 are included in the level set solutions the idea of surrounding the evolving surface by a family of hypersurfaces with a certain property is related to the barriers of de giorgi an introduction to the theory of barriers and his connection to the level set solutions is also provided

on the existence of local smooth repulsive stabilizing feedbacks in dimension three abstract given an affine control system in r 3 subject to the hormander s condition at the origin we prove the existence of a local smooth repulsive stabilizing feedback at the origin our construction is based on the classical homogenization procedure on the existence of a semiconcave control lyapunov function and on the classification of singularities of semiconcave functions in dimension two

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

viscosity solutions for hamilton jacobi equations with unbounded nonlinear term a simplified approach abstract in a previous paper the author defined a notion for hamilton jacobi equations with an unbounded nonlinear term ax du and for such solutions proved existence and uniqueness results here we define an equivalent notion of solution for such equations which in addition has the advantage of being simpler and more intuitive

decentralized measurement feedback stabilization of large scale systems via control vector lyapunov functions abstract this paper studies the problem of decentralized measurement feedback stabilization of nonlinear interconnected systems as a natural extension of the recent development on control vector lyapunov functions the notion of output control vector lyapunov function ocvlf is introduced for investigating decentralized measurement feedback stabilization problems sufficient conditions on local stabilizability are discussed which are based on the proposed notion of ocvlf it is shown that a decentralized controller for a nonlinear interconnected system can be constructed using these conditions under an additional vector dissipation like condition to illustrate the proposed method two examples are given

a stochastic jurdjevic quinn theorem the purpose of this paper is to state sufficient conditions for the stabilizability of stochastic differential systems when both the drift and diffusion terms are affine in the control this result extends to stochastic differential systems the well known theorem of jurdjevic quinn j differential equations 28 1978 pp 381 389 and incorporates earlier stabilization results proved in p florchinger stochastic anal appl 12 1994 pp 473 480 and r chabour and m oumoun stochastic anal appl 16 1998 pp 43 50

a lyapunov like characterization of asymptotic controllability it is shown that a control system in bf r n is asymptotically controllable to the origin if and only if there exists a positive definite continuous functional of the states whose derivative can be made negative by appropriate choices of controls

asymptotic controllability implies feedback stabilization it is shown that every asymptotically controllable system can be globally stabilized by means of some discontinuous feedback law the stabilizing strategy is based on pointwise optimization of a smoothed version of a control lyapunov function iteratively sending trajectories into smaller and smaller neighborhoods of a desired equilibrium a major technical problem and one of the contributions of the present paper concerns the precise meaning of solution when using a discontinuous controller

robust controller design for stochastic nonlinear systems via convex optimization this paper presents a new nonlinear control design framework called convex optimization based stochastic steady state tracking error minimization cvstem for a class of ito stochastic nonlinear systems including stochastic lagrangian systems this state feedback controller design involves the use of multiple state dependent coefficient sdc models of a nonlinear system equation and computes its controller parameters by solving a convex optimization problem in order to minimize an upper bound of the steady state tracking error this problem is shown to be equivalent to the original nonlinear optimization problem of minimizing the upper bound the exponential stability and robustness properties due to the state dependent riccati inequality are proven using stochastic contraction analysis discrete time stochastic incremental contraction analysis with respect to a state and time dependent metric is also presented along with its explicit connection to continuous time contraction analysis a result of simulation is used to validate the superiority of the proposed controller compared to a known nonlinear controller a pid controller and an h infinity controller

a stochastic differential game for the inhomogeneous laplace equation given a bounded c 2 domain gr m and functions g 2 c g r and h 2 c g r 0 let u denote the unique viscosity solution to the equation 2 u h in g with boundary data g we provide a representation for u as the value of a two player zero sum stochastic differential game ams 2000 subject classifications 91a15 91a23 35j70 49l20 g r 0 be given we study a two player zero sum stochastic differential game sdg defined in terms of an m dimensional state process that is driven by a one dimensional brow nian motion played until the state exits the domain the functions g and h serve as terminal and respectively running payoffs the players controls enter in a diffusion coefficient and in an unbounded drift coefficient of the state process the dynamics are degenerate in that it is possible for the players to completely switch off the brownian motion we show that the game has value and characterize the value function as the unique viscosity solution u uniqueness of solutions is known from 9 of the equation 2 u h in g

state space approximation of multi input multi output systems with stochastic exogenous inputs instead of the usual ar ma x or var vector autoregressive modelling procedures will be described to obtain approximate balanced state space models and steady state kalman filters with prewhitened inputs these state space models and kalman filters can be used for prediction and also for control whenever the output and input variables are target and control variables respectively

stochastic versions of the lasalle theorem the main aim of this paper is to establish stochastic versions of the well known lasalle stability theorem from these stochastic versions follow many classical results on stochastic stability this shows clearly the power of our new results

stabilization with relaxed controls cannot in general be stabilized using a continuous closed loop control u x even if each state separately can be driven asymptotically to the origin an example is analyzed in section 2 in this paper we examine the possibility of stabilizing such systems with a continuous closed loop relaxed control we find indeed that the family of systems stabilizable with relaxed controls is larger than the family of those stabilizable with ordinary controls an even larger class is obtained if the continuity of the closed loop at the origin is not required the latter class includes all one dimensional systems for which states can be driven asymptotically to the origin this result does not hold in two dimensional systems and we provide a counter example it should be pointed that relaxed control type stabilization is used both in theory and in practice the method is known as dither we shall comment on the similarities lyapunov functions for the system 1 help us in the construction of the continuous closed loop stabilizers in fact we find that the existence of a smooth lyapunov function is equivalent to the existence of a stabilizing closed loop which is continuous except possibly at the origin an additional condition on the lyapunov function implies the continuity at the origin as well we present these results in section 4 after a brief introduction of closed loop relaxed controls notations and terminology in section 3 prior to that in section 2 we discuss an example illustrating the power of relaxed controls in the particular case of systems linear in the controls relaxed controls can be replaced by ordinary controls this is discussed in section 5 the role of lyapunov functions in the stability and stabilization theories is of course well known examples of systems with lyapunov functions are available in the literature we display some in section 6 along with general comments on the construction applications and counterexamples including one which cannot be continuously stabilized yet possesses a nonsmooth lyapunov function closed loop stabilization with ordinary controls is analyzed extensively in the literature see sontag 8 sussmann ll and references therein lyapunov functions techniques in stabil

a deterministic control based approach motion by curvature the level set formulation of motion by mean curvature is a degenerate parabolic equation we show that its solution can be interpreted as the value function of a deterministic two person game more precisely we give a family of discrete time two person games whose value functions converge in the continuous time limit to the solution of the motion by curvature pde for a convex domain the boundary s first arrival time solves a degenerate elliptic equation this corresponds in our game theoretic setting to a minimum exit time problem for a nonconvex domain the two person game still makes sense we draw a connection between its minimum exit time and the evolution of curves with velocity equal to the positive part of the curvature these results are unexpected because the value function of a deterministic control problem is normally the solution of a first order hamilton jacobi equation our situation is different because the usual first order calculation is singular 2005 wiley periodicals inc

sufficient conditions for asymptotic controllability and hybrid feedback stabilization for a class of time varying systems for a class of time varying systems including those whose dynamics have triangular structure we provide sufficient conditions for global asymptotic controllability and global asymptotic stabilization by means of a hybrid time varying feedback

invariant sets and lyapunov pairs for differential inclusions with maximal monotone operators we give different conditions for the invariance of closed sets with respect to differential inclusions governed by a maximal monotone operator defined on hilbert spaces which is subject to a lipschitz continuous perturbation depending on the state these sets are not necessarily weakly closed as in 3 4 while the invariance criteria are still written by using only the data of the system so no need to the explicit knowledge of neither the solution of this differential inclusion nor the semi group generated by the maximal monotone operator these invariant viability results are next applied to derive explicit criteria for a lyapunov pairs of lower semi continuous not necessarily weakly lsc functions associated to these differential inclusions the lack of differentiability of the candidate lyapunov functions and the consideration of general invariant sets possibly not convex or smooth are carried out by using techniques from nonsmooth analysis

lyapunov pairs for continuous perturbations of nonlinear evolutions abstract this paper deals with the lyapunov pairs for a nonlinear evolution system in banach spaces applications on a priori estimates finite extinction time null controllability and regularity of the minimal time function are given

logarithmic diffusion and porous media equations a unified description in this work we present the logarithmic diffusion equation as a limit case when the index that characterizes a nonlinear fokker planck equation in its diffusive term goes to zero a linear drift and a source term are considered in this equation its solution has a lorentzian form consequently this equation characterizes a superdiffusion like a levy kind in addition an equation that unifies the porous media and the logarithmic diffusion equations including a generalized diffusion equation in fractal dimension is obtained this unification is performed in the nonextensive thermostatistics context and increases the possibilities about the description of anomalous diffusive processes

stability of dynamical systems with competitive controls the degenerate case abstract we study the stability of a dynamical system with two competitive controls to a general closed set the stability is proven to be equivalent to the continuity on such set of the value function of a differential game with maximum cost functional the existence of a suitably defined and continuous lyapunov function is proven to imply the stability our lyapunov function satisfies a differential inequality in the sense of the theory of viscosity solutions

a realization approach to stochastic model reduction the problem of discrete time stochastic model reduction approximation is considered using the canonical correlation analysis approach of akaike 1975 a new order reduction algorithm is developed furthermore it is shown that the inverse of the reduced order realization is asymptotically stable next an explicit relationship between canonical variables and the linear least squares estimate of the state vector is established using this a more direct approach for order reduction is presented and also a new design for reduced order kalman filters is developed finally the uniqueness and symmetry properties for the new realization the balanced stochastic realization along with a simulation result are presented

multifractal detrended fluctuation analysis of nonstationary time series abstract we develop a method for the multifractal characterization of nonstationary time series which is based on a generalization of the detrended fluctuation analysis dfa we relate our multifractal dfa method to the standard partition function based multifractal formalism and prove that both approaches are equivalent for stationary signals with compact support by analyzing several examples we show that the new method can reliably determine the multifractal scaling behavior of time series by comparing the multifractal dfa results for original series with those for shuffled series we can distinguish multifractality due to long range correlations from multifractality due to a broad probability density function we also compare our results with the wavelet transform modulus maxima method and show that the results are equivalent

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

a deterministic control based approach to fully nonlinear parabolic and elliptic equations we show that a broad class of fully nonlinear second order parabolic or elliptic pde s can be realized as the hamilton jacobi bellman equations of deterministic twoperson games more precisely given the pde we identify a deterministic discretetime two person game whose value function converges in the continuous time limit to the viscosity solution of the desired equation our game is roughly speaking a deterministic analogue of the stochastic representation recently introduced by cheridito soner touzi and victoir comm pure appl math 60 2006 1081 1110 in the parabolic setting with no u dependence it amounts to a semidiscrete numerical scheme whose timestep is a min max our result is interesting because the usual control based interpretations of second order pde s involve stochastic rather than deterministic control

a stochastic representation for mean curvature type geometric flows a smooth solution t t 0 t rd of a parabolic geometric flow is characterized as the reachability set of a stochastic target problem in this control problem the controller tries to steer the state process into a given deterministic set tc with probability one the reachability set v t for the target problem is the set of all initial data x from which the state process xx t tc for some control process this representation is proved by studying the squared distance function to t for the codimension k mean curvature flow the state process is dx t 2 pdw t where w t is a d dimensional brownian motion and the control p is any projection matrix onto a d k dimensional plane smooth solutions of the inverse mean curvature flow and a discussion of non smooth solutions are also given

lyapunov functions for infinite dimensional systems abstract we study lyapunov functions for infinite dimensional dynamical systems governed by general maximal monotone operators we obtain a characterization of lyapunov pairs by means of the associated hamilton jacobi partial differential equations whose solutions are meant in the viscosity sense as evolved in works of tataru and crandall lions our approach also leads to a new sufficient condition for lyapunov pairs generalizing a classical result of pazy

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

hurst exponents for interacting random walkers obeying nonlinear fokker planck equations abstract anomalous diffusion of random walks has been extensively studied for the case of non interacting particles here we study the evolution of nonlinear partial differential equations by interpreting them as fokker planck equations arising from interactions among random walkers we extend the formalism of generalized hurst exponents to the study of nonlinear evolution equations and apply it to several illustrative examples they include an analytically solvable case of a nonlinear diffusion constant and three nonlinear equations which are not analytically solvable the usual fisher equation which contains a quadratic nonlinearity a generalization of the fisher equation with density dependent diffusion constant and the nagumo equation which incorporates a cubic rather than a quadratic nonlinearity we estimate the generalized hurst exponents

performance bounds for linear stochastic control we develop computational bounds on performance for causal state feedback stochastic control with linear dynamics arbitrary noise distribution and arbitrary input constraint set this can be very useful as a comparison with the performance of suboptimal control policies which we can evaluate using monte carlo simulation our method involves solving a semidefinite program a linear optimization problem with linear matrix inequality constraints a convex optimization problem which can be efficiently solved numerical experiments show that the lower bound obtained by our method is often close to the performance achieved by several widely used suboptimal control policies which shows that both are nearly optimal as a by product our performance bound yields approximate value functions that can be used as control lyapunov functions for suboptimal control policies

differential games and nonlinear boldmath cal h_ infty control in infinite dimensions this paper studies the cal h infty control problem for a nonlinear unbounded infinite dimensional system with state constraints we characterize the solvability of the problem by means of a hamilton jacobi isaacs hji equation proving that the cal h infty problem can be solved if and only if the hji equation has a positive definite viscosity supersolution vanishing and continuous at the origin in order to do so the standard definition of the cal h infty problem has to be relaxed by using the theory of differential games we apply our results to the one phase stefan problem

lyapunov like techniques for stochastic stability the purpose of this paper is to study the stabilizability problem for control stochastic nonlinear systems driven by a wiener process sufficient conditions for the existence of stabilizing feedback laws that are smooth except possibly at the equilibrium point of the system are provided by means of stochastic lyapunov like techniques the notion of dynamic asymptotic stability in probability of control stochastic differential systems is introduced and the stabilization by means of dynamic controllers is studied

kinetic equations for reaction subdiffusion systems derivation and stability analysis we derive general kinetic equations for reacting and subdiffusing entities based on a nonlinear continuous time random walk formalism proposed by vlad and ross phys rev e 66 061908 2002 reaction and diffusion processes are separable in a typical reaction diffusion system and their combined influence on the evolution of the density of a species is a simple sum our derivation shows that this is no longer true for subdiffusive entities undergoing reactions the strong memory effects in the transport process i e the non markovian nature of subdiffusion results in a nontrivial combination of reactions and spatial dispersal which we discuss in detail we carry out a linear stability analysis of the derived reaction subdiffusion system to understand the effects of memory on pattern formation we find that the turing instability persists in the subdiffusive system however the memory modifies the turing threshold and the characteristics of the band of unstable modes close to this threshold

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

feedback stabilization of affine in the control stochastic differential systems by the control lyapunov function method the purpose of this paper is to study the asymptotic stability in probability of affine in the control stochastic differential systems sufficient conditions for the existence of control lyapunov functions leading to the existence of stabilizing feedback laws which are smooth except possibly at the equilibrium point of the system are provided

physical pictures of transport in heterogeneous media advection dispersion random walk and fractional derivative formulations 1 the basic conceptual picture and theoretical basis for development of transport equations in porous media are examined the general form of the governing equations is derived for conservative chemical transport in heterogeneous geological formations for single realizations and for ensemble averages of the domain the application of these transport equations is focused on accounting for the appearance of non fickian anomalous transport behavior the general ensemble averaged transport equation is shown to be equivalent to a continuous time random walk ctrw and reduces to the conventional forms of the advection dispersion equation ade under highly restrictive conditions fractional derivative formulations of the transport equations both temporal and spatial emerge as special cases of the ctrw in particular the use in this context of levy flights is critically examined in order to determine chemical transport in field scale situations the ctrw approach is generalized to nonstationary systems we outline a practical numerical scheme similar to those used with extended geological models to account for the often important effects of unresolved heterogeneities

on the existence of optimal controls the optimal control problem where the state is governed by an ito stochastic differential equation possibly just an ordinary differential equation is formulated in martingale terms under a coercivity condition which is weaker than compactness of the control set a convexity condition and mild continuity hypotheses on the data it is shown by the direct method that optimal controls exist hard and soft constraints are allowed in the absence of soft constraints it is shown that there exists an optimal control that is a function only of the present time and state i e the synthesis problem has a solution the main tool here is krylov s markovian selection theorem

an algebraic approach to bounded controllability of linear systems in this paper we present an algebraic approach to the proof that a linear system with matrices a b is null controllable using bounded inputs if and only if it is null controllable with unbounde

a universal construction of artstein s theorem on nonlinear stabilization abstract this note presents an explicit proof of the theorem due to artstein which states that the existence of a smooth control lyapunov function implies smooth stabilizability moreover the result is extended to the real analytic and rational cases as well the proof uses a universal formula given by an algebraic function of lie derivatives this formula originates in the solution of a simple riccati equation

sampled data feedback practical semi global controllability and stabilization for time varying systems we deal with the problems of the closed loop controllability and stabilization for time varying systems whose dynamics are in general unbounded with respect to time it is shown that a general notion of global asymptotic controllability being in general non uniform with respect to initial values of time implies existence of a family of continuous control lyapunov functions and this enables the establishment of practical semi global closed loop controllability by means of a sampled data time varying feedback moreover the usual concept of global asymptotic controllability being uniform with respect to the initial values of time implies practical semi global stabilization by sampled data feedback the work is a continuation of a recent paper of the author in the same journal and extends krichman s 2000 result relying practical stabilization of time varying systems with bounded in time dynamics

superreplication under gamma constraints in a financial market consisting of a nonrisky asset and a risky one we study the minimal initial capital needed in order to superreplicate a given contingent claim under a gamma constraint this is a constraint on the unbounded variation part of the hedging portfolio we first consider the case in which the prices are given as general markov diffusion processes and prove a verification theorem which characterizes the superreplication cost as the unique solution of a quasi variational inequality in the context of the black scholes model i e when volatility is constant this theorem allows us to derive an explicit solution of the problem these results are based on a new dynamic programming principle for general stochastic target problems

front propagation and phase field theory the connection between the weak theories for a class of geometric equations and the asymptotics of appropriately rescaled reaction diffusion equations is rigorously established two different scalings are studied in the first the limiting geometric equation is a first order equation in the second it is a generalization of the mean curvature equation intrinsic definitions for the geometric equations are obtained and uniqueness under a geometric condition on the initial surface is proved in particular in the case of the mean curvature equation this condition is satisfied by surfaces that are strictly starshaped that have positive mean curvature or that satisfy a condition that interpolates between the positive mean curvature and the starshape conditions

forecasting international growth rates with leading indicators a system theoretic approach abstract an approach to estimating multivariate time invariant state space models for armax type processes is considered it is based on estimated markov parameters unit impulse responses and involves singular value decomposition techniques and leads to internally balanced state space representations the method is applied to forecast the growth rate of real output of nine countries using leading indicators the forecasting performance of this state space approach is compared to the performance of autoregressive models with leading indicators

differential games and zubov s method in this paper we provide generalizations of zubov s equation to differential games without the isaacs condition we show that both generalizations of zubov s equation which we call the min max and max min zubov equation respectively possess unique viscosity solutions which characterize the respective controllability domains as a consequence we show that under the usual isaacs condition the respective controllability domains as well as the local controllability assumptions coincide

viscosity solutions of hamilton jacobi equations with unbounded nonlinear terms abstract the aim of this paper is to introduce a new definition of viscosity solutions for hamilton jacobi equations in infinite dimensions with unbounded nonlinear terms we prove existence and uniqueness results under usual assumptions regarding the bounded part of the hamiltonian

conditions for local almost sure asymptotic stability abstract in this paper we investigate local asymptotic stability ensured by the addition of gaussian white noise into dynamical systems there are different stability notions for stochastic systems such as asymptotic stability in probability asip and uniform almost sure asymptotic stability uasas the local asip property is incapable of ensuring that sample paths converge to the origin with probability one whereas the local uasas property is capable of it however in general the local uasas property requires tight conditions here we provide our notion of local almost sure asymptotic stability local asas to relax the conditions with both almost sure convergence of sample paths to the origin and the existence of bounded weak invariant sets we find that the addition of gaussian white noise always prevents the origin from being locally uasas as long as we consider smooth lyapunov functions however it is possible to make the origin locally asas the result is confirmed by a simple example of elimination of unstable equilibria by deliberately adding gaussian white noise

technical communique stabilizing controllers design for switched nonlinear systems in strict feedback form this paper considers the stabilization problem for a class of switched nonlinear systems under arbitrary switching based on the backstepping method and the control lyapunov function approach it is shown that under a simultaneous domination assumption a switched nonlinear system in strict feedback form can be globally uniformly asymptotically stabilized by a continuous state feedback controller a universal formula for constructing stabilizing feedback laws is presented one example is included for verifying the obtained results

a semi lagrangian scheme for the curve shortening flow in codimension 2 we consider the model problem where a curve in r 3 moves according to the mean curvature flow the curve shortening flow we construct a semi lagrangian scheme based on the feynman kac representation formula of the solutions of the related level set geometric equation the first step is to obtain an approximation of the associated codimension 1 problem formulated by ambrosio and soner where the squared distance from the initial curve is used as initial condition since the e sublevel of this evolution contains the curve the next step is to extract the curve itself by following an optimal trajectory inside each e sublevel we show that this procedure is robust and accurate as long as the fattening phenomenon does not occur moreover it can still single out the physically meaningful solution when it occurs

flow by mean curvature of convex surfaces into spheres the motion of surfaces by their mean curvature has been studied by brakke 1 from the viewpoint of geometric measure theory other authors investigated the corresponding nonparametric problem 2 5 9 a reason for this interest is that evolutionary surfaces of prescribed mean curvature model the behavior of grain boundaries in annealing pure metal in this paper we take a more classical point of view consider a compact uniformly convex w dimensional surface m mo without boundary which is smoothly imbedded in r let mo be represented locally by a diffeomorphism

on near optimal trajectories for a game associated with the laplacian a two player stochastic differential game representation has recently been obtained for solutions of the equation u h in a mathcal c 2 domain with dirichlet boundary condition where h is continuous and takes values in mathbb r setminus 0 under appropriate assumptions including smoothness of u we identify a family of diffusion processes that may arise as the vanishing limit law of the state process when both players play optimally we also identify the limit law of the state process under a sequence of near saddle points

turing pattern formation in the brusselator model with superdiffusion the effect of superdiffusion on pattern formation and pattern selection in the brusselator model is studied our linear stability analysis shows in particular that unlike the case of normal diffusion the turing instability can occur even when diffusion of the inhibitor is slower than that of the initiator a weakly nonlinear analysis yields a system of amplitude equations analysis of which predicts parameter regimes where hexagons stripes and their coexistence are expected numerical computations of the original brusselator model near the stability boundaries confirm the results of the analysis in addition further from the stability boundaries we find a regime of self replicating spots

almost sure stabilizability of controlled degenerate diffusions we develop a direct lyapunov method for the almost sure open loop stabilizability and asymptotic stabilizability of controlled degenerate diffusion processes the infinitesimal decrease condition for a lyapunov function is a new form of hamilton jacobi bellman partial differential inequality of second order we give local and global versions of the first and second lyapunov theorems assuming the existence of a lower semicontinuous lyapunov function satisfying such an inequality in the viscosity sense an explicit formula for a stabilizing feedback is provided for affine systems with smooth lyapunov function several examples illustrate the theory

a converse lyapunov theorem for almost sure stabilizability we prove a converse lyapunov theorem for almost sure stabilizability and almost sure asymptotic stabilizability of controlled diffusions given a stochastic system a s stochastic open loop stabilizable at the origin we construct a lower semicontinuous positive definite function whose level sets form a local basis of viable neighborhoods of the equilibrium this result provides with the direct lyapunov theorems proved in a companion paper a complete lyapunov like characterization of the a s stabilizability

a viscosity approach to infinite dimensional hamilton jacobi equations arising in optimal control with state constraints we consider nonlinear optimal control problems with state constraints and nonnegative cost in infinite dimensions where the constraint is a closed set possibly with empty interior for a class of systems with a maximal monotone operator and satisfying certain stability properties of the set of trajectories that allow the value function to be lower semicontinuous we prove that the value function is a viscosity solution of the bellman equation and is in fact the minimal nonnegative supersolution

the heat equation shrinks embedded plane curves to round points soit c 0 s 1 r 2 une courbe lisse plongee dans le plan alors c s 1 0 t r 2 existe en satisfaisant c t k n ou k est la courbure de c et n est son vecteur unite normal entrant c t est lisse pour tout t il converge vers un point quand t t et sa forme limite quand t t est un cercle rond avec convergence dans norme c

stabilization by artificial wiener processes in this technical note we restage the positive effects of gaussian white noises for continuous time driftless input affine nonholonomic systems compared with previous studies we found that the positive effects are hidden in the shadows of nonuniqueness of stochastic integrals over multi dimensional wiener processes furthermore we found that the positive effects are artificially restaged by numerical calculations with use of euler scheme for ordinary differential equations our results demonstrate that the positive effects act as hidden independent control inputs for nonholonomic systems this implies that our results provide a constructive procedure for designing control inputs such that the origins become p th moment exponentially or finite time stable

characterization of lyapunov pairs in the nonlinear case and applications abstract the paper presents a characterization of the lyapunov pairs for a general initial value problem with a possibly multivalued m accretive operator on an arbitrary banach space by means of the contingent derivative related to the operator the proof is based on tangency and flow invariance arguments combined with a priori estimates and approximation the abstract results are applied to obtain precise a priori estimates for the mild solutions they readily lead to the existence of global solutions and various controllability properties important lyapunov pairs are pointed out in connection with specific problems

motion of level sets by mean curvature i we continue our investigation of the level set technique for describing the generalized evolution of hypersurfaces moving according to their mean curvature the principal assertion of this paper is a kind of reconciliation with the geometric measure theoretic approach pioneered by k brakke we prove that almost every level set of the solution to the mean curvature evolution pde is in fact aunit density varifold moving according to its mean curvature in particular a e level set is endowed with a kind of geometric structure the proof utilizes compensated compactness methods to pass to limits in various geometric expressions

stochastic lyapunov functions without differentiability at supposed equilibria abstract stochastic calculus indicates that allowing a lyapunov candidate function to be non differentiable at the origin helps its expected value to decrease to utilize this observation this paper investigates stability of stochastic systems supposing that the origin of the state space is an equilibrium before influenced by gaussian white noises for identifying such stability properties notions of instantaneous points and almost sure equilibria which are mutually exclusive are introduced this paper clarifies the relationship between the stability properties and develops their lyapunov type characterizations without differentiability at the origin discussions are given in relation to the notion of noise to state stability and this paper proposes a method to address both transient and global in time properties simultaneously with a single positive definite function even when stochastic noises prevent the origin from being an equilibrium

the heat equation shrinking convex plane curves soient m et m des varietes de riemann et f m m une application reguliere si m est une courbe convexe plongee dans le plan r 2 l equation de la chaleur contracte m a un point

stochastic target problems dynamic programming and viscosity solutions in this paper we define and study a new class of optimal stochastic control problems which is closely related to the theory of backward sdes and forward backward sdes the controlled process x nu y nu takes values in mathbb r d times mathbb r and a given initial data for x nu 0 then the control problem is to find the minimal initial data for y nu so that it reaches a stochastic target at a specified terminal time t the main application is from financial mathematics in which the process x nu is related to stock price y nu is the wealth process and nu is the portfolio we introduce a new dynamic programming principle and prove that the value function of the stochastic target problem is a discontinuous viscosity solution of the associated dynamic programming equation the boundary conditions are also shown to solve a first order variational inequality in the discontinuous viscosity sense this provides a unique characterization of the value function which is the minimal initial data for y nu

stabilization of stochastic nonlinear systems driven by noise of unknown covariance this paper poses and solves a new problem of stochastic nonlinear disturbance attenuation where the task is to make the system solution bounded by a monotone function of the supremum of the covariance of the noise this is a natural stochastic counterpart of the problem of input to state stabilization in the sense of sontag 1989 our development starts with a set of new global stochastic lyapunov theorems for an exemplary class of stochastic strict feedback systems with vanishing nonlinearities where the equilibrium is preserved in the presence of noise we develop an adaptive stabilization scheme based on tuning functions that requires no a priori knowledge of a bound on the covariance next we introduce a control lyapunov function formula for stochastic disturbance attenuation finally we address optimality and solve a differential game problem with the control and the noise covariance as opposing players for strict feedback systems the resulting isaacs equation has a closed form solution

lyapunov s second method for random dynamical systems abstract the method of lyapunov functions lyapunov s second or direct method was originally developed for studying the stability of a fixed point of an autonomous or non autonomous differential equation it was then extended from fixed points to sets from differential equations to dynamical systems and to stochastic differential equations we go one step further and develop lyapunov s second method for random dynamical systems and random sets together with matching notions of attraction and stability as a consequence lyapunov functions will also be random our test is that the extension be coherent in the sense that it reduces to the deterministic theory in case the noise is absent and that we can prove that a random set is asymptotically stable if and only if it has a lyapunov function several examples are treated including the stochastic lorenz system

viscosity solutions for a class of hamilton jacobi equations in hilbert spaces we study hamilton jacobi equations with an unbounded term in hilbert spaces we introduce a new variant of the notion of viscosity solution for a class of hamilton jacobi equations and obtain comparison and existence results for viscosity solutions we also examine the usefulness of the notion of viscosity solution in optimal control

some properties of viscosity solutions of hamilton jacobi equations abstract recently m g crandall and p l lions introduced the notion of viscosity solutions of scalar nonlinear first order partial differential equations viscosity solutions need not be differentiable anywhere and thus are not sensitive to the classical problem of the crossing of characteristics the value of this concept is established by the fact that very general existence uniqueness and continuous dependence results hold for viscosity solutions of many problems arising in fields of application the notion of a viscosity solution admits several equivalent formulations here we look more closely at two of these equivalent criteria and exhibit their virtues by both proving several new facts and reproving various known results in a simpler manner moreover by forsaking technical generality we hereby provide a more congenial introduction to this subject than the original paper author

fattening and comparison principle for level set equations of mean curvature type in this paper we give several applications of the discrete game approach to partial differential equations pdes we first present a rigorous game theoretic proof of fattening phenomenon for motion by curvature with figure eight shaped initial curves without using parabolic pde theory the proof is based on a comparison between the game value and its inverse accompanied by the example of figure eight our second result shows for the stationary equation of mean curvature type in an arbitrary region omega that fattening of positive curvature flow with initial surface partial omega causes loss of the weak comparison principle which partially answers an open question posed by kohn and serfaty in 2006 in addition we prove the existence of solutions of the stationary problem and its game approximation in the absence of comparison principles but under regularity conditions of the flow the main difference between our games and those in other papers is that we take the domain perturbation into consideration

a geometrical approach to the study of unbounded solutions of quasilinear parabolic equations in this article we are interested in the existence and uniqueness of solutions for quasilinear parabolic equations set in the whole space n we consider in particular cases when there is no restriction on the growth or the behavior of these solutions at infinity our model equation is the mean curvature equation for graphs for which ecker and huisken have shown the existence of smooth solutions for any locally lipschitz continuous initial data we use a geometrical approach which consists in seeing the evolution of the graph of a solution as a geometric motion which is then studied by the so called level set approach after determining the right class of quasilinear parabolic pdes which can be taken into account by this approach we show how the uniqueness for the original pde is related to fattening phenomena in the level set approach existence of solutions is proved using a local l bound obtained by using in an essential way the level set approach finally we apply these results to convex initial data and prove existence and comparison results in full generality i e without restriction on their growth at infinity

turing bifurcation in a reaction diffusion system with density dependent dispersal abstract motivated by the recent finding n kumar g m viswanathan v m kenkre physica a 388 2009 3687 that the dynamics of particles undergoing density dependent nonlinear diffusion shows sub diffusive behaviour we study the turing bifurcation in a two variable system with this kind of dispersal we perform a linear stability analysis of the uniform steady state to find the conditions for the turing bifurcation and compare it with the standard turing condition in a reaction diffusion system where dispersal is described by simple fickian diffusion while activator inhibitor kinetics are a necessary condition for the turing instability as in standard two variable systems the instability can occur even if the diffusion constant of the inhibitor is equal to or smaller than that of the activator we apply these results to two model systems the brusselator and the gierer meinhardt model

waiting and propagating fronts in nonlinear diffusion abstract a nonlinear diffusion equation is considered where the nonlinearity arises due to the variation of the diffusion coefficient with concentration if this diffusion coefficient vanishes at a certain value of the concentration this produces a number of interesting interfacial effects such as the finite propagation speed of interfaces in contrast to the infinite propagation speed of linear diffusion and waiting time behavior in which the interfaces can remain stationary for a finite time before beginning to move a relatively simple singular perturbation analysis is given to explain these effects

lyapunov pairs for multi valued semi linear evolutions abstract we give here a characterization of a lyapunov pair for a multi valued semi linear evolution equation on a banach space by means of an appropriate lower contingent derivative the contingent derivative introduced in this paper is related to a new concept of tangency introduced recently in o c rj m necula i i vrabie necessary and sufficient conditions for viability for semilinear differential inclusions trans amer math soc 361 2009 343 390 as an application we give a controllability result and a lipschitz estimate of the corresponding minimum time function under a petrov like condition