darboux transformation with dihedral reduction group we construct the darboux transformation with dihedral reduction group for the 2 dimensional generalisation of the periodic volterra lattice the resulting backlund transformation can be viewed as a nonevolutionary integrable differential difference equation we also find its generalised symmetry and the lax representation for this symmetry using formal diagonalisation of the darboux matrix we obtain local conservation laws of the system

multiphase semiclassical approximation of an electron in a one dimensional crystalline lattice i homogeneous problems we present a computational approach for the wkb approximation of the wave function of an electron moving in a periodic one dimensional crystal lattice we derive a nonstrictly hyperbolic system for the phase and the intensity where the flux functions originate from the bloch spectrum of the schrodinger operator relying on the framework of the multibranch entropy solutions introduced by brenier and corrias we compute efficiently multiphase solutions using well adapted and simple numerical schemes in this first part we present computational results for vanishing exterior potentials which demonstrate the effectiveness of the proposed method

finite fuel problem in nonlinear singular stochastic control we investigate via the dynamic programming approach a finite fuel nonlinear singular stochastic control problem of bolza type we prove that the associated value function is continuous and that its continuous extension to the closure of the domain coincides with the value function of a nonsingular control problem for which we prove the existence of an optimal control moreover such a continuous extension is characterized as the unique viscosity solution of a quasi variational inequality with suitable boundary conditions of mixed type

convergence rate for a curse of dimensionality free method for a class of hjb pdes in previous work of the first author and others max plus methods have been explored for solution of first order nonlinear hamilton jacobi bellman partial differential equations hjb pdes and corresponding nonlinear control problems although max plus basis expansion and max plus finite element methods can provide substantial computational speed advantages they still generally suffer from the curse of dimensionality here we consider hjb pdes where the hamiltonian takes the form of a pointwise maximum of linear quadratic forms the approach to solution will be rather general but in order to ground the work we consider only constituent hamiltonians corresponding to long run average cost per unit time optimal control problems for the development we consider a previously obtained numerical method not subject to the curse of dimensionality the method is based on construction of the dual space semigroup corresponding to the hjb pde this dual space semigroup is constructed from the dual space semigroups corresponding to the constituent linear quadratic hamiltonians the dual space semigroup is particularly useful due to its form as a max plus integral operator with kernel obtained from the originating semigroup one considers repeated application of the dual space semigroup to obtain the solution although previous work indicated that the method was not subject to the curse of dimensionality it did not indicate any error bounds or convergence rate here we obtain specific error bounds

dynamic programming for nonlinear systems driven by ordinaryand impulsive controls a dynamic programming approach is considered for a class of minimum problems with impulses the minimization domain consists of trajectories satisfying an ordinary differential equation whose right hand side depends not only on a measurable control v but also on a second control u and on its time derivative dot u for this reason the control u and the differential equation are called impulsive the value function of the considered minimum problem turns out to depend on the time the state the u variable and the variation allowed to the impulsive control it is shown that the value function satisfies in a generalized sense a dynamic programming equation dpe which is obtained from a dynamic programming principle involving space time trajectories moreover the value function is the unique map solving equation dpe satisfying either an inequality condition or a supersolution condition at each point of the boundary incidentally this extends a result by barron jensen and menaldi nonlinear anal 21 1993 pp 241 268 where the impulsive control is scalar monotone and the corresponding vector field is independent of the state variable next a maximum principle is proved and the well known relationship between adjoint variables and value function is suitably extended to impulsive control systems a fully elaborated example concludes the paper

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

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

numerical approximations for stochastic differential games the markov chain approximation method is a widely used robust relatively easy to use and efficient family of methods for the bulk of stochastic control problems in continuous time for reflected jump diffusion type models it has been shown to converge under broad conditions and there are good algorithms for solving the numerical problems if the dimension is not too high versions of these methods have been used in applications to various two player differential and stochastic dynamic games for a long time and proofs of convergence are available for some cases mainly using pde type techniques in this paper purely probabilistic proofs of convergence are given for a broad class of such problems where the controls for the two players are separated in the dynamics and cost function and which cover a substantial class not dealt with in previous works discounted and stopping time cost functions are considered finite horizon problems and problems where the process is stopped on first hitting an a priori given boundary can be dealt with by adapting the methods of h j kushner and p dupuis numerical methods for stochastic control problems in continuous time 2nd ed springer verlag berlin new york 2001 as done in this paper for the treated problems the essential conditions are the weak sense existence and uniqueness of solutions an almost everywhere continuity condition and that a weak local consistency condition holds almost everywhere for the numerical approximations just as for the control problem there are extensions to problems with controlled variance and jumps

continuity of the upper and lower value of slow growth differential games abstract some continuity properties are proved for a zero sum differential game with an unbounded control and no coercivity assumptions

optimal switching for ordinary differential equations we consider the problem of controlling an ordinary differential equation subject to positive switching costs and show in particular that the value functions form the viscosity solution cf 6 7 of the dynamic programming quasi variational inequalities this interpretation allows for a rigorous application of various dynamic programming techniques

repeated games with lack of information on one side we introduce the dual differential game of a repeated game with lack of information on one side as the natural continuous time version of the dual game introduced by de meyer 1996 a traditional way to study the value of differential games is through discrete time approximations here we follow the opposite approach we identify the limit value of a repeated game in discrete time as the value of a differential game namely we use the recursive structure for the finitely repeated version of the dual game to construct a differential game for which the upper values of the uniform discretization satisfy precisely the same property the value of the dual differential game exists and is the unique viscosity solution of a first order derivative equation which appears implicitely in de meyer and rosenberg 1999 with a limit condition we identify the solution by translating viscosity properties in the primal

reduction groups and related integrable difference systems of nonlinear schrodinger type we extend the reduction group method to the lax darboux schemes associated with nonlinear schrodinger type equations we consider all possible finite reduction groups and construct corresponding lax operators darboux transformations hierarchies of integrable differential difference equations integrable partial difference systems and associated scalar partial difference equations

robust optimal switching control for nonlinear systems we formulate a robust optimal control problem for a general nonlinear system with finitely many admissible control settings and with costs assigned to switching of controls we formulate the problem both in an l2 gain dissipative system framework and in a game theoretic framework we show that under appropriate assumptions a continuous switching storage function is characterized as a viscosity supersolution of the appropriate system of quasi variational inequalities the appropriate generalization of the hamilton jacobi bellman isaacs equation for this context and that the minimal such switching storage function is equal to the continuous switching lower value function for the game finally we show how a prototypical example with one dimensional state space can be solved by a direct geometric construction

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

automorphic lie algebras with dihedral symmetry the concept of automorphic lie algebras arises in the context of reduction groups introduced in the early 1980s in the field of integrable systems automorphic lie algebras are obtained by imposing a discrete group symmetry on a current algebra of krichever novikov type past work shows remarkable uniformity between algebras associated to different reduction groups for example if the base lie algebra is sl2 c and the poles of the automorphic lie algebra are restricted to an exceptional orbit of the symmetry group changing the reduction group does not affect the lie algebra structure in this research we fix the reduction group to be the dihedral group and vary the orbit of poles as well as the group action on the base lie algebra we find a uniform description of automorphic lie algebras with dihedral symmetry valid for poles at exceptional and generic orbits

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

comparison theorems for viscosity solutions of a system of quasivariational inequalities with application to optimal control with switching costs in this paper we prove a comparison principle between a viscosity sub and supersolution for a system of quasivariational inequalities and apply it to show that a continuous lower value vector function of an optimal switching cost control problem is characterized as the minimal nonnegative continuous viscosity supersolution of the sqvi

some applications of nevanlinna theory to mathematical logic identities of exponential functions in this paper we study identities between certain functions of many variables that are constructed by using the elementary functions of addition x y multiplication x v and two place exponentiation x for a restricted class of such functions we show that every true identity follows from the natural set of eleven axioms the rates of growth of such functions in the case of a single independent variable x as x are also studied and we give an algorithm for the hardy relation of eventual domination again for a restricted class of functions value distribution of analytic functions of one and of several complex variables especially the nevanlinna characteristic plays a major role in our proofs introduction to make precise the questions we consider it is necessary to introduce a formal language containing the expressions which define the elementary functions to be studied this language has variables x x2 a set s of constants representing a given set of fixed numbers and function symbols for addition multiplication and exponentiation the terms that is the function defining expres sions are obtained inductively starting with the variables and constants and continuing according to the rule if t and s are terms then so are t s t s and t we follow some standard conventions for dropping parentheses and below let us refer to these as the exponential terms over s in most cases considered here the constants in s represent positive real numbers in such a case every term t represents a function which is defined for all positive real values of the variables which occur in it we write t s to mean that t and s are terms which define exactly the same function for positive real values of the variables appearing in t or s it is necessary to distinguish between this function equality relation t s and a formal expression of equality between t and s which we write in the form t s the relation t s is a mathematical relation between certain elementary functions the expression t s is a purely formal expression which is susceptible to formal proof within various axiomatic systems a closely related topic concerns the orders of growth of functions of one variable defined by exponential terms if t and s are such terms which contain only the single

adaptation and optimal chemotactic strategy for e coli extending the classic works of berg and purcell on the biophysics of bacterial chemotaxis we find the optimal chemotactic strategy for the peritrichous bacterium ital e coli in the high and low signal to noise ratio limits the optimal strategy depends on properties of the environment and properties of the individual bacterium and is therefore highly adaptive we review experiments relevant to testing both the form of the proposed strategy and its adaptability and propose extensions of them which could test the limits of the adaptability in this simplest sensory processing system copyright ital 1998 ital the american physical society

l sub 2 gain analysis of nonlinear systems and nonlinear state feedback h sub infinity control previously obtained results on l2 gain analysis of smooth nonlinear systems are unified and extended using an approach based on hamilton jacobi equations and inequalities and their relation to invariant manifolds of an associated hamiltonian vector field on the basis of these results a nonlinear analog is obtained of the simplest part of a state space approach to linear h sub infinity control namely the state feedback h sub infinity optimal control problem furthermore the relation with h sub infinity control of the linearized system is dealt with

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

inverse optimal design of input to state stabilizing nonlinear controllers we show that input to state stabilizability as defined by sontag 1989 1995 is both necessary and sufficient for the solvability of a hamilton jacobi isaacs equation associated with a meaningful differential game problem similar to but more general than the nonlinear h sub spl infin problem the significance of the result stems from the fact that constructive solutions to the input to state stabilization problem are available presented in the paper and that as shown here inverse optimal controllers possess margins on input to state stability against a certain class of input unmodeled dynamics rather than completion of squares the main tools in our analysis are legendre fenchel transformations and the general form of young s inequality

darboux transformations finite reduction groups and related yang baxter maps in this paper we construct yang baxter yb maps using darboux matrices which are invariant under the action of finite reduction groups we present six dimensional yb maps corresponding to darboux transformations for the nonlinear schrodinger nls equation and the derivative nonlinear schrodinger dnls equation these yb maps can be restricted to four dimensional yb maps on invariant leaves the former are completely integrable and they also have applications to a recent theory of maps preserving functions with symmetries fordy a and kassotakis a 2013 j phys a math theor 46 205201 we give a six dimensional yb map corresponding to the darboux transformation for a deformation of the dnls equation we also consider vector generalizations of the yb maps corresponding to the nls and dnls equations

risk sensitive control with hara utility a control methodology based on the hyperbolic absolute risk averse hara utility function is presented as an alternative to the exponential of an integral approach to finding robust controllers the work is inspired by the intuition that hara controllers while being robust may give better performance than exponential controllers in normal situations the hara problem is shown to be equivalent to a certain differential game and the asymptotic properties of the hara problem and this differential game are studied as an example a linear quadratic hara problem is studied where the problem of finding a robust hara controller is proved to be equivalent to solving a standard linear quadratic problem for a system with a higher noise intensity this reveals an interesting relationship between robustness and uncertainty

computation of the h norm for nonlinear systems a convergence result in this paper we show that h norm for a class of nonlinear continuous time systems is the limit of the h norm for approximating discrete time systems the convergence result is proved by means of a characterization of the h norm in terms of the value for an ergodic control problem

on singular stochastic control problems for diffusion with jumps we consider some cases of control problems for diffusion processes with jumps when the payoff functional does not depend explicitly on the control we prove the continuity of the optimal cost with a quasi variational inequality interpreting the problems as the limit of an impulse control problem when the cost of impulse tends to zero moreover we show the existence of an optimal control for some particular situations

additive control of stochastic linear systems with finite horizon we consider a dynamic system whose state is governed by a linear stochastic differential equation with time dependent coefficients the control acts additively on the state of the system our objective is to minimize an integral cost which depends upon the evolution of the state and the total variation of the control process it is proved that the optimal cost is the unique solution of an appropriate free boundary problem in a space time domain by using some decomposition arguments the problems of a two sided control i e optimal corrections and the case with constraints on the resources i e finite fuel can be reduced to a simpler case of only one sided control i e a monotone follower these results are applied to solving some examples by the so called method of similarity solutions

reductions of n wave interactions related to low rank simple lie algebras i z2 reductions the analysis and the classification of all reductions for the nonlinear evolution equations solvable by the inverse scattering method is an interesting and still open problem we show how the second order reductions of the n wave interactions related to low rank simple lie algebras g can be embedded also in the weyl group of g this allows us to display along with the well known ones a number of new types of integrable n wave systems some of the reduced systems find applications to nonlinear optics

twisted hierarchies associated with the generalized sine gordon equation twisted u and twisted u k hierarchies are soliton hierarchies introduced by terng to find higher flows of the generalized sine gordon equation twisted o j j o j o j hierarchies are among the most important classes of twisted hierarchies in this paper we derive explicit interesting first and higher flows of twisted o j j o j o j hierarchies justify that the one dimensional systems of twisted o j j o j o j hierarchies for j iq n q 1 q n 1 called the generalized sinh gordon equations are the gauss codazzi equations for n dimensional timelike submanifolds with constant sectional curvature 1 and index q in pseudo euclidean 2n 1 dimensional space r2q 12n 1 with index 2q 1 furthermore a unified treatment of the inverse scattering theory for twisted o j j o j o j hierarchies is provided

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

a multistate multicontrol problem with unbounded controls this paper gives the optimal synthesis for a two dimensional singular control problem of the vidale wolfe type the controls take values in an unbounded set moreover the optimal feedback control turns out to be impulsive on certain regions and the order in which these impulses occur is important a parameterization of time is introduced into the problem this helps to elucidate the optimal synthesis to prove its optimality by the verification method and to design suboptimal physical approximations to the optimal impulsive control

existence of solutions to the hamilton jacobi bellman equation under quadratic growth conditions abstract in this paper we show the existence of solutions with quadratic growth to hamilton jacobi bellman equations we assume that the hamiltonian has the quadratic growth both in x and p we obtain the explicit lipschitz bound of solutions in the weighted sup norm also we discuss the vanishing viscosity limit of solutions and show the existence of viscosity solutions to the resulting hamilton jacobi equation

analytical approximation methods for the stabilizing solution of the hamilton jacobi equation in this paper two methods for approximating the stabilizing solution of the hamilton jacobi equation are proposed using symplectic geometry and a hamiltonian perturbation technique as well as stable manifold theory the first method uses the fact that the hamiltonian lifted system of an integrable system is also integrable and regards the corresponding hamiltonian system of the hamilton jacobi equation as an integrable hamiltonian system with a perturbation caused by control the second method directly approximates the stable flow of the hamiltonian systems using a modification of stable manifold theory both methods provide analytical approximations of the stable lagrangian submanifold from which the stabilizing solution is derived two examples illustrate the effectiveness of the methods

on the classification of automorphic lie algebras the problem of reduction of integrable equations can be formulated in a uniform way using the theory of invariants this provides a powerful tool of analysis and it opens the road to new applications of automorphic lie algebras beyond the con text of integrable systems in this paper it is shown that sl2 c based automorphic lie algebras associated to the icosahedral group i the octahedral group o the tetrahedral group t and the dihedral group dn are isomorphic the proof is based on techniques from classical invariant theory and makes use of clebsch gordan decomposition and transvectants molienfunctionsandthetrace form thisresultprovidesacompleteclas sification of sl2 c based automorphic lie algebras associated to finite groups when the group representations are chosen to be the same and it is a crucial step towards the complete classification of automorphic lie algebras

a maximum principle for optimal processes with discontinuous trajectories a maximum principle is proved which governs solutions to dynamic optimization problems in which the controls driving the system may be impulsive and give rise to discontinuous trajectories the approach which involves approximating the problem by a conventional one and using ekeland s theorem is new it permits us to weaken very considerably the hypotheses under which maximum principles for such problems have previously been proved

blow up time involved with perturbed hamilton jacobi equations in this paper we consider the evolution of positive bounded uniformly continuous data u 0 by perturbed equations like under general assumptions on u 0 existence uniqueness and regularity of the evolution u in the set are studied where the blow up function is given by the exact blow up rate of u is obtained uniqueness regularity decay at infinity of the function as well a s a representation formula for the case m 1 are also proved

generalized viscosity solutions for hamilton jacobi equations with time measurable hamiltonians the objective of this paper is to extend the idea of viscosity solution for nonlinear first order hamilton jacobi equations u h t x d u 0 with time continuous h to time measurable hamiltonians we are primarily motivated by the fact that control problems for which the value function is the viscosity solution of a hamilton jacobi equation should not be restric ted to continuity in time in barron and jensen 1 we were confronted with a linear hamilton jacobi equation arising in the proof of the pon tryagin which had time measurable coeflicients the existing theory of viscosity solutions did not apply to even this case moreover important models of controlled first order hamiltonjacobi equations require time measurable controls for example models of controlled traffic flow or queueing processes might be of this type we are about then to extend the definition of viscosity solution to generalized viscosity solution applying to time measurable hamiltonians in this paper we give the definition and derive the corresponding uni queness result with an implicit domain of dependence consequent the fun damental technique used in the proof is the so called blow up method which has also been used elsewhere s although not in the generality discussed here this method effectively freezes the point to x0 under consideration and essentially removes it from the problem of course t is

a uniqueness result for p monotone viscosity solutions of hamilton jacobi equations in bounded domains we consider a class of hamilton jacobi equations h x du x 0 with no u dependence and with continuity properties consistent with recent applications in queueing theory continuous viscosity solutions are considered in a compact polyhedral domain with oblique derivative neumann type boundary conditions comparison and uniqueness results are presented which use monotonicity of h x p in the p variable for values of p in the appropriate sub and superdifferential sets of the solution u x several examples illustrate the results

ℒ2 gain tuning of variable structure siso systems of relative degree n abstractthe present paper extends the l2 gain analysis towards variable structure systems vss with sliding mode sm dynamics the developed analysis is applied to feedback linearisable vss driven by standard sm algorithms of the first order constructive parameter tuning of the resulting closed loop system to possess the l2 gain less than a desired attenuation level is additionally deduced the proposed tuning is shown to be capable not only of rejecting matched uniformly bounded disturbances but also of attenuating unbounded ones including mismatched disturbances numerical simulations of an inverted pendulum driven by a first order sm controller are involved to support the theory in the presence of the position measurement inaccuracy being a source of a mismatched disturbance

h control stabilization and input output stability of nonlinear systems with homogeneous properties new relations are established between h control and stabilization and between input output stability and lyapunov stability of nonlinear systems with homogeneous properties these results can be applied to h controller design and stability analysis

module structure of the center of the universal central extension of a genus zero krichever novikov algebra abstract we describe how the center of the universal central extension of the genus zero krichever novikov current algebra decomposes as a direct sum of irreducible modules for automorphism group of the coordinate ring of this algebra

hamilton jacobi bellman equation and feedback synthesis for impulsive control there is an increasing number of applications whose trajectories are better modeled by discontinuous or impulsive trajectories thus we explore optimality conditions for impulsive control system expressed in terms of an hamilton jacobi bellman equation we use a measure driven differential inclusion to model the impulsive behavior since it provides a formal framework in which the control space is complete additionally we use the impulsive euler solution and invariance results to derive feedback optimal control synthesis

geometric existence theory for the control affine h problem abstract we consider the differential game formulation of the nonlinear state feedback h control problem in which the control term enters linearly in the dynamics and quadratically in the cost under well known conditions on the linearisation of this problem around the equilibrium point at the origin there exists a stable lagrangian manifold this manifold has a generating function s quadratic at infinity a lusternick schnirelman minimax construction produces from s a lipschitz function w over state space we show that for problems in general position w is the lower value function for the h problem and prove existence of a weak globally optimal set valued feedback solution in terms of w the generalised gradient of w this feedback generalises to a maximal region over which is simply connected the classical smooth feedback defined on the neighbourhood of the origin over which has a well defined projection onto state space

brief paper robust stabilization of nonlinear systems via normalized coprime factor representations a reasonably complete theory for the synthesis of robust controllers for a broad class of nonlinear systems is now available we use this theory to generalize the linear theory of normalized coprime factor robustness optimization to the case of affine input nonlinear systems in particular we show that the equilibrium controller may be characterized in terms of the stabilizing and destabilizing solutions of the hamilton jacobi equation used to calculate the normalized right coprime factors of the plant we also show that the optimal robustness margin email protected 1 mn 2 hgeneralizes to the nonlinear case in preparation for the nonlinear analysis we review the linear case in a way which motivates our approach to the nonlinear case and highlights the parallels with it

persistence and stability of solutions of hamilton jacobi equations abstract given a convergent sequence of hamiltonians h n and a convergent sequence of initial data g n for the first order evolutionary hamilton jacobi equation we look for conditions ensuring that the sequences u n and v n of lax solutions and hopf solutions respectively converge the convergences we deal with are variational convergences we take advantage of several recent results giving criteria for the continuity of usual operations

a twisted integrable hierarchy with d2 symmetry a loop algebra approach to the gerdjikov mikhailov valchev gmv equation is provided to exploit the associated twisted integrable structure and a new twisted integrable hierarchy is discovered using the twisted loop algebra structure we obtain a transparent treatment of the associated scattering and inverse scattering theory and solve the initial value problem for the gmv equation

interplay of chemotaxis and chemokinesis mechanisms in bacterial dynamics motivated by observations of the dynamics of myxococcus xanthus we present a self interacting random walk model that describes the competition between chemokinesis and chemotaxis cells are constrained to move in one dimension but release a chemical chemoattractant at a steady state the bacteria sense the chemical that they produce the probability of direction reversals is modeled as a function of both the absolute level of chemoattractant sensed directly under each cell as well as the gradient sensed across the length of the cell if the chemical does not degrade or diffuse rapidly the one dimensional trajectory depends on the entire past history of the trajectory we derive the corresponding fokker planck equations use an iterative mean field approach that we solve numerically for short times and perform extensive monte carlo simulations of the model cell positional distributions and the associated moments are computed in this feedback system average drift and mean squared displacements are found crossover behaviors among different diffusion regimes are found

darboux transformations with tetrahedral reduction group and related integrable systems in this paper we derive new two component integrable differential difference and partial difference systems by applying a lax darboux scheme to an operator formed from an x1d530 x1d529 3 based automorphic lie algebra the integrability of the found systems is demonstrated via lax pairs and generalised symmetries

existence theorems for optimal control and calculus of variations problems where the states can jump we examine optimal control and calculus of variations problems where the states can be discontinuous in the control theory setting this is caused by allowing impulse controls conditions under which optimal solutions will exist for these problems are determined after we derive suitable objective functionals that can handle the jump terms

the pontryagin maximum principle from dynamic programming and viscosity solutions to first order partial differential equations on demontre le principe de maximum de pontryagin pour le probleme de lagrange de commande optimale en utilisant le fait que la fonction valeur du probleme est la solution viscosite de l equation de hamilton jacobi bellman associee

reduction groups and automorphic lie algebras we study a new class of infinite dimensional lie algebras which has important applications to the theory of integrable equations the construction of these algebras is very similar to the one for automorphic functions and this motivates the name automorphic lie algebras for automorphic lie algebras we present bases in which they are quasigraded and all structure constants can be written out explicitly these algebras have useful factorisations on two subalgebras similar to the factorisation of the current algebra on the positive and negative parts

asymptotic behavior of the first order obstacle problem on considere l equation aux derivees partielles d ordre 1 non lineaire max u x x u x g x du x f x 0 x ou est un sous ensemble ouvert de r d est un parametre positif g g 1 g d est un champ vectoriel sur r d du est le gradient de la fonction inconnue u et f et sont des fonctions a valeurs reelles donnees on etudie le comportement de cette equation quand 0

randomly curved runs interrupted by tumbling a model for bacterial motion received 23 february 2005 published 24 august 2005small bacteria are strongly buffeted by brownian forces that make completely straight runs impossible amodel for bacterial motion is formulated in which the effects of uctuational forces and torques on the runphase are taken into account by using coupled langevin equations an integrated description of the motion including runs and tumbles is then obtained by the use of convolution and laplace transforms the propertiesof the velocity velocity correlation function of the mean displacement and of the two relevant diffusioncoef cients are examined in terms of the bacterial sizes and of the magnitude of the propelling forces forbacteria smaller than

a variational inequality for measurement feedback almost dissipative control in this paper an optimal measurement feedback control problem that yields an almost or practically dissipative closed loop system is considered that is the aim is to consider optimal control problems which when solved yield a closed loop system which almost satisfies a dissipation inequality the main idea is that by weakening the required dissipation inequality a broader class of open loop systems and controllers are admissible leading to broader application in obtaining the main results of this paper dynamic programming is applied to the optimal control problem of interest to derive a variational inequality that generalizes the information state based partial differential equation associated with measurement feedback nonlinear dissipative control this variational inequality can in principle be used to derive the optimal controller in the special case of certainty equivalence an explicit solution of the variational inequality exists and is a functional of the solution of the corresponding optimal state feedback almost dissipative control problem

max plus summation of fenchel transformed semigroups for solution of nonlinear bellman equations abstract max plus methods have been explored for solution of first order nonlinear hamilton jacobi bellman partial differential equations hjb pdes and corresponding nonlinear control problems these methods exploit the max plus linearity of the associated semigroups in particular although the problems are nonlinear the semigroups are linear in the max plus sense these methods have been used successfully to compute solutions although they provide certain advantages they still generally suffer from the curse of dimensionality the natural analog to the laplace transform in ordinary spaces is the fenchel transform over max plus spaces the range space being referred to as the dual space one can transform the semigroup operators into operators on the dual space there are natural operations on the transformed operators which may be used to construct solutions of nonlinear control problems natural building blocks correspond to transforms of operators for linear quadratic problems in this paper a method for exploiting operations in the fenchel transform space is used to develop a method for certain problems such that the computational growth in the most time consuming portion of the computations can be hugely reduced although the curse of dimensionality is not entirely avoided the computational cost reductions are very high for some classes of problems

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

max plus eigenvector representations for solution of nonlinear h sub spl infin problems basic concepts the h sub spl infin problem for a nonlinear system is considered the corresponding dynamic programming equation is a fully nonlinear first order steady state partial differential equation pde possessing a term which is quadratic in the gradient the solutions are typically nonsmooth and further there is nonuniqueness among the class of viscosity solutions in the case where one tests a feedback control to see if it yields an h sub spl infin controller the pde is a hamilton jacobi bellman equation in the case where the optimal feedback control is being determined as well the problem takes the form of a differential game and the pde is in general an isaacs equation the computation of the solution of a nonlinear steady state first order pde is typically quite difficult in this paper we develop an entirely new class of methods for obtaining the correct solution of such pdes these methods are based on the linearity of the associated semigroup over the max plus or in some cases min plus algebra in particular solution of the pde is reduced to solution of a max plus or min plus eigenvector problem for known unique eigenvalue 0 the max plus multiplicative identity it is demonstrated that the eigenvector is unique and that the power method converges to it an example is included

on some cheap control problems for diffusion processes on considere plusieurs cas de problemes de commande de processus de diffusion dans lesquels la fonctionnelle de cout ne depend pas explicitement de la commande on demontre la continuite de la fonction cout optimale et on en donne une caracterisation a l aide d une inegalite quasivariationnelle interpretant le probleme comme la limite d un probleme de commande impulsionnelle dont le cout tend vers zero

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

viability property for a backward stochastic differential equation and applications to partial differential equations in the present paper we study conditions under which the solutions of a backward stochastic differential equation remains in a given set of constraints this property is the so called viability property in a separate section this condition is translated to a class of partial differential equations

reductions of integrable equations dihedral group we discuss the algebraic and analytic structure of rational lax operators with algebraic reductions of lax equations we associate a reduction group a group of automorphisms of the corresponding infinite dimensional lie algebra we present a complete study of dihedral reductions for sl 2 c lax operators with simple poles and corresponding integrable equations in the last section we give three examples of dihedral reductions for sl n c lax operators

deterministic impulse control problems we prove that the optimal cost function of a deterministic impulse control problem is the unique viscosity solution of a first order hamilton jacobi quasi variational inequality in mathbb r n