optimal switching for partial differential equations i abstract optimal switching for an abstract cauchy problem is considered a type of approximating problems with value functions u 0 is studied we prove that each u is the unique viscosity solution of corresponding hamilton jacobi system that u converges to the value function u of the original problem as goes to and that via u an optimal control for the switching problem is determined

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

infinite horizon value functions in the wasserstein spaces abstract we perform a systematic study of optimization problems in the wasserstein spaces that are analogs of infinite horizon deterministic control problems we derive necessary conditions on action minimizing paths and present a sufficient condition for their existence we also verify that the corresponding generalized value functions are a type of viscosity solution of a time independent hamilton jacobi equation in the space of probability measures finally we prove a special case of a conjecture involving the subdifferential of generalized value functions and their relation to action minimizing paths

a comparison principle for hamilton jacobi equations related to controlled gradient flows in infinite dimensions we develop new comparison principles for viscosity solutions of hamilton jacobi equations associated with controlled gradient flows in function spaces as well as the space of probability measures our examples are optimal control of ginzburg landau and fokker planck equations they arise in limit considerations of externally forced non equilibrium statistical mechanics models or through the large deviation principle for interacting particle systems our approach is based on two key ingredients an appropriate choice of geometric structure defining the gradient flow and a free energy inequality resulting from such gradient flow structure the approach allows us to handle hamiltonians with singular state dependency in the nonlinear term as well as hamiltonians with a state space which does not satisfy the radon nikodym property in the case where the state space is a hilbert space the method simplifies existing theories by avoiding the perturbed optimization principle

a maximum principle for evolution hamilton jacobi equations on riemannian manifolds abstract we establish a maximum principle for viscosity subsolutions and supersolutions of equations of the form u t f t d x u 0 u 0 x u 0 x where u 0 m r is a bounded uniformly continuous function m is a riemannian manifold and f 0 t m r this yields uniqueness of the viscosity solutions of such hamilton jacobi equations

some results on non linear optimal control problems and hamilton jacobi equations in infinite dimensions abstract the optimal control of a distributed parameter system is connected to the solution of the corresponding hamilton jacobi equation this is a first order equation in infinite dimensions with discontinuous coefficients we study the hamilton jacobi equation of a system governed either by a semilinear or by a monotone dynamics replacing the unbounded terms of this equation by their yosida approximations we prove convergence of the approximate solutions to the value function of the original problem by using the uniqueness result for viscosity solutions examples and applications are included

nonsmooth analysis and hamilton jacobi equations on riemannian manifolds we establish some perturbed minimization principles and we develop a theory of subdifferential calculus for functions defined on riemannian manifolds then we apply these results to show existence and uniqueness of viscosity solutions to hamilton jacobi equations defined on riemannian manifolds

hamilton jacobi equations in infinite dimensions i uniqueness of viscosity solutions soient un espace de banach v son dual v et des solutions d une equation h x u du 0 dans etablies dans un sous ensemble de v h v r v r est continue on demontre que des solutions classiques sont determinees de facon unique par leurs valeurs limites on montre que l extension de la notion de solutions viscosite aux espaces de banach reussit dans des espaces munis de la propriete de radon nikodym

boundary region based rough sets and uncertainty measures in the approximation space the approximation operators play an important role in rough set theory which are mainly defined by means of neighborhood systems in this paper firstly we try to propose a class of novel definitions of the approximation operators via a predefined boundary region based on a binary relation then we compare the proposed concepts with the originals the necessary and sufficient conditions of their equipollence are investigated secondly we give the definitions of boundary region based on a covering by employing the boundary a class of novel definitions of the approximation operators based on a covering are proposed it is shown that the proposed operators are equivalent to a class of covering approximation operators introduced by zakowski thirdly the relationship between general binary relations and coverings based approximation operators is investigated with the aid of the novel boundary region finally the more reasonable characterizations of the accuracy and the roughness are proposed by employing the boundary operators meanwhile we study uncertainty measures of approximation spaces based on a partition and a covering

microscopic derivations of several hamilton jacobi equations in infinite dimensions and large deviation of stochastic systems abstract we consider hamilton jacobi equations which characterize optimal controlled partial differential equations of the following types the allen cahn equation the cahn hilliard equation a nonlinear fokker planck equation and a vlasov fokker planck equation in each of the examples the optimal control problem and its associated cost functional can be derived as limit from a microscopically defined stochastic system using the probabilistic theory of large deviation the physical context here makes it natural to derive a free energy inequality which is very useful in proving the well posedness of the hamilton jacobi equation the article is written using informal arguments rigorous results will appear elsewhere

viscosity solutions of hamilton jacobi equations in infinite dimensions iv hamiltonians with unbounded linear terms in this work we show how to interpret s and e in the viscosity sense when the solutions may be expected to be weakly sequentially continuous in particular we introduce notions which allow us to handle the terms and for which we can obtain existence and uniqueness results in a variety of circumstances

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

on a class of first order hamilton jacobi equations in metric spaces abstract we establish well posedness of a class of first order hamilton jacobi equation in geodesic metric spaces the result is then applied to solve a hamilton jacobi equation in the wasserstein space of probability measures which arises from the variational formulation of a compressible euler equation

hamilton jacobi equations in infinite dimensions iii on etudie des equations de la forme f x u du 0 dans ou est un sous ensemble ouvert d un espace de banach v la fonction inconnue u r est continue du x est la derivee de frechet de u en x et f est une fonction non lineaire qui est une application continue de r v dans r on introduit la notion de solution viscosite stricte afin d eliminer l hypothese que v a la propriete de radon nikodym

viscosity solutions of fully nonlinear second order equations and optimal stochastic control in infinite dimensions iii uniqueness of viscosity solutions for general second order equations we prove here the uniqueness and existence of viscosity solutions for a general class of fully nonlinear second order equations in an infinite dimensional hilbert space the uniqueness is proved using the so called inf and sup convolutions which allow some reduction to finite dimensional problems

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

infinite dimensional hamilton jacobi equations with large zeroth order coefficient abstract we establish the existence and uniqueness of the c 1 1 solution of an infinite dimensional hamilton jacobi equation defined on the space of probability measures on r d when the zeroth order coefficient is sufficiently large

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

optimal switching for partial differential equations ii abstract optimal switching problem for systems governed by abstract semilinear evolution equations is considered the generator of semigroups is allowed to depend on the decision parameter and the nonlinear term in the evolution equation and the integral in the index are allowed to be unbounded under some mild conditions we prove that the value function of the optimal switching problem is the limit of the value functions of approximation problems we also discuss the finite horizontal case and the problem of approximating continuous controls by switching

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

hamilton jacobi equations in infinite dimensions ii existence of viscosity solutions on demontre des resultats d existence generaux pour 2 problemes 1 le probleme de cauchy u t h x t u du 0 dans v 0 t u x 0 x dans v et 2 le probleme stationnaire u h x u du 0 dans v ou h et sont donnees

semigroups of relations and subdifferentials of the minimal time function and of the geodesic distance given a closed subset s of a banach space x we study the minimal time function to reach a point x of x starting from s we relate this problem to the study of the geodesic distance from x to s associated with a riemannian metric or a finsler metric it appears that both cases can be treated simultaneously and yield solutions to a hamilton jacobi equation we detect simple links between the general framework we propose and multivalued semigroups new concepts of infinitesimal generators of such semigroups are considered

differential games and directional derivatives of viscosity solutions of issacs equations ii recent work by the authors this journal 23 1985 pp 566 583 has demonstrated the con nections between the notion of viscosity sub and super solutions of first order dynamic programming pde and the optimality principle of dynamic programming as well as the directional derivatives of viscosity solutions of the above equations at an arbitrary point the present note contains a remark and a counterexample which complement the results cited

convergence of semidiscrete approximations to optimal control problems in hilbert spaces a counterexample in this paper we give and discuss a counterexample related to the approximation of non quadratic optimal control problems for linear distributed parameter systems more precisely we show that the assumption 1 7 which has been proved in a previous paper to be sufficient is also necessary for the convergence at least in the most general setting of the problem

viscosity solutions of hamilton jacobi equations in infinite dimensions v unbounded linear terms and b continuous solutions we continue the systematic study of first order hamilton jacobi equations in infinite dimensions begun in parts i iv of this series 121 our approach relies on as before an appropriate interpretation of the notion of viscosity solutions as introduced in 11 lo in the interim the strong development of viscosity solution theory in many directions has continued unabated in addition to the many references given in earlier papers in this series which we will not repeat here see 12 we would like to point out that a complete theory of second order fully nonlinear and possibly degenerate elliptic equations in finite dimensions has been made possible by the use of viscosity solutions s 9 18 20 22 19 32 331 for example and there is now a substantial body of work concerning second order equations in infinite dimensions 25 271 however in this work we come back to the particular situation studied in 12 part iv namely we come back to hamilton jacobi equations in

on existence and uniqueness of solutions of hamilton jacobi equations abstract the theory of scalar first order nonlinear partial differential equations has been enjoying a rapid development in the last few years this development occurred because the authors established uniqueness criteria for generalized solutions called viscosity solutions which correctly identify the solutions sought in areas of application including control theory differential games and the calculus of variations the concept of viscosity solutions is relatively easy to work with and many formally heuristic or difficult proofs have been made rigorous or simple using this concept a feedback process has begun and the experience recently gained in working with viscosity solution has suggested new existence and uniqueness results the current paper continues this interaction by establishing new existence ane uniqueness results in a natural generality suggested by earlier proofs it is also felt that the presentation of the comparison results which imply uniqueness continuous dependence and are used to estimate moduli of continuity has something to offer over earlier presentations in special cases author

the necessary conditions for optimal control in hilbert spaces obtention du principe maximum de pontryagin pour une classe generale de problemes de commande de type lagragien afin qu associe avec un ensemble de conditions necessaires une commande soit satisfaite le principe est demontre en utilisant la programmation dynamique et l equation resultante d hamilton jacobi bellman de dimension finie

a class of linear phase fir filters for decimation interpolation and narrow band filtering a new class of linear phase finite impulse response fir digital filters for decimation and interpolation is discussed the transfer function of these filters is of the form h z a zd b z where d is the decimation or interpolation ratio and a zd is realized at the lower sampling rate the polynomial a zd is adjusted so that the overall magnitude response presents an equiripple behavior in the pass band whereas b z provides an equiripple stopband response an efficient iterative procedure for the design of the filters of this type is presented and the optimal selection of the orders of a zd and b z is considered several examples show how the new decimators and interpolators require significantly fewer multiplications per second than equivalent conventional linear phase fir and elliptic designs the performance of these linear phase filters is about the same as that of the recursive filters of martinez and parks with only powers of zdin the denominator some characteristics of the new filters such as zero positions coefficient sensitivity and output noise variance due to roundoff errors are discussed and compared with conventional fir designs furthermore it is shown that by cascading multistage new decimators and interpolators we obtain efficient narrow band filters requiring considerably fewer multiplications per output sample than equivalent elliptic designs

on stegall s smooth variational principle abstract assume that every coercive continuous and bounded below function defined on a real banach space attains after a suitable linear perturbation a not necessarily strong minimum then the space is dentable

delay optimal control and viscosity solutions to associated hamilton jacobi bellman equations abstractin this article optimal control problems of differential equations with delays are investigated for which the associated hamilton jacobi bellman hjb equations are nonlinear partial diffe

dynamical shape control of the heat equation abstract we study a shape optimization problem in one space dimension by using dynamic programming we obtain the existence and uniqueness of the generalized viscosity solution to the hamilton jacobi equation corresponding to our problem as well as the feedback formula

path dependent hamilton jacobi equations in infinite dimensions abstract we propose notions of minimax and viscosity solutions for a class of fully nonlinear path dependent pdes with nonlinear monotone and coercive operators on hilbert space our main result is well posedness existence uniqueness and stability for minimax solutions a particular novelty is a suitable combination of minimax and viscosity solution techniques in the proof of the comparison principle one of the main difficulties the lack of compactness in infinite dimensional hilbert spaces is circumvented by working with suitable compact subsets of our path space as an application our theory makes it possible to employ the dynamic programming approach to study optimal control problems for a fairly general class of delay evolution equations in the variational framework furthermore differential games associated to such evolution equations can be investigated following the krasovski subbotin approach similarly as in finite dimensions

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