control of the stochastic burgers model of turbulence we consider a control problem for a stochastic burgers equation this problem is motivated by a model from the control of turbulence see choi et al j fluid mech 253 1993 pp 509 543 we study a sequence of approximated hamilton jacobi equations by using dynamic programming

bellman equations associated to the optimal feedback control of stochastic navier stokes equations in this paper we study infinite dimensional second order hamilton jaco bi bellman equations associated to the feedback synthesis of stochastic navier stokes equations forced by space time white noise uniqueness and existence of viscosity solutions are proven for these infinite dimensional partial differential equations c 2004 wiley periodicals inc

on projection algorithms for solving convex feasibility problems due to their extraordinary utility and broad applicability in many areas of classical mathematics and modern physical sciences most notably computerized tomography algorithms for solving convex feasibility problems continue to receive great attention to unify generalize and review some of these algorithms a very broad and flexible framework is investigated several crucial new concepts which allow a systematic discussion of questions on behaviour in general hilbert spaces and on the quality of convergence are brought out numerous examples are given

direct numerical simulations of flow past an array of distributed roughness elements direct numerical simulation was used to describe the subsonic flow past an array of distributed cylindrical roughness elements mounted on a flat plate solutions were obtained for element heights corresponding to a roughness based reynolds number re k of both 202 and 334 the numerical method used a sixth order accurate centered compact finite difference scheme to represent spatial derivatives which was used in conjunction with a tenth order low pass pade type nondispersive filter operator to maintain stability an implicit approximately factored time marching algorithm was employed and newton like subiterations were applied to achieve second order temporal accuracy calculations were carried out on a massively parallel computing platform using domain decomposition to distribute subzones on individual processors a high order overset grid approach preserved spatial accuracy on the mesh system used to represent the roughness elements features of the flowfields are described and results of the computations are compared with experimentally measured velocity components of the time mean flowfield which are available only for re k 202 flow about the elements is characterized by a system of two weak corotating horseshoe vortices for re k 334 an unstable shear layer emanating from the top of the cylindrical element generated nonlinear unsteady disturbances of sufficient amplitude to produce explosive bypass transition downstream of the array the re k 202 case displayed exponential growth of turbulence energy in the streamwise direction which may eventually result in transition

direct numerical simulation of the transitional boundary layer flow induced by an isolated hemispherical roughness element the joint application of direct numerical simulation dns and a combined multiple direct forcing and immersed boundary method mdf ibm is proposed to investigate the transitional boundary layer flow induced by three dimensional roughness elements the multi direct forcing technique is used to calculate the interacting force between the solid surface of roughness elements and the fluid and let the no slip boundary conditions be satisfied in order to validate the efficiency of these numerical methods a flow past an isolated three dimensional hemispherical roughness element mounted on a flat plate is simulated the evolutional process of the discrete hairpin vortex and the formation of two kinds of secondary vortex structures are captured the comparisons of profiles of streamwise mean velocity and velocity fluctuation between the simulated results and the experimental ones show great quantitative agreement the evolution of disturbances and the growth of steady disturbance energy prove the transient growth mechanisms underlying the transitional flow induced by moderate amplitude isolated three dimensional roughness elements numerical methods used in this investigation can be extended to the simulation of the transitional boundary layer flows induced by randomly distributed three dimensional roughness elements

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

optimal control of a stochastic delay heat equation with boundary noise and boundary control in this paper a controlled stochastic delay heat equation with neumann boundary noise and boundary control is considered the existence and uniqueness of the mild solution for the associated hamilton jacobi bellman equations are obtained by means of the backward stochastic differential equations which is applied to the optimal control problem

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

infinite horizon backward stochastic differential equations and elliptic equations in hilbert spaces solutions of semilinear elliptic differential equations in infinite dimensional spaces are obtained by means of forward and backward infinite dimensional stochastic evolution equations the backward equation is considered on an infinite time horizon and a suitable growth condition replaces the final condition elliptic equations are intended in a mild sense suitable also for applications to optimal control we finally notice that due to the lack of smoothing properties the elliptic partial differential equation considered here could not be treated by analytic methods

the alternating algorithm in a uniformly convex and uniformly smooth banach space abstract let x be a uniformly convex and uniformly smooth banach space assume that the m i i 1 r are closed linear subspaces of x p m i is the best approximation operator to the linear subspace m i and m m 1 m r we prove that if m is closed then the alternating algorithm given by repeated iterations of i p m r i p m r 1 i p m 1 applied to any x x converges to x p m x where p m is the best approximation operator to the linear subspace m this result in the case r 2 was proven in deutsch 4

on uniqueness and existence of viscosity solutions of fully nonlinear second order elliptic pde s we prove several comparison and existence theorems for viscosity solutions of fully nonlinear degenerate elliptic equations one of them extends some recent uniqueness results by jensen some establish the uniqueness of solutions for second order isaacs equations and hence include the uniqueness results for bellman equations by p l lions our comparison results apply even for discontinuous solutions and so perron s method readily yields the existence of continuous solutions

on viscosity solutions of path dependent pdes in this paper we propose a notion of viscosity solutions for path dependent semi linear parabolic pdes this can also be viewed as viscosity solutions of non markovian backward sdes and thus extends the well known nonlinear feynman kac formula to non markovian case we shall prove the existence uniqueness stability and comparison principle for the viscosity solutions the key ingredient of our approach is a functional ito calculus recently introduced by dupire functional ito calculus 2009 preprint

partial regularity of viscosity solutions for a class of kolmogorov equations arising from mathematical finance we study value functions which are viscosity solutions of certain kolmogorov equations using pde techniques we prove that they are c1 regular on special finite dimensional subspaces the problem has origins in hedging derivatives of risky assets in mathematical finance

transient growth and transition induced by random distributed roughness recent experiments on transient disturbances generated by three dimensional roughness have used spanwise periodic arrays of geometrically simple cylindrical roughness elements connecting these laboratory experiments to more realistic situations requires the study of surfaces with distributed roughness this is accomplished in this work by numerically generating quasi random rough surfaces and manufacturing these surfaces using rapid prototyping technology measurements of the disturbances that the rough surface creates in a blasius boundary layer are obtained for three test configurations corresponding to roughness based reynolds numbers of re k 164 227 and 301 the two lower values give laminar flow the highest value results in localized transition approximately 140 mm downstream of the leading edge of the roughness all three configurations exhibit transient growth of steady disturbances unsteady fluctuations indicate that transition in the re k 301 configuration is likely an example of a bypass transition mechanism in which the unsteady disturbance growth outpaces the stabilizing relaxation of the steady flow measurements above the roughness surface in the re k 227 configuration provide a phenomenological model for distributed receptivity

dynamic programming for optimal control of stochastic mckean vlasov dynamics we study the optimal control of general stochastic mckean vlasov equation such problem is motivated originally from the asymptotic formulation of cooperative equilibrium for a large population of particles players in mean field interaction under common noise our first main result is to state a dynamic programming principle for the value function in the wasserstein space of probability measures which is proved from a flow property of the conditional law of the controlled state process next by relying on the notion of differentiability with respect to probability measures due to p l lions 32 and ito s formula along a flow of conditional measures we derive the dynamic programming hamilton jacobi bellman equation and prove the viscosity property together with a uniqueness result for the value function finally we solve explicitly the linear quadratic stochastic mckean vlasov control problem and give an application to an interbank systemic risk model with common noise

on the rate of convergence of iterated bregman projections and of the alternating algorithm abstract we study the alternating algorithm for the computation of the metric projection onto the closed sum of two closed subspaces in uniformly convex and uniformly smooth banach spaces for banach spaces which are convex and smooth of power type we exhibit a condition which implies linear convergence of this method we show these convergence results for iterates of bregman projections onto closed linear subspaces using an intimate connection between the metric projection onto a closed linear subspace and the bregman projection onto its annihilator we deduce the convergence rate results for the alternating algorithm from the corresponding results for the iterated bregman projection method

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

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

partially observed control of markov processes iv abstract a class of degenerate elliptic nonlinear second order equations is considered these equations arising from a related class of probabilistic variational problems are defined on the space of probability measures on rd the existence of c1 1 viscosity solutions is established as well as the existence uniqueness and feedback characterization of the minima of the variational problems

optimal control of semilinear stochastic evolution equations the aim of this paper is to initiate a semigroup theory based approach to characterization of optimal markov controls for controlled semilinear stochastic evolution equations it may be recalled that markov controls are those that depend only on the current state at each time for finite dimensional controlled stochastic differential equations with a nondegenerate diffusion matrix this task is traditionally achieved through the hamilton jacobi bellman equation of dynamic programming associated with the problem and an accompanying verification theorem the latter states that an optimal markov control can be explicitly obtained by the pointwise minimization of a hamiltonian derivable from the solution of the hjb equation moreover any optimal markov control is obtainable in this manner

viscosity solutions of fully nonlinear second order equations and optimal stochastic control in infinite dimensions part i the case of bounded stochastic evolutions on etudie des equations elliptiques degenerees d ordre 2 non lineaires de la forme f d 2 u du u x 0 dans h ou h est un espace de hilbert separable x un point generique dans h u une fonction de h r

mild solutions of semilinear elliptic equations in hilbert spaces abstract this paper extends the theory of regular solutions c 1 in a suitable sense for a class of semilinear elliptic equations in hilbert spaces the notion of regularity is based on the concept of g derivative which is introduced and discussed a result of existence and uniqueness of solutions is stated and proved under the assumption that the transition semigroup associated to the linear part of the equation has a smoothing property that is it maps continuous functions into g differentiable ones the validity of this smoothing assumption is fully discussed for the case of the ornstein uhlenbeck transition semigroup and for the case of invertible diffusion coefficient covering cases not previously addressed by the literature it is shown that the results apply to hamilton jacobi bellman hjb equations associated to infinite horizon optimal stochastic control problems in infinite dimension and that in particular they cover examples of optimal boundary control of the heat equation that were not treatable with the approaches developed in the literature up to now

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 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

the maximum principle for viscosity solutions of fully nonlinear second order partial differential equations we prove that viscosity solutions in w1 of the second order fully nonlinear equation f d2u du u 0 are unique when i f is degenerate elliptic and decreasing in u or ii f is uniformly elliptic and nonincreasing in u we do not assume that f is convex the method of proof involves constructing nonlinear approximation operators which map viscosity subsolutions and supersolutions onto viscosity subsolutions and supersolutions respectively this method is completely different from that used in lions 8 9 for second order problems with f convex in d 2 u and from that used by crandall lions 3 and crandall evans lions 2 for fully nonlinear first order problems

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

regularity for obstacle problems in infinite dimensional hilbert spaces in this paper we study fully nonlinear obstacle type problems in hilbert spaces we introduce the notion of q elliptic equation and prove existence uniqueness and regularity of viscosity solutions of q elliptic obstacle problems in particular we show that solutions of concave problems with semiconvex obstacles are in the space

regularity results for the minimum time function of a class of semilinear evolution equations of parabolic type under suitable controllability and smoothness assumptions the minimum time function t x of a semilinear control system is proved to be locally lipschitz continuous and semiconcave on the controllable set these properties are then applied to derive optimality conditions relating optimal trajectories to the superdifferential of t

optimal control problems for stochastic reaction diffusion systems with non lipschitz coefficients by using the dynamic programming approach we study a control problem for a class of stochastic reaction diffusion systems with coefficients having polynomial growth in the cost functional a non lipschitz term appears and this allows us to treat the quadratic case which is of interest in the applications the corresponding hamilton jacobi bellman equation is first resolved by a fixed point argument in a small time interval and then is extended to arbitrary time intervals by suitable a priori estimates the main ingredient in the proof is the smoothing effect of the transition semigroup associated with the uncontrolled system

numerical investigation of synthetic jet flowfields the flowflelds surrounding a synthetic jet actuating device are investigated numerically by direct simulation solutions are obtained to the unsteady compressible navier stokes equations for both the interior of the actuator cavity and for the external jet flowfield the interior results are generated on an overset deforming zonal mesh system whereas the jet flowfield is obtained by a high order compact difference scheme newton like subiterations are employed to achieve second order temporal accuracy details of the computations are summarized and the quality of the results is assessed via grid resolution and time step size studies several aspects of the actuator configuration are investigated including cavity geometry and reynolds number differences between two dimensional and three dimensional external unsteady flowfields are elucidated and comparison is made with experimental data in terms of the mean and fluctuating components of the jet velocity

generalized solutions of hjb equations applied to stochastic control on hilbert space abstract in this paper we prove the existence of borel measurable optimal control laws for infinite dimensional nonstationary stochastic control problems on hilbert space this is based on some recent results on existence of generalized solutions of hjb equations on hilbert space l2 h where is a suitable reference measure on h for illustration some examples are presented

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

high order compact finite difference methods on general overset grids this work investigates the coupling of a very high order finite difference algorithm for the solution of conservation laws on general curvilinear meshes with overset grid techniques originally developed to address complex geometric configurations the solver portion of the algorithm is based on pade type compact finite differences of up to sixth order with up to 10th order filters employed to remove spurious waves generated by grid non uniformities boundary conditions and flow non linearities the overset grid approach is utilized as both a domain decomposition paradigm for implementation of the algorithm on massively parallel machines and as a means for handling geometric complexity in the computational domain two key features have been implemented in the current work the ability of the high order algorithm to accommodate holes cut in grids by the overset grid approach and the use of high order interpolation at non coincident grid overlaps several high order high accuracy interpolation methods were considered and a high order explicit non optimized lagrangian method was found to be the most accurate and robust for this application several two dimensional benchmark problems were examined to validate the interpolation methods and the overall algorithm these included grid to grid interpolation of analytic test functions the inviscid convection of a vortex laminar flow over single and double cylinder configurations and the scattering of acoustic waves from one and three cylinder configurations the employment of the overset grid techniques coupled with high order interpolation at overset boundaries was found to be an effective way of employing the high order algorithm for more complex geometries than was previously possible

regular solutions of second order stationary hamilton jacobi equations abstract we study a second order stationary hamilton jacobi equation in infinite dimension this equation is nonlinear and convex with respect to the first order term we use properties of the transition semigroup associated to the linear equation to write the hamilton jacobi equation in integral form and we prove existence uniqueness and regularity of a solution by the theory of maximal monotone operators we also prove that this solution is the pointwise limit of a uniformly bounded sequence of classical solutions of approximating problems finally the solution is the value function of the associated optimal stochastic control problem some examples are given

structure of laminar juncture flows a computational study of both steady and periodic laminar horseshoe vortex flows generated upstream of a cylinder flat plate juncture is presented the flowfields are simulated using the full three dimensional unsteady navier stokes equations and a time accurate implicit algorithm a new type of laminar horseshoe vortex topology is identified for the case of a single primary vortex this new topology is found to be independent of the computational grid and is also supported by recent experimental flow visualizations the flat plate skin friction portraits corresponding to the new and to the standard horseshoe vortex topologies are equivalent pointing out the nonunique relation between the wall limiting streamline pattern and the three dimensional flow above the plate for the new topology the foremost line of coalescense is an attachment rather than a separation line this unusual feature illustrates the fact that convergence of skin friction lines is a necessary but not sufficient condition for separation as the reynolds number increases the flow topology evolves from a single to multiple primary horseshoe vortices in agreement with experimental observations at least two different types of triple horseshoe vortex systems are shown to be possible above a certain value of the reynolds number the juncture flow becomes unsteady and periodic at a frequency that increases with reynolds number the unsteady horseshoe vortex process upstream of the cylinder is found in qualitative agreement with experiment horseshoe vortices are periodically generated and convected toward the juncture vorticity intensification by vortex stretching and the eruption of vorticity from the plate surface are observed

numerical simulation of separation control for transitional highly loaded low pressure turbines the subsonic flow through highly loaded low pressure turbines is simulated numerically using a high order method the configuration approximates cascade experiments that were conducted to investigate a reduction in turbine stage blade count which can decrease both weight and mechanical complexity at a nominal reynolds number of 25 10 3 based upon axial chord and inlet conditions massive separation occurs on the suction surface of each blade as a result of uncovered turning pulsed injection vortex generator jets were then used to help mitigate separation thereby reducing wake losses computations were performed for both uncontrolled and controlled cases and reproduced the transitional flow occurring in the aft blade and near wake regions the numerical method utilizes a centered compact finite difference scheme to represent spatial derivatives which is used in conjunction with a low pass pade type nondispersive filter operator to maintain stability an implicit approximately factored time marching algorithm is employed and newton like subiterations are applied to achieve second order temporal accuracy calculations were carried out on a massively parallel computing platform using domain decomposition to distribute subzones on individual processors a high order overset grid approach preserved spatial accuracy in locally refined embedded regions features of the flowfields are elucidated and simulations are compared with each other and with available experimental data relative to the uncontrolled case it was found that pulsed injection maintained attached flow over an additional 15 of the blade chord resulting in a 22 decrease of the wake total pressure loss coefficient

optimal control problems for stochastic delay evolution equations in banach spaces we consider the optimal control for a banach space valued stochastic delay evolution equation the existence and uniqueness of the mild solution for the associated hamilton jacobi bellman equations are obtained by means of backward stochastic differential equations an application to optimal control of stochastic delay partial differential equations is also given

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

necessary and sufficient conditions for j spectral factorizations with a j lossless property for infinite dimensional systems in continuous and discrete time necessary and sufficient conditions for j spectral factorizations are given in terms of the existence of a self adjoint stabilizing solution of an appropriate riccati equation in infinite dimensions furthermore it is shown that a certain j lossless property holds if and only if the stabilizing solution to the riccati equation is nonnegative definite we prove these results first for the continuous time case and thereby derive analogous results for the discrete time case by means of a bilinear transformation of the complex plane

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

numerical simulation and theoretical analysis of perturbations in hypersonic boundary layers direct numerical simulations of receptivity in a boundary layer over a flat plate and a sharp wedge were carried out with two dimensional perturbations introduced into the flow by periodic in time blowing suction through a slot the free stream mach numbers are equal to 5 92 and 8 in the cases of adiabatic flat plate and sharp wedge respectively the perturbation flow field was decomposed into normal modes with the help of the multimode decomposition technique based on the spatial biorthogonal eigenfunction system the decomposition allows filtering out the stable and unstable modes hidden behind perturbations having another physical nature

explicit solutions for continuous time mean variance portfolio selection with nonlinear wealth equations abstract this paper concerns the continuous time mean variance portfolio selection problem with a special nonlinear wealth equation this nonlinear wealth equation has a nonsmooth coefficient and the dual method developed in ji 2010 does not work we invoke the hjb equation of this problem and give an explicit viscosity solution of the hjb equation furthermore via this explicit viscosity solution we obtain explicitly the efficient portfolio strategy and efficient frontier for this problem finally we show that our nonlinear wealth equation can cover three important cases

hamilton jacobi bellman equations for the optimal control of the duncan mortensen zakai equation abstract we study a class of hamilton jacobi bellman hjb equations associated to stochastic optimal control of the duncan mortensen zakai equation the equations are investigated in weighted l 2 spaces we introduce an appropriate notion of weak viscosity solution of such equations and prove that the value function is the unique solution of the hjb equation we apply the results to stochastic optimal control problems with partial observation and correlated noise

the existence and uniqueness of the solution for nonlinear kolmogorov equations abstract by means of backward stochastic differential equations the existence and uniqueness of the mild solution are obtained for the nonlinear kolmogorov equations associated with stochastic delay evolution equations applications to optimal control are also given

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

stationary hamilton jacobi equations in hilbert spaces and applications to a stochastic optimal control problem we study an infinite horizon stochastic control problem associated with a class of stochastic reaction diffusion systems with coefficients having polynomial growth the hamiltonian is assumed to be only locally lipschitz continuous so that the quadratic case can be covered we prove that the value function v corresponding to the control problem is given by the solution of the stationary hamilton jacobi equation associated with the state system to this purpose we write the hamilton jacobi equation in integral form and by using the smoothing properties of the transition semigroup relative to the state system and the theory of m dissipative operators we show that it admits a unique solution moreover the value function v is obtained as the limit of minima for some approximating control problems which admit unique optimal controls and states

infinite horizon bsdes with dissipative coefficients in hilbert spaces and applications in this paper we study a class of infinite horizon backward stochastic differential equations bsdes of the form dy t y t dt f t y t z t dt z t dw t 0 t in a real separable hilbert space where is a given real parameter and the coefficient f is dissipative in y and lipschitz in z by yosida approximation to dissipative mappings we show existence and uniqueness of the solutions for these equations this result is applied to construct unique viscosity solutions to semilinear elliptic partial differential equations pdes

optimal control problem for stochastic evolution equations in hilbert spaces in this article we consider an optimal control problem in which the controlled state dynamics is governed by a stochastic evolution equation in hilbert spaces and the cost functional has a quadratic growth the existence and uniqueness of the optimal control are obtained by the means of an associated backward stochastic differential equations with a quadratic growth and an unbounded terminal value as an application an optimal control of stochastic partial differential equations with dynamical boundary conditions is also given to illustrate our results

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

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

analytic system problems and j lossless coprime factorization for infinite dimensional linear systems abstract this paper extends the coprime factorization approach to the synthesis of internally stabilizing controllers satisfying an h norm bound to a class of systems with irrational transfer matrices using the coprime factorization description the h control problem can be reduced to two stable analytic system problems such problems have solutions if and only if a certain j lossless factorization exists the full h synthesis problem is shown to be equivalent to the solution of two nested j lossless factorizations if the irrational transfer matrix has a state space realization then the known state space formulas for the h control problem may be recovered using the relationship between j lossless factorizations and solutions of riccati equations however the results derived here are valid for a larger class of infinite dimensional systems

smooth fit principle for impulse control of multidimensional diffusion processes value functions of impulse control problems are known to satisfy quasi variational inequalities qvis a bensoussan and j l lions impulse control and quasivariational inequalities heyden son philadelphia 1984 translation of controle impulsionnel et inequations quasi variationnelles gauthier villars paris 1982 this paper proves the smooth fit c 1 property of the value function for multidimensional controlled diffusions using a viscosity solution approach we show by examples how to exploit this regularity property to derive explicitly optimal policy and value functions

comparison of viscosity solutions of semi linear path dependent pdes this paper provides a probabilistic proof of the comparison result for viscosity solutions of path dependent semilinear pdes we consider the notion of viscosity solutions introduced in 8 which considers as test functions all those smooth processes which are tangent in mean when restricted to the markovian case this definition induces a larger set of test functions and reduces to the notion of stochastic viscosity solutions analyzed in 1 2 our main result takes advantage of this enlargement of the test functions and provides an easier proof of comparison this is most remarkable in the context of the linear path dependent heat equation as a key ingredient for our methodology we introduce a notion of punctual differentiation similar to the corresponding concept in the standard viscosity solutions 3 and we prove that semimartingales are almost everywhere punctually differentiable this smoothness result can be viewed as the counterpart of the aleksandroff smoothness result for convex functions a similar comparison result was established earlier in 8 the result of this paper is more general and more importantly the arguments that we develop do not rely on any representation of the solution

nonlinear semigroups associated with optimal stopping of controlled diffusions under partial observation abstract we treat optimal stopping problems of controlled diffusions under partial observation two kinds of admissibility strict sense and wide sense are considered using the girsanov transformation the unnormalized conditional probability turns out the state process subject to the zakai equation and we can derive a nonlinear semigroup which presents the bellman principle in our problems in the diffusion case an optimal stopping time is given appealing to a snell envelope