abstract volterra integrodifferential equations with applications to parabolic models with memory in this paper we are concerned with local existence regularity and continuous dependence upon the initial data of epsilon regular mild solutions for the abstract integrodifferential equation 1 2 we also present a result on unique continuation and a blow up alternative for an epsilon regular mild solution of 1 2 finally we apply our results to three interesting models navier stokes equations with memory diffusion equations with memory and a strongly damped plate equation with memory

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

sobolev regularity for a class of second order elliptic pde s in infinite dimension we consider an elliptic kolmogorov equationu ku f in a separable hilbert space h the kolmogorov operator k is associated to an infinite dimensional convex gradient system dx ax du x dt dw t where a is a self adjoint operator in h and u is a convex lower semicontinuous function under mild assumptions we prove that for 0 and f 2 l 2 h the weak solution u belongs to the sobolev space w 2 2 h whereis the log concave probability measure of the system moreover maximal estimates on the gradient of u are proved the maximal regularity results are used in the study of perturbed non gradient systems for which we prove that there exists an invariant measure the general results are applied to kolmogorov equations associated to reaction diffusion and cahn hilliard stochastic pdes

spectrum of ornstein uhlenbeck operators in lp spaces with respect to invariant measures abstract let a i j 1 n q ij d ij i j 1 n b ij x j d i be a possibly degenerate ornstein uhlenbeck operator in r n and assume that the associated markov semigroup has an invariant measure we compute the spectrum of a in l p for 1 p

numerical approximations of algebraic riccati equations for abstract systems modelled by analytic semigroups and applications this paper provides a numerical approximation theory of algebraic riccati operator equations with unbounded coefficient operators a and b such as arise in the study of optimal quadratic cost problems over the time interval 0 oo for the abstract dynamics j ay bu here a is the generator of a strongly continuous analytic semigroup and b is an unbounded operator with any degree of unboundedness less than that of a convergence results are provided for the riccati operators as well as for all the other relevant quantities which enter into the dynamic optimization problem the present numerical theory is the counterpart of a known continuous theory several examples of partial differential equations with boundary point control where all the required assumptions are verified illustrate the theory they include parabolic equations with l2 dirichlet control as well as plate equations with a strong degree of damping and point control

regular densities of invariant measures in hilbert spaces abstract we consider a nonlinear differential stochastical equation in a hilbert space that is a lipschitzian perturbation of a linear equation we prove that under suitable hypotheses both equations have invariant measures and 0 respectively and that is absolutely continuous with respect to 0 we also give several regularity results on the density d d 0

vector valued stochastic delay equations a weak solution and its markovian representation abstract a class of stochastic delay equations in a type 2 umd banach space e driven by the cylindrical wiener process is studied we investigate two concepts of solutions weak and generalised strong and give conditions under which they are equivalent we present an evolution equation approach in a banach space e p e l p 1 0 e proving that the solutions can be reformulated as e p valued markov processes based on the markovian representation we prove the existence and continuity of the solutions the results are applied to stochastic reaction diffusion equations with delays

some properties of invariant measures of non symmetric dissipative stochastic systems we consider transition semigroups generated by stochastic partial differential equations with dissipative nonlinear terms we prove an integration by part formula and a logarithmic sobolev inequality for the invariant measure no symmetry or reversibility assumptions are made furthemore we prove some compactness results on the transition semigroup and on the embedding of the sobolev spaces based on the invariant measure we use these results to derive asymptotic properties for a stochastic reaction diffusion equation

a systematic method for constructing sparse gaussian integer sequences with ideal periodic autocorrelation functions a gaussian integer is a complex number whose real and imaginary parts are both integers meanwhile a sequence is defined as perfect if and only if it has an ideal periodic auto correlation function this paper proposes a method for constructing sparse perfect gaussian integer sequences spgiss in which most of the sequence elements are zero the proposed spgiss are obtained by linearly combining four base sequences or their cyclic shift equivalents using nonzero gaussian integer coefficients of equal magnitudes each base sequence contains four nonzero elements belonging to the set pm 1 pm j the number of nonzero elements of the constructed spgiss depends on the choice of complex coefficients and cyclic shifts however each spgis has at most 16 nonzero elements irrespective of the sequence length a systematic investigation is performed into the properties of the spgiss and their fourier dual equivalents finally a general expression is derived for a perfect gaussian integer sequence pgis of length 4n where n is any positive integer and most of the sequence elements are nonzero

existence uniqueness and invariant measures for stochastic semilinear equations on hilbert spaces a class of stochastic evolution equations with additive noise and weakly continuous drift is considered first regularity properties of the corresponding ornstein uhlenbeck transition semigrouprt are obtained we show thatrt is a compactc0 semigroup in all sobolev spaceswn p which are built on its invariant measure then we show the existence uniqueness compactness and smoothing properties of the transition semigroup for semilinear equations inlp spaces and spacesw1 p as a consequence we prove the uniquencess of martingale solutions to the stochastic equation and the existence of a unique invariant measure equivalent to it is shown also that the density of this measure with respect to is inlp for allp 1

on the mathematical model for linear elastic systems with analytic damping in the present paper we investigate linear elastic systems with damping ddot y b dot y ay 0 in hilbert spaces where a is a positive definite unbounded linear operator and b is a closed linear operator related in various ways to a alpha frac 1 2 leqq alpha leqq 1 we discuss the spectral property of these systems and obtain some fundamental results for the holomorphic property and the exponential stability of the semigroups associated with these systems

linear wave equations with time dependent propagation speed and strong damping abstract we consider a second order linear equation with a time dependent coefficient c t in front of the elastic operator for these equations it is well known that a higher space regularity of initial data compensates a lower time regularity of c t in this paper we investigate the influence of a strong dissipation namely a friction term which depends on a power of the elastic operator what we discover is a threshold effect when the exponent of the elastic operator in the friction term is greater than 1 2 the damping prevails and the equation behaves as if the coefficient c t were constant when the exponent is less than 1 2 the time regularity of c t comes into play if c t is regular enough once again the damping prevails on the contrary when c t is not regular enough the damping might be ineffective and there are examples in which the dissipative equation behaves as the non dissipative one as expected the stronger is the damping the lower is the time regularity threshold we also provide counterexamples showing the optimality of our results

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

strongly damped wave problems bootstrapping and regularity of solutions abstract the aim of the article is to present a unified approach to the existence uniqueness and regularity of solutions to problems belonging to a class of second order in time semilinear partial differential equations in banach spaces our results are applied next to a number of examples appearing in literature which fall into the class of strongly damped semilinear wave equations the present work essentially extends the results on the existence and regularity of solutions to such problems previously these problems have been considered mostly within the hilbert space setting and with the main part operators being selfadjoint in this article we present a more general approach involving sectorial operators in reflexive banach spaces

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

complex hadamard matrices related to bent sequences hadamard matrices are often used for some applications such as error correcting codes and spread sequences this article gives the construction of complex hadamard matrices of order p sup n where p is prime and n is even the complex hadamard matrices include bi phase hadamard matrices whose elements take 1 1 and four phase hadamard matrices whose elements take spl plusmn 1 spl plusmn j with j spl radic 1

optimal stopping time problem for stochastic navier stokes equations and infinite dimensional variational inequalities we prove the existence of a solution for the obstacle problem associated with the kolmogorov operator corresponding to the stopping time problem for stochastic navier stokes equations in 2 d 2005 elsevier ltd all rights reserved

invariant measures and regularity properties of perturbed ornstein uhlenbeck semigroups abstract this paper deals with perturbations of the ornstein uhlenbeck operator on l 2 spaces with respect to a gaussian measure we perturb the generator of the ornstein uhlenbeck semigroup by a certain unbounded non linear drift and show various properties of the perturbed semigroup such as compactness and positivity strong feller property existence and uniqueness of an invariant measure are discussed as well

matrix methods for wave equations in analogy to a characterisation of operator matrices generating c0 semigroups due to r nagel 13 we give conditions on its entries in order that a 2 2 operator matrix generates a cosine operator function we apply this to systems of wave equations to second order initial boundary value problems and to overdamped wave equations

strong feller property and irreducibility for diffusions on hilbert spaces it is shown that the transition semigroup p t t 0 corresponding to a nonlinear stochastic evolution equation is strong feller and irreducible provided the nonlinearities are lipschitz continuous and the diffusion term is nondegenerate this result ensures the uniqueness of the invariant measure for p t t 0

discretisation of an infinite delay equation in this paper a banach phase space containing bc 0 and contained in c 0 is defined with which existence of a solution and convergence of a discrete scheme are proved for an infinite delay differential equation

gaussian integer sequences with ideal periodic autocorrelation functions a gaussian integer is a complex number whose real and imaginary parts are both integers meanwhile a sequence is defined as perfect if and only if the out of phase value of the periodic autocorrelation function is equal to zero this paper presents two novel classes of perfect sequences constructed using two groups of base sequences the nonzero elements of these base sequences belong to the set 1 j a perfect sequence can be obtained by linearly combining these base sequences or their cyclic shift equivalents with arbitrary nonzero complex coefficients of equal magnitudes in general the elements of the constructed sequences are not gaussian integers however if the complex coefficients are gaussian integers then the resulting perfect sequences will be gaussian integer perfect sequences gipss in addition a periodic cross correlation function is derived which has the same mathematical expression as the investigated sequences finally the maximal energy efficiency of the proposed gipss is investigated

real interpolation and compact linear operators we prove that if t a0 b0 is compact and t a1 b1 is compact or t a1 b1 is bounded and then given any and q with 0 1 and 0 q it follows that t a0 a1 0 q b0 b1 0 q is also compact here a0 a1 0 q and b0 b1 0 q are the usual real interpolation spaces

self adjointness of some infinite dimensional elliptic operators and application to stochastic quantization we consider an operator k l in a hilbert space h where l is an ornstein uhlenbeck operator u w 1 4 h and is the invariant measure associated with l we show that k is essentially self adjoint in the space l 2 h where is the gibbs measure dx z 1 e 2u x dx an application to stochastic quantization is given

pointwise pressure observations of a canonical structural acoustics model in this paper we study regularity properties of a physically motivated observation associated with a controlled partial differential equation pde model on a rectangular domain this particular pde describes the active control of acoustic pressure in a chamber by means of piezoelectric activation and involves a wave coupled to a structurally damped beam equation with the coupling being accomplished by boundary trace terms the observation here consists of pointwise in space measurements of the velocity of the wave component with this velocity representing the acoustic pressure of the system as defined then the observation of the system like the control imput is modeled by an unbounded operator the main result of this paper is the discovery of an energy space such that for initial data taken from this space the desired observation of the acoustic pressure is well defined

existence uniqueness and asymptotic behavior of mild solutions to stochastic functional differential equations in hilbert spaces abstract in this paper we shall consider the existence uniqueness and asymptotic behavior of mild solutions to stochastic partial functional differential equations with finite delay r 0 dx t ax t f t x t dt g t x t dw t where we assume that a is a closed densely defined linear operator and the generator of a certain analytic semigroup f c h g c l 0 2 k h are two locally lipschitz continuous functions where c c r 0 d a l 0 2 k h are two proper infinite dimensional spaces 0 w t is a given k valued wiener process and both h and k are separable hilbert spaces

existence and uniqueness of invariant measures for a class of transition semigroups on hilbert spaces we prove a smoothing property and the irreducibility of transition semigroups corresponding to a class of semilinear stochastic equations on a separable hilbert space h existence and uniqueness of invariant measures are discussed as well

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

stability of semilinear stochastic evolution equations stability of moments of the mild solution of a semilinear stochastic evolution equation is studied and sufficient conditions are given for the exponential stability of the pth moment in terms of liapunov function sufficient conditions for sample continuity of the solution are also obtained and the exponential stability of sample paths is proved three examples are given to illustrate the theory

local and global smoothing effects for some linear hyperbolic equations with a strong dissipation we consider an abstract second order linear equation with a strong dissipation namely a friction term which depends on a power of the elastic operator in the homogeneous case we investigate the phase spaces in which the initial value problem gives rise to a semigroup and the further regularity of solutions in the nonhomogeneous case we study how the regularity of solutions depends on the regularity of forcing terms and we characterize the spaces where a bounded forcing term yields a bounded solution what we discover is a variety of different regimes with completely different behaviors depending on the exponent in the friction term we also provide counterexamples in order to show the optimality of our results mathematics subject classification 2010 msc2010 35l10 35l15 35l20

nonsymmetric ornstein uhlenbeck semigroups in banach spaces letebe a separable real banach space and letq l e e be positive and symmetric lets s t t 0be ac0 semigroup onewe study the relations between the reproducing kernel hilbert spaces associated with the operatorsqt t0 s s qs s ds under the assumption that these operators are the covariances of centered gaussian measures tone we also study equivalence t sfor different values of s and t and we calculate their radon nikodym derivatives

simplified conditions of initial observability for infinite dimensional second order damped dynamical systems abstract in the paper the initial observability of abstract second order infinite dimensional dynamical systems is considered two cases of observation operator are explored first the observation operator is assumed to be bounded then results are extended to more general class of observations related to the observation operators that are assumed to be bounded with respect to square root of elasticity operator which covers the case of some unboundedness in observation using the frequency domain method it is proved that initial observability of second order system under discussion can be verified by the initial observability conditions for the corresponding simplified first order system general results are then applied for initial observability investigation of infinite dimensional systems described by partial differential equations modelling flexible mechanical structures some special cases are also discussed and practical comments and remarks are given the paper extends earlier results on initial observability for a class of infinite dimensional second order abstract dynamical systems

on the dual boundary conditions and the nonnegative square root of the second derivative operator abstract a dual relationship between the boundary conditions of the second derivative operator au u is established some dual properties determined by the dual boundary conditions are considered furthermore the singularities of the nonnegative square root of the above second derivative operator are studied our results show that although a 1 2 u is not u it is a harmonic conjugate of u in some sense

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

long term optimal investment strategies in the presence of adjustment costs we consider the problem of determining in a dynamical way the optimal capacity level of an investment project that operates within a random economic environment in particular we consider an investment project that yields payoff at a rate that depends on its installed capacity level and on a random economic indicator such as the price of the project s output commodity we model this economic indicator by means of a general one dimensional ergodic diffusion at any time the project s capacity level can be increased or decreased at given proportional costs the aim is to maximize an ergodic performance criterion that reflects the long term average payoff resulting from the project s management we solve this genuinely two dimensional stochastic control problem by constructing an explicit solution to an appropriate hamilton jacobi bellman equation and by fully characterizing an optimal investment strategy

adaptive boundary control for unstable parabolic pdes part i lyapunov design we develop adaptive controllers for parabolic partial differential equations pdes controlled from a boundary and containing unknown destabilizing parameters affecting the interior of the domain these are the first adaptive controllers for unstable pdes without relative degree limitations open loop stability assumptions or domain wide actuation it is the first necessary step towards developing adaptive controllers for physical systems such as fluid thermal and chemical dynamics where actuation can be only applied non intrusively the dynamics are unstable and the parameters such as the reynolds rayleigh prandtl or peclet numbers are unknown because they vary with operating conditions our method builds upon our explicitly parametrized control formulae to avoid solving riccati or bezout equations at each time step most of the designs we present are state feedback but we present two benchmark designs with output feedback which have infinite relative degree

adaptive boundary and point control of linear stochastic distributed parameter systems an adaptive control problem for the boundary or the point control of a linear stochastic distributed parameter system is formulated and solved in this paper the distributed parameter system is modeled by an evolution equation with an infinitesimal generator for an analytic semigroup since there is boundary or point control the linear transformation for the control in the state equation is also an unbounded operator the unknown parameters in the model appear affinely in both the infinitesimal generator of the semigroup and the linear transformation of the control strong consistency is verified for a family of least squares estimates of the unknown parameters an ito formula is established for smooth functions of the solution of this linear stochastic distributed parameter system with boundary or point control the certainty equivalence adaptive control is shown to be self tuning by using the continuity of the solution of a stationary riccati equation as a function of parameters in a uniform operator topology for a quadratic cost functional of the state and the control the certainty equivalence control is shown to be self optimizing that is the family of average costs converges to the optimal ergodic cost some examples of stochastic parabolic problems with boundary control and a structurally damped plate with random loading and point control are described that satisfy the assumptions for the adaptive control problem solved in this paper

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

controllability of second order infinite dimensional systems in the paper the approximate controllability of linear abstract second order infinite dimensional dynamical systems is considered it is proved using the frequency domain method that approximate controllability of second order system can be verified by the approximate controllability conditions for the corresponding simplified first order system general results are then applied for approximate controllability investigation of a vibratory dynamical system modeling flexible mechanical structure some special cases are also considered moreover remarks and comments on the relationships between different concepts of controllability are given the paper extends earlier results on approximate controllability of second order abstract dynamical systems

regularity of solutions on the global attractor for a semilinear damped wave equation we consider attractors a 0 1 corresponding to a singularly perturbed damped wave equation utt 2 a 1 2 ut aut au f u

regularity of some structurally damped problems with point control and with boundary control abstract we study the regularity of structurally damped abstract systems with application to point control and boundary control problems for plate equations in general multidimensional domains in rn our treatment includes in particular both the cases where the damping operator is equal to the square root of the original differential operator and the case where the damping operator is only comparable to such a square root

sufficient conditions for the eventual strong feller property for degenerate stochastic evolutions abstract the strong feller property is an important quality of markov semigroups which helps for example in establishing uniqueness of invariant measure unfortunately degenerate stochastic evolutions such as stochastic delay equations do not possess this property however the eventual strong feller property is sufficient in establishing uniqueness of invariant probability measure in this paper we provide operator theoretic conditions under which a stochastic evolution equation with additive noise possesses the eventual strong feller property the results are used to establish uniqueness of invariant probability measure for stochastic delay equations and stochastic partial differential equations with delay with an application in neural networks

carleman estimate for an adjoint of a damped beam equation and an application to null controllability abstract in this article we consider a control problem of a linear euler bernoulli damped beam equation with potential in dimension one with periodic boundary conditions we derive a new carleman estimate for an adjoint of the equation under consideration then using a well known duality argument we obtain explicitly the control function which can be used to drive the solution trajectory of the control problem to zero state

liapunov razumikhin pairs and the location of positive limit sets for precompact functional differential equations with infinite delay in a recent paper l haddock and terjeki addressed the problem of locating positive limit sets for autonomous functional differential equations with infinite delay the authors obtained invariance principles and other results concerning asymptotic behavior of solutions by employing liapunov razumikhin techniques the autonomous nature of the equations they considered was central to their work the purpose of this paper is to extend their results to a class of nonautonomous equations such a task was undertaken for finite delay equations in 2 here as in 2 the assumption of autonomy is replaced by a continuity condition and our invariance principles involve limiting equations of our original equation we feel that the setting for our theory is quite natural and that our methods illustrate the role of limiting equations in extending results from the autonomous theory to a significantly larger class of equations several examples are given to illustrate our techniques

a non autonomous strongly damped wave equation existence and continuity of the pullback attractor abstract in this paper we consider the strongly damped wave equation with time dependent terms u t t u t u t e t u t f u in a bounded domain r n under some restrictions on e t t and growth restrictions on the nonlinear term f the function e t depends on a parameter e e t e 0 0 we will prove under suitable assumptions local and global well posedness using the uniform sectorial operators theory the existence and regularity of pullback attractors a e t t r uniform bounds for these pullback attractors characterization of these pullback attractors and their upper and lower semicontinuity at 0

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

ergodic bsdes and optimal ergodic control in banach spaces in this paper we introduce a new kind of backward stochastic differential equations called ergodic bsdes which arise naturally in the study of optimal ergodic control we study the existence uniqueness and regularity of solution to ergodic bsdes then we apply these results to the optimal ergodic control of a banach valued stochastic state equation we also establish the link between the ergodic bsdes and the associated hamilton jacobi bellman equation applications are given to the optimal ergodic control of stochastic partial differential equations

balanced quadriphase sequences with optimal periodic correlation properties constructed by real valued bent functions the real valued bent function was previously introduced by the authors 1991 as a generalization of the usual p ary bent function p a prime in such a way that the range of the function is the set of real numbers i e not restricted to gf p the real valued bent function was used to construct a family of 2 sup n 2 balanced quadriphase sequences of period 2 sup n 1 with optimal periodic correlation properties where n is a multiple of four a class of real valued bent functions that map the set of all the m tuples over gf 2 into the set 0 1 2 1 3 2 for an arbitrary m is given this is applied to generalize a previous construction to the case where n is even i e not restricted to a multiple of four it is also shown that the quadriphase sequences given by t novosad can be considered as one kind of sequence constructed by real valued bent functions conditions are given for some families of the quadriphase sequences constructed by some real valued bent functions to be balanced the exact distributions of the periodic correlation values are derived for the families of the balanced quadriphase sequences

adaptive control for semilinear stochastic systems an adaptive ergodic cost stochastic control problem for a partially known semilinear stochastic system in an infinite dimensional space is formulated and solved the solutions of the hamilton jacobi bellman equations for the discounted cost and the ergodic cost stochastic control problems require some special interpretations because they do not typically exist in the usual sense the solutions of the parameter dependent ergodic hamilton jacobi bellman equations are obtained from some corresponding discounted cost control problems as the discount rate tends to zero the solutions of the ergodic hamilton jacobi bellman equations are shown to depend continuously on the parameter a certainty equivalence adaptive control is given that is based on the optimal controls from the solutions of the ergodic hamilton jacobi bellman equations and a strongly consistent family of estimates of the unknown parameter this adaptive control is shown to achieve the optimal ergodic cost for the known system

kolmogorov equation associated to a stochastic navier stokes equation abstract a direct solution of the kolmogorov equation associated to a stochastic navier stokes equation is given with restriction to two space dimensions and periodic boundary conditions the existence of a variational solution is proved using a special property of the nonlinear operator

optimal adaptive control contradiction in terms or a matter of choosing the right cost functional approaching the problem of optimal adaptive control as ldquooptimal control made adaptive rdquo namely as a certainty equivalence combination of linear quadratic optimal control and standard parameter estimation fails on two counts numerical as it requires a solution to a riccati equation at each time step and conceptual as the combination actually does not possess any optimality property in this note we present a particular form of optimality achievable in lyapunov based adaptive control state and control are subject to positive definite penalties whereas the parameter estimation error is penalized through an exponential of its square which means that no attempt is made to enforce the parameter convergence but the estimation transients are penalized simultaneously with the state and control transients the form of optimality we reveal here is different from our work in z h li and m krstic ldquooptimal design of adaptive tracking controllers for nonlinear systems rdquo automatica vol 33 pp 1459 1473 1997 where only the terminal value of the parameter error was penalized we present our optimality concept on a partial differential equation pde example boundary control of a particular parabolic pde with an unknown reaction coefficient two technical ideas are central to the developments in the note a nonquadratic lyapunov function and a normalization in the lyapunov based update law the optimal adaptive control problem is fundamentally nonlinear and we explore this aspect through several examples that highlight the interplay between the non quadratic cost and value functions

on adaptive boundary control for stochastic parabolic systems with unknown potential coefficient an adaptive boundary control problem for a stochastic heat diffusion equation is studied the considered system contains an unknown potential coefficient which is a function of the spatial variables the estimation algorithm for the unknown potential coefficient is proposed by using the stochastic approximation technique after showing the strong consistency of the estimated parameter the cost for the adaptive control scheme presented here is shown to converge to the optimal ergodic cost finally some numerical examples are shown

on the location of positive limit sets for autonomous functional differential equations with infinite delay in 1983 the authors published a paper lo on an invariance principle for finite delay functional differential equations fdes in 9 t krisztin and the authors extended many of the ideas of lo to include fdes with infinite delay the basic results in both 9 and lo centered around the use of liapunov razumikhin techniques and the location of positive limit sets of precompact orbits the main purpose of this paper is to extend the results of 9 and to provide several examples to illustrate the main theorems of particular interest will be to establish asymptotic constancy of solutions of equations for which each constant function is itself a solution this idea was one of the main themes in lo but was not examined in 9 likewise we will see how a family of spaces as opposed to a single phase space can be useful in examining certain equations cf example 3 4 in considering the use of limit sets in conjunction with razumikhin techniques and invariance principles for infinite delay equations there are several difficulties that are encountered these include i choice of space s ii precompactness of orbits and iii appropriate comparison principles these topics are discussed individually in 11 7 and 8

on line parameter estimation for infinite dimensional dynamical systems the on line or adaptive identification of parameters in abstract linear and nonlinear infinite dimensional dynamical systems is considered an estimator in the form of an infinite dimensional linear evolution system having the state and parameter estimates as its states is defined convergence of the state estimator is established via a lyapunov estimate the finite dimensional notion of a plant being sufficiently rich or persistently excited is extended to infinite dimensions convergence of the parameter estimates is established under the additional assumption that the plant is persistently excited a finite dimensional approximation theory is developed and convergence results are established numerical results for examples involving the estimation of both constant and functional parameters in one dimensional linear and nonlinear heat or diffusion equations and the estimation of stiffness and damping parameters in a one dimensional wave equation with kelvin voigt viscoelastic damping are presented

homogenized dynamics of stochastic partial differential equations with dynamical boundary conditions a microscopic heterogeneous system under random influence is considered the randomness enters the system at physical boundary of small scale obstacles as well as at the interior of the physical medium this system is modeled by a stochastic partial differential equation defined on a domain perforated with small holes obstacles or heterogeneities together with random dynamical boundary conditions on the boundaries of these small holes

on the nonnegative square root of the fourth derivative operators abstract the singularities of the nonnegative square root of the fourth derivative operator au u 4 are studied

ergodic control of semilinear stochastic equations and the hamilton jacobi equation abstract in this paper we consider optimal control of stochastic semilinear equations with lipschitz continuous drift and cylindrical noise we show existence and uniqueness up to an additive constant of solutions to the stationary hamilton jacobi equation associated with the cost functional given by the asymptotic average per unit time cost as a consequence we find the optimizing controls given in the feedback form to obtain these results we prove also some new results on the transition semigroups of semilinear diffusion acting in the spaces of continuous function with the weighted sup norms and on the optimal control of semilinear diffusions for the discounted cost functional

existence results for some damped second order volterra integro differential equations abstract in this paper we make a subtle use of operator theory techniques and the well known schauder fixed point principle to establish the existence of pseudo almost automorphic solutions to some second order damped integro differential equations with pseudo almost automorphic coefficients in order to illustrate our main results we will study the existence of pseudo almost automorphic solutions to a structurally damped plate like boundary value problem

second order hamilton jacobi equations in infinite dimensions some second order hamilton jacobi equations connected to stochastic optimal control problems for infinite dimensional systems driven by a white noise are studied a direct method to prove existence and uniqueness of mild solutions is developed then this solution is identified as the value function of the related stochastic control problem and a feedback formula for optimal controls is derived

stochastic initial boundary value problems subject to distributed and boundary noise and their optimal control abstract in this paper we consider a class of stochastic evolution equations arising from initial boundary value problems with both boundary and distributed noise we prove existence and regularity of mild solutions then we consider a controlled version of the model and prove the existence of optimal controls and develop the necessary conditions of optimality for partially observed problems using relaxed controls

ergodicity of the 2 d navier stokes equation under random perturbations a 2 dimensional navier stokes equation perturbed by a sufficiently distributed white noise is considered existence of invariant measures is known from previous works the aim is to prove uniqueness of the invariant measures strong law of large numbers and convergence to equilibrium

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

second order hamilton jacobi equations in hilbert spaces and stochastic boundary control the paper is concerned with fully nonlinear second order hamilton jacobi bellman isaacs equations of elliptic type in separable hilbert spaces which have unbounded first and second order terms the viscosity solution approach is adapted to the equations under consideration and the existence and uniqueness of viscosity solutions are proved a stochastic optimal control problem driven by a parabolic stochastic pde with control of dirichlet type on the boundary is considered it is proved that the value function of this problem is the unique viscosity solution of the associated hamilton jacobi bellman equation

maximum principle for an optimal control problem associated to a spde with nonlinear boundary conditions abstract we study a control problem where the state equation is a nonlinear partial differential equation of the calculus of variation in a bounded domain perturbed by noise we allow the control to act on the boundary and set boundary conditions which result in a stochastic differential equation for the trace of the solution on the boundary this work provides necessary and sufficient conditions of optimality in the form of a maximum principle we also provide a result of existence for the optimal control in the case where the control acts linearly

probabilistic approach to the strong feller property a new probabilistic method based on the girsanov theorem for establishing the strong feller property of diffusion processes in both finite and infinite dimensional spaces is proposed applications to second order stochastic differential equations stochastic delay equations and stochastic partial differential equations of parabolic type are discussed with a twofold aim both to extend some older results usually by weakening the assumptions on the drift term and to obtain simpler proofs of them

non explosion boundedness and ergodicity for stochastic semilinear equations abstract sufficient conditions for non explosion boundedness and ergodicity of solutions of a class of semilinear stochastic equations are given applications of the results are based on a new regularity theorem for solutions of the linear problem

adaptive boundary control for unstable parabolic pdes part ii estimation based designs the certainty equivalence approach to adaptive control is commonly used with two types of identifiers passivity based identifiers and swapping identifiers the passive also known as observer based approach is the prevalent identification technique in existing results on adaptive control for pdes but has so far not been used in boundary control problems the swapping approach prevalent in finite dimensional adaptive control is employed here for the first time in adaptive control of pdes for a class of unstable parabolic pdes we prove a separation principle result for both the passive and swapping identifiers combined with the backstepping boundary controllers the result is applicable in any dimension for physical reasons we restrict our attention to dimensions no higher than three the results of the paper are illustrated by simulations

ergodic boundary point control of stochastic semilinear systems a controlled markov process in a hilbert space and an ergodic cost functional are given for a control problem that is solved where the process is a solution of a parameter dependent semilinear stochastic differential equation and the control can occur only on the boundary or at discrete points in the domain the linear term of the semilinear differential equation is the infinitesimal generator of an analytic semigroup the noise for the stochastic differential equation can be distributed boundary and point some ergodic properties of the controlled markov process are shown to be uniform in the control and the parameter the existence of an optimal control is verified to solve the ergodic control problem the optimal cost is shown to depend continuously on the system parameter

an analytic approach to existence and uniqueness for martingale problems in infinite dimensions we prove existence and uniqueness for a class of martingale problems in a hilbert space we solve the associated kolmogorov equation and prove that the corresponding semigroup is determined by a kernel of measures if a schauder type regularity is satisfied

characterization of domains of fractional powers of certain operators arising in elastic systems and applications with x a hilbert space where a the elastic operator and b the damping operator are closed operators in x with dense domains q a and 9 b respectively moreover a is throughout assumed to be self adjoint and strictly positive stimulated by two conjectures raised in crl the most general of which referred to the case lx f described below we have already studied the operators and sb defined respectively by