absence of absolutely continuous spectrum of floquet operators the spectrum of the floquet operator associated with time periodic perturbations of discrete hamiltonians is considered if the gap between successive eigenvalues j of the unperturbed hamiltonian grows as j j 1 j and the multiplicity of j grows asj with 0 asj tends to infinity then the corresponding floquet operator possesses no absolutely continuous spectrum provided the perturbation is smooth enough

eigenvalues for infinite matrices this paper is concerned with the problem of determining the location of eigenvalues for diagonally dominant infinite matrices upper and lower bounds for eigenvalues are established for tridiagonal matrices a numerical procedure for improving the bounds is given and the approximation of the eigenvectors is also discussed the techniques are illustrated for the solution of the well known mathieu s equation

perturbation of an eigenvalue from a dense point spectrum an example we study an example of a perturbed floquet hamiltonian depending on a coupling constant the spectrum is pure point and dense we pick up an eigenvalue namely and show the existence of a function defined on such that for all 0 is a point of density for the set i and the rayleigh schrodinger perturbation series represents an asymptotic series for the function all ideas are developed and demonstrated when treating the explicit example but some of them are expected to have an essentially wider range of application

stochastic optimal control with delay in the control ii verification theorem and optimal feedbacks we consider a stochastic optimal control problem governed by a stochastic differential equation with delay in the control using an existence and uniqueness result of a sufficiently regular mild solution of the associated hamilton jacobi bellman equation see the companion paper f gozzi and f masiero siam j control optim 55 2017 pp 2981 3012 we solve the control problem by proving a verification theorem and the existence of optimal feedback controls

spurious poles in pade approximation in the theory of pade approximation locally uniform convergence has been proved only for special classes of functions for much larger classes convergence in capacity has been shown to hold true the reason for one type of convergence to hold true but the other one not can be found in poles of the approximants that may occur apparently anywhere in the complex plane because of their unwanted nature and since they do not correspond to singularities of the function f to be approximated these poles are called spurious the denominators of pade approximants satisfy orthogonality relations and consequently the location and distribution of spurious poles depend on properties of the orthogonality relations in the present paper the possibility of spurious poles is studied from the perspective of these orthogonal relations

decay rates for inverses of band matrices spectral theory and classical approximation theory are used to give a new proof of the exponential decay of the entries of the inverse of band matrices the rate of decay of a 1 can be bounded in terms of the essential spectrum of aa for general a and in terms of the essential spectrum of a for positive definite a in the positive definite case the bound can be attained these results are used to establish the exponential decay for a class of generalized eigenvalue problems and to establish exponential decay for certain sparse but nonbanded matrices we also establish decay rates for certain generalized inverses

dynamic programming for optimal control problems with delays in the control variable we study a class of optimal control problems with state constraint where the state equation is a differential equation with delays in the control variable this class of problems arises in some economic applications in particular in optimal advertising problems the optimal control problem is embedded in a suitable hilbert space and the associated hamilton jacobi bellman hjb equation is considered in this space it is proved that the value function is continuous with respect to a weak norm and that it solves in the viscosity sense the associated hjb equation the main results are the proof of a directional c 1 regularity for the value function and the feedback characterization of optimal controls

normal families new perspectives this paper surveys some surprising applications of a lemma characterizing normal families of meromorphic functions on plane domains these include short and efficient proofs of generalizations of i the picard theorems ii gol dberg s theorem a meromorphic function on c which is the solution of a first order algebraic differential equation has finite order and iii the fatou julia theorem the julia set of a rational function of degree d 2 is the closure of the repelling periodic points we also discuss bloch s principle and provide simple solutions to some problems of hayman connected with this principle over twenty years ago on the way to a partial explication of the phenomenon known as bloch s principle i proved a little lemma characterizing normal families of holomorphic and meromorphic functions on plane domains 68 over the years the lemma has grown and in dextrous hands proved amazingly versatile with applications to a wide variety of topics in function theory and related areas with the renewed interest in normal families arising largely from the important role they play in complex dynamics it seems sensible to survey some of the most striking of these applications to the one variable theory with the aim of making this technique available to as broad an audience as possible that is the purpose of this report one pleasant aspect of the theory is that judicious application of the lemma often leads to proofs which seem almost magical in their brevity in such cases we have made no effort to resist the temptation to write out complete proofs hardly anything beyond a basic knowledge of function theory is required to understand what follows so the reader is urged to take courage and plough on through and now we turn to our tale 1 let d be a domain in the complex plane c we shall be concerned with analytic maps i e meromorphic functions f d r2 received by the editors october 15 1997 and in revised form may 26 1998 1991 mathematics subject classification primary 30d45 secondary 30d35 34a20 58f23

orthogonal polynomials associated polynomials and functions of the second kind abstract a survey is given of the interaction between orthogonal polynomials associated polynomials and functions of the second kind with an emphasis on asymptotic results various formulas are presented in a unified way in terms of wronskians of solutions of linear recurrence relations some of these formulas are classical and go back to the previous century but usually they are hard to locate in the literature some new formulas are also given in particular a formula expressing the derivative of an orthogonal polynomial in terms of the orthogonal polynomials and the associated polynomials

self adjointness of unbounded tridiagonal operators and spectra of their finite truncations abstract this paper addresses two different but related questions regarding an unbounded symmetric tridiagonal operator its self adjointness and the approximation of its spectrum by the eigenvalues of its finite truncations the sufficient conditions given in both cases improve and generalize previously known results it turns out that not only self adjointness helps to study limit points of eigenvalues of truncated operators but the analysis of such limit points is a key help to prove self adjointness several examples show the advantages of these new results compared with previous ones besides an application to the theory of continued fractions is pointed out

gevrey formal power series of wannier stark ladders we consider time independent schr odinger equations in one dimension with both periodic and stark potentials by means of an iterative procedure we obtain a formal power series for the wannier stark ladders in the case of strongly singular periodic potentials we prove that such a formal power series is of gevrey type

extensions of the preiss differentiability theorem abstract the preiss differentiability theorem for lipschitz functions on banach spaces is generalized to locally lower upper semi lipschitz functions and several extensions are presented

floquet spectrum for two level systems in quasiperiodic time dependent fields we study the time evolution ofn level quantum systems under quasiperiodic time dependent perturbations the problem is formulated in terms of the spectral properties of a quasienergy operator defined in an enlarged hilbert space or equivalently of a generalized floquet operator we discuss criteria for the appearance of pure point as well as continuous spectrum corresponding respectively to stable quasiperiodic dynamics and to unstable chaotic behavior we discuss two types of mechanisms that lead to instability the first one is due to near resonances while the second one is of topological nature and can be present for arbitrary ratios between the frequencies of the perturbation we treat explicitly an example of this type the stability of the pure point spectrum under small perturbations is proven using kam techniques

uniform approximation of convex function in smooth banach spaces abstract this article is devoted to smooth approximation of convex functions on banach spaces with smooth norm we prove that if x is a smooth space and f is a w lower semicontinuous lipschitzian convex function on x then there exist two w lower semicontinuous g teaux differentiable convex function sequences f n n 1 and g n n 1 such that 1 f n f n 1 f and f g n 1 g n 2 f n f and g n f uniformly on x 3 cl x x d f n x x cl x x d g n x x x

limit points of eigenvalues of truncated tridiagonal operators let t be the tridiagonal operator ten a n e n 1 a n 1 e n 1 b n e n te 1 a 1 e 2 b 1 e 1 acting on a fixed orthonormal basis e n n 1 2 of a hilbert space h let p n be the orthogonal projection on the finite dimensional space h n spanned by the elements e 1 e 2 e n and let t n be the truncated operator t n p n tp n if t has a unique self adjoint extension then the set t there exists a sequence of eigenvalues n of t n with the property n contains the spectrum t of t and examples show that in general t t for many reasons the knowledge of the equality t a t is important in this paper sufficient conditions are presented such that t t

the characteristic function for jacobi matrices with applications abstract we introduce a class of jacobi operators with discrete spectra which is characterized by a simple convergence condition with any operator j from this class we associate a characteristic function as an analytic function on a suitable domain and show that its zero set actually coincides with the set of eigenvalues of j in that domain as an application we construct several examples of jacobi matrices for which the characteristic function can be expressed in terms of special functions in more detail we study the example where the diagonal sequence of j is linear while the neighboring parallels to the diagonal are constant

hjb equations for the optimal control of differential equations with delays and state constraints i regularity of viscosity solutions we study a class of optimal control problems with state constraints where the state equation is a differential equation with delays this class includes some problems arising in economics in particular the so called models with time to build see p k asea and p j zak j econom dynam control 23 1999 pp 1155 1175 m bambi j econom dynam control 32 2008 pp 1015 1040 f e kydland and e c prescott econometrica 50 1982 pp 1345 1370 we embed the problem in a suitable hilbert space h and consider the associated hamilton jacobi bellman hjb equation this kind of infinite dimensional hjb equation has not been previously studied and is difficult due to the presence of state constraints and the lack of smoothing properties of the state equation our main result on the regularity of solutions to such an hjb equation seems to be entirely new more precisely we prove that the value function is continuous in a sufficiently big open set of h that it solves in the viscosity sense the associated hjb equation and that it has continuous classical derivative in the direction of the present this regularity result is the starting point to define a feedback map in the classical sense which gives rise to a candidate optimal feedback strategy

on the eigenvalue problem for a particular class of finite jacobi matrices abstract a function f with simple and nice algebraic properties is defined on a subset of the space of complex sequences some special functions are expressible in terms of f first of all the bessel functions of first kind a compact formula in terms of the function f is given for the determinant of a jacobi matrix further we focus on the particular class of jacobi matrices of odd dimension whose parallels to the diagonal are constant and whose diagonal depends linearly on the index a formula is derived for the characteristic function yet another formula is presented in which the characteristic function is expressed in terms of the function f in a simple and compact manner a special basis is constructed in which the jacobi matrix becomes a sum of a diagonal matrix and a rank one matrix operator a vector valued function on the complex plain is constructed having the property that its values on spectral points of the jacobi matrix are equal to corresponding eigenvectors

optimization of globally convex functions convex functions have nice properties with respect to both minimization and maximization similar properties are established here for functions that are permitted to have bad local behavior but are globally convex in the sense that they behave convexly on triples of collinear points that are widely dispersed the results illustrate a development that seems desirable in the interest of more realistic mathematical modeling the globalization of important function properties in connection with the maximization of globally convex functions over convex bodies in a given finite dimensional normed space e there is interest in estimating the maximum for points c of bodies c subset e of the ratio between two measures of how close c comes to being an extreme point of c good estimates are obtained for the cases in which e is euclidean or has the max norm

on the common zeros of bessel functions using a functional analytic method we give some results concerning common zeros of the ordinary bessel functions j z of first kind with respect to and fixed z a lower bound for the common zero z of the bessel functions j z j z where are known is also given

quadratic control of evolution equations with delays in control the quadratic cost problem of evolution equations with delays in control is considered a semigroup model which involves no explicit delays in control but contains an unbounded control operator is introduced with the aid of a family of approximating systems it is shown that the optimal feedback control and the minimum cost are characterized by the solution of a riccati equation three examples are given to illustrate the theory the filtering problem of evolution equations with observation delays is also solved through the duality relation

continuity of bethe solutions with respect to chain length n and winding numbers λl bethe solutions for r reversed spins are characterized by a set of winding numbers 1 2 r such classification is however not unique since the same sequences can describe different solutions and different sequences yield essentially equivalent states these ambiguities should find their resolution in a complete configuration we demonstrate here that in general a solution with a fixed sequence of winding numbers evolves in a quasicontinuous way as the function of n the number of spins this property could be disturbed in some cases at special transition point ntr we explain analytically the origin of this discontinuity consideration was addressed for three and four spin deviations

toeplitz matrix techniques and convergence of complex weight pade approximants one considers diagonal pade approximants about of functions of the form f z email protected 1 1 z x 1w x dx email protected 1 1 where w is an integrable possibly complex valued function defined on 1 1 convergence of the sequence of diagonal pade approximants towards f is established under the condition that there exists a weight w positive almost everywhere on 1 1 such that g x w x w x is continuous and not vanishing on 1 1 the rate of decrease of the error is also described the proof proceeds by establishing the link between the pade denominators and the orthogonal polynomials related to w in terms of the toeplitz matrix of symbol g cos f

a smooth variational principle with applications to subdifferentiability and to differentiability of convex functions we show that typically lower semicontinuous functions on a banach space densely inherit lower subderivatives of the same degree of smoothness as the norm in particular every continuous convex function on a space with a g teaux weak hadamard frechet smooth renorm is densely g teaux weak hadamard frechet differentiable our technique relies on a more powerful analogue of ekeland s variational principle in which the function is perturbed by a quadratic like function this smooth variational principle has very broad applicability in problems of nonsmooth analysis

the eigenvalue problem for infinite compact complex symmetric matrices with application to the numerical computation of complex zeros of j0 z ij1 z and of bessel functions jm z of any real order m abstract consider computing simple eigenvalues of a given compact infinite matrix re garded as operating in the complex hilbert space l 2 by computing the eigenvalues of the truncated finite matrices and taking an obvious limit in this paper we deal with a special case where the given matrix is compact complex and symmetric but not necessarily hermitian two examples of application are studied the first is con cerned with the equation j 0 z ij 1 z 0 appearing in the analysis of the solitary wave runup on a sloping beach and the second with the zeros of the bessel function j m z of any real order m in each case the problem is reformulated as an eigenvalue problem for a compact complex symmetric tridiagonal matrix operator in l 2 whose eigenvalues are all simple a complete error analysis for the numerical solution by truncation is given based on the general theorems proved in this paper where the usefulness of the seldom used generalized rayleigh quotient is demonstrated

semi analytical karhunen loeve representation of irregular waves based on the prolate spheroidal wave functions abstract a new semi analytical approach is presented to solving the matrix eigenvalue problem or the integral equation in karhunen loeve k l representation of random data such as irregular ocean waves instead of direct numerical approach to this matrix eigenvalue problem which may suffer from the computational inaccuracy for big data a pair of integral and differential equations are considered which are related to the so called prolate spheroidal wave functions pswf first the pswf is expressed as a summation of a small number of the analytical legendre functions after substituting them into the pswf differential equation a much smaller size matrix eigenvalue problem is obtained than the direct numerical k l matrix eigenvalue problem by solving this with a minimal numerical effort the pswf and the associated eigenvalue of the pswf differential equation are obtained then the eigenvalue of the pswf integral equation is analytically expressed by the functional values of the pswf and the eigenvalues obtained in the pswf differential equation finally the analytically expressed pswfs and the eigenvalues in the pwsf integral equation are used to form the kernel matrix in the k l integral equation for the representation of exemplary wave data such as ordinary irregular waves it is found that with the same accuracy the required memory size of the present method is smaller than that of the direct numerical k l representation and the computation time of the present method is shorter than that of the semi analytical method based on the sinusoidal functions

orthogonal polynomials associated with coulomb wave functions abstract a new class of orthogonal polynomials associated with coulomb wave functions is introduced these polynomials play a role analogous to that the lommel polynomials have in the theory of bessel functions the orthogonality measure for this new class is described in detail in addition the orthogonality measure problem is discussed on a more general level apart from this various identities derived for the new orthogonal polynomials may be viewed as generalizations of certain formulas known from the theory of bessel functions a key role in these derivations is played by a jacobi tridiagonal matrix j l whose eigenvalues coincide with the reciprocal values of the zeros of the regular coulomb wave function f l the spectral zeta function corresponding to the regular coulomb wave function or more precisely to the respective tridiagonal matrix is studied as well

mathieu functions for purely imaginary parameters for the mathieu differential equation y x a 2qcos x y x 0 with purely imaginary parameter q is the characteristic value a exhibits branching points we analyze the properties of the mathieu functions and their fourier coefficients in the vicinity of the branching points symmetry relations for the mathieu functions as well as the fourier coefficients behind a branching point are given a numerical method to compute mathieu functions for all values of the parameter s is presented

optimal control of heterogeneous systems basic theory a general model of a heterogeneous control system is introduced in the form of a first order distributed system with nonlocal dynamics and exogenous side conditions the heterogeneity is represented by a parameter taking values in an abstract measurable space so that both continuous and discrete heterogeneity as well as probabilistic heterogeneity without density are included a distributed and a lumped controls are involved the latter appearing also in the side conditions an existence theorem is proved for the uncontrolled system and the sensitivity of the solution with respect to the control variables is estimated the main result is an optimality condition in the form of the pontryagin local maximum principle a global maximum principle holds for the distributed control under an additional condition that rules out discrete measurable heterogeneity spaces a number of possible applications are outlined age structured systems size structured systems nonlocal advection reaction equations static parametric heterogeneity in epidemiology and two stage control systems with uncertain switching time

the infinite time quadratic control problem for linear systems with state and control delays an evolution equation approach we show how a linear differential delay equation with delays in the control can be reformulated as an evolution equation with bounded input operator as a simple application we solve an infinite time quadratic cost control problem for the delay differential equation we also point out how our results even when specialized to the case where delays in the control disappear extend results previously published in the literature

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

two characterizations of super reflexive banach spaces by the behaviour of differences of convex functions abstract a function defined on a banach space x is called convex if it can be represented as a difference of two continuous convex functions in this work we study the relationship between some geometrical properties of a banach space x and the behaviour of the class of all convex functions defined on it more precisely we provide two new characterizations of super reflexivity in terms convex functions

multiple orthogonal polynomials results on multiple orthogonal polynomials will be surveyed multiple orthogonal polynomials are intimately related to hermite pade approximants and often they are also called hermite pade polynomials special attention will be paid to an application of multiple orthogonal polynomials and to analytic theory of two model families of general multiple orthogonal polynomials referred to as angelesco and nikishin systems among the applications the number theory special functions and spectral analysis of nonsymmetric band operators will be highlighted in the analytic theory results and methods for the study of multiple orthogonal polynomials asymptotics will be reviewed new results on strong asymptotics of multiple orthogonal polynomials for nikishin system will be presented

generic frechet differentiability of convex functions dominated by a lower semicontinuous convex function abstract in this paper an extended real valued proper lower semicontinuous convex function f on a banach space is said to have the frechet differentiability property fdp if every proper lower semicontinuous convex function g with g f is frechet differentiable on a dense g subset of int dom g the interior of the effective domain of g we show that f has the fdp if and only if the w closed convex hull of the image of the subdifferential map of f has the radon nikod m property this is a generalization of the main theorem in a paper by lixin and shuzhong to appear according to this result it also gives several new criteria of asplund spaces

special functions and spectrum of jacobi matrices abstract several examples of jacobi matrices with an explicitly solvable spectral problem are worked out in detail in all discussed cases the spectrum is discrete and coincides with the set of zeros of a special function moreover the components of corresponding eigenvectors are expressible in terms of special functions as well our approach is based on a recently developed formalism providing us with explicit expressions for the characteristic function and eigenvectors of jacobi matrices this is done under an assumption of a simple convergence condition on matrix entries among the treated special functions there are regular coulomb wave functions confluent hypergeometric functions q bessel functions and q confluent hypergeometric functions in addition in the case of q bessel functions we derive several useful identities

smooth approximation of convex functions in banach spaces abstract this paper shows that every extended real valued lower semi continuous proper respectively lipschitzian convex function defined on an asplund space can be represented as the point wise limit respectively uniform limit on every bounded set of a sequence of lipschitzian convex functions which are locally affine hence c at all points of a dense open subset and shows an analogous for w lower semi continuous proper respectively lipschitzian convex functions defined on dual spaces whose pre duals have the radon nikodym property

quantum averaging for driven systems with resonances abstract we discuss the effects of resonances in driven quantum systems within the context of quantum averaging techniques in the floquet representation we consider in particular iterative methods of kam type and the extensions needed to take into account resonances the approach consists in separating the coupling terms into resonant and nonresonant components at a given scale of time and intensity the nonresonant part can be treated with perturbative techniques which we formulate in terms of kam type unitary transformations that are close to the identity these can be interpreted as averaging procedures with respect to the dynamics defined by effective uncoupled hamiltonians the resonant parts are treated with a different kind of unitary transfomations that are not close to the identity and are adapted to the structure of the resonances they can be interpreted as a renormalization of the uncoupled hamiltonian that yields an effective dressed hamiltonian around which a perturbation expansion can be developed the combination of these two ingredients provides a strongly improved approximation technique

computation of multiple eigenvalues of infinite tridiagonal matrices in this paper it is first given as a necessary and sufficient condition that infinite matrices of a certain type have double eigenvalues the computation of such double eigenvalues is enabled by the newton method of two variables the three term recurrence relations obtained from its eigenvalue problem evp subsume the well known relations of a the zeros of j v z b the zeros of zj f v z hj v z c the evp of the mathieu differential equation and d the evp of the spheroidal wave equation the results of experiments are shown for the three cases a c for the computation of their double pairs

on the zeros of a class of polynomials defined by a three term recurrence relation abstract let p n 1 x be the polynomial which is defined recursively by p 0 x 0 p 1 x 1 and n p n 1 x n 1 p n 1 x b n p n x xd n p n x n 1 2 n where n b n d n are real sequences with n 0 for every n 1 2 n and k terms of the sequence d n n 1 0 k n are equal to zero it is proved that if b n 0 and the sequence 2 n b n b n 1 n 1 is a chain sequence then the polynomial p n 1 x is of degree n k and has real and simple zeros different from zero this result generalizes and simplifies previously known results with respect to the class of polynomials whose zeros are real and simple

hamilton jacobi equations arising from boundary control problems with state constraints the analysis of a class of infinite dimensional hamilton jacobi bellman hjb equations is undertaken related to linear convex boundary control problems for pdes with constraints on the state a definition of generalized solution namely weak solution of the hjb equation is provided weak is the limit of strong while strong is the limit of classical as defined in s faggian appl math optim 51 2005 pp 123 162 consequently an existence and uniqueness result is provided for weak solutions of the hjb equation as well as existence of an optimal control being the limit of optimal controls of approximating problems the study then describes an economic application to optimal investment with vintage capital having positivity constraints on the capital

c algebras and numerical linear algebra given a self adjoint operator a on a hilbert space suppose that one wishes to compute the spectrum of a numerically in practice these problems often arise in such a way that the matrix of a relative to a natural basis is sparse for example discretized second order differential operators can be represented by doubly infinite tridiagonal matrices in these cases it is easy and natural to compute the eigenvalues of large n n submatrices of the infinite operator matrix and to hope that if n is large enough then the resulting distribution of eigenvalues will give a good approximation to the spectrum of a numerical analysts call this the galerkin method while this hope is often realized in practice it often fails as well and it can fail in spectacular ways the sequence of eigenvalue distributions may not converge as n or they may converge to something that has little to do with the original operator a at another level even the meaning of convergence has not been made precise in general in this paper we determine the proper general setting in which one can expect convergence and we describe the asymptotic behavior of the n n eigenvalue distributions in all but the most pathological cases under appropriate hypotheses we establish a precise limit theorem which shows how the spectrum of a is recovered from the sequence of eigenvalues of the n n compressions in broader terms our results have led us to the conclusion that numerical problems involving infinite dimensional operators require a reformulation in terms ofc algebras indeed it is only when the single operator a is viewed as an element of an appropriate c algebra a that one can see the precise nature of the limit of the n n eigenvalue distributions the limit is associated with a tracial state on a normally a is highly noncommutative and in our main applications it is a simple c algebra having a unique tracial state we obtain precise asymptotic results for operators which represent discretized hamiltonians of one dimensional quantum systems with arbitrary continuous potentials

some results for an optimal control problem with semilinear state equation a control problem governed by a semilinear state eqution depending on a small parameter varepsilon is considered it is proven that for varepsilon sufficiently small the associated hamilton jacobi equation has a unique strict solution consequently the control problem can be solved by dynamic programming method

a pontryagin maximum principle for infinite dimensional problems a basic idea of the classical approach for obtaining necessary optimality conditions in optimal control is to construct suitable needle like control variations we use this idea to prove the main result of the present paper a pontryagin maximum principle for infinite dimensional optimal control problems with pointwise terminal constraints in arbitrary banach state space by refining the classical variational technique we are able to replace the differentiability of the norm of the state space guaranteed by the strict convexity of its dual norm which is assumed in the known results by a separation argument we also drop another key assumption which is common in the existing literature on infinite dimensional control problems that the set of variations in the state space of the state trajectory s endpoint resulting from the control variations be finite codimensional instead we require only that it has nonempty interior in its closed affine hull as an application of the abstract result we present an illustrative example an optimal control problem for an age structured system with pointwise terminal state constraints

the eigenvalue problem for infinite complex symmetric tridiagonal matrices with application abstract we consider an infinite complex symmetric not necessarily hermitian tridiagonal matrix t whose diagonal elements diverge to in modulus and whose off diagonal elements are bounded we regard t as a linear operator mapping a maximal domain in the hilbert space l2 into l2 assuming the existence of t 1 we consider the problem of approximating a given simple eigenvalue of t by an eigenvalue n of tn and nth order principal submatrix of t let x x 1 x 2 t be an eigenvector corresponding to assuming xtx 0 and f n 1 x n 1 x n 0 as n we show that there exists a sequence n of tn such that n f n 1 x n 1 1 o 1 x t x 0 where fn 1 represents the n n 1 element of t application to the following problems is included a solve jv z 0 for v given z 0 and b compute the eigenvalues of the mathieu equation fortunately the existence of t 1 need not be verified for these examples since we may show that t i with taken appropriately has an inverse

generic frechet differentiability of convex functions on non asplund spaces abstract let f be a continuous convex function on a banach space e this paper shows that every proper convex function g on e with g f is generically frechet differentiable if and only if the image of the subdifferential map f of f has the radon nikod m property and in this case it is equivalent to showing that the image of f is separable on each separable subspace of e

strong solutions for differential equations in abstract spaces abstract let e f be a locally convex space we denote the bounded elements of e by e b x e x f sup f x in this paper we prove that if b e b is relatively compact with respect to the f topology and f i e b e b is a measurable family of f continuous maps then for each x 0 e b there exists a norm differentiable i e differentiable with respect to the f norm local solution to the initial valued problem u t t f t u t u t 0 x 0 all of this machinery is developed to study the lipschitz stability of a nonlinear differential equation involving the hardy littlewood maximal operator

a smooth variational principle with applications to hamilton jacobi equations in infinite dimensions abstract we prove that if x is a banach space which admits a smooth lipschitzian bump function then for every lower semicontinuous bounded below function there exists a lipschitzian smooth function g on x such that f g attains its strong minimum on x thus extending a result of borwein and preiss we then show how the above result can be used to obtain existence and uniqueness results of viscosity solutions of hamilton jacobi equations in infinite dimensional banach spaces a without assuming the radon nikodym property

differentiability of lipschitz functions on banach spaces abstract the main result of this note says that if the norm of a banach space e is differentiable frechet gateaux or in some intermediate sense away from the origin then every locally lipschitz function defined on an open subset g of e is differentiable in the same sense at every point of some subset of g dense in g as the main corollary we prove that every lipschitz function on an asplund space is frechet differentiable on a dense set moreover in all these results the mean value theorem for the points of differentiability which is equivalent to the possibility of computing the clarke subdifferential from the points of differentiability is shown to hold

variable metric random pursuit we consider unconstrained randomized optimization of smooth convex objective functions in the gradient free setting we analyze random pursuit rp algorithms with fixed f rp and variable metric v rp the algorithms only use zeroth order information about the objective function and compute an approximate solution by repeated optimization over randomly chosen one dimensional subspaces the distribution of search directions is dictated by the chosen metric variable metric rp uses novel variants of a randomized zeroth order hessian approximation scheme recently introduced by leventhal and lewis optimization 60 3 329 345 2011 doi 10 1080 02331930903100141 we here present 1 a refined analysis of the expected single step progress of rp algorithms and their global convergence on strictly convex functions and 2 novel convergence bounds for v rp on strongly convex functions we also quantify how well the employed metric needs to match the local geometry of the function in order for the rp algorithms to converge with the best possible rate our theoretical results are accompanied by numerical experiments comparing v rp with the derivative free schemes cma es implicit filtering nelder mead newuoa pattern search and nesterov s gradient free algorithms

stochastic optimal control with delay in the control i solving the hjb equation through partial smoothing stochastic optimal control problems governed by delay equations with delay in the control are usually more difficult to study than those in which the delay appears only in the state this is particularly true when we look at the associated hamilton jacobi bellman hjb equation indeed even in the simplified setting introduced first by vinter and kwong siam j control optim 19 1981 pp 139 153 for the deterministic case the hjb equation is an infinite dimensional second order semilinear partial differential equation that does not satisfy the so called structure condition which substantially means that the control can act on the system modifying its dynamics at most along the same directions upon which the noise acts the absence of such a condition together with the lack of smoothing properties a common feature of problems with delay prevents the use of known techniques based on backward stochastic differential equations or on the smoothing properties of the linear part to prove the

optimal control of age structured systems with mixed state control constraints abstract the paper deals with a general optimal control problem for age structured systems a necessary optimality condition of pontryagin type is obtained where the novelty is in that mixed control state constraints are present the proof uses an abstract lagrange multiplier theorem and the main difficulty is to obtain regularity of the lagrange multipliers in the particular problem at hand

common zeros of two bessel functions ii approximations and tables in 1 it was shown that two bessel functions j x je xx could have two zeros which were common to both functions and a computer program was made which takes approximate values of v y and jv k jy h and jvk jy h and from them computes the exact values here it will be shown how to find the necessary approximate values to initiate the computation a table of the smaller ratios m n with the orders of the functions less than one hundred is given

necessary conditions for a weak minimum in optimal control problems with integral equations subject to state and mixed constraints the first order necessary optimality conditions for a weak minimum are derived for optimal control problems with volterra type integral equations considered on a fixed time interval subject to endpoint constraints of equality and inequality type mixed state control constraints of inequality and equality type and pure state constraints of inequality type the main assumption is the uniform linear positive independence of the gradients of active mixed constraints with respect to the control the conditions obtained generalize the euler lagrange equation as a stationarity condition for the lagrange problem in the classical calculus of variations with ordinary differential equations

an alternative proof of a theorem of stieltjes and related results abstract let p n x n 1 be the orthogonal polynomials defined by a n p n 1 x a n 1 p n 1 x b n p n x xp n x p 0 x 0 p 1 x 1 where both sequences a n and b n are bounded and a n 0 assume that x is the unique up to a constant distribution function which corresponds to the measure of orthogonality of p n x and denote by s the spectrum of x alternative proofs of a theorem due to stieltjes and of a conjecture due to maki concerning the limit points of s are given a typical example to the maki s conjecture together with a general result concerning the density of the zeros of the polynomials p n x covers as a particular case a theorem of chihara which generalizes the well known theorem of blumenthal

hjb equations for the optimal control of differential equations with delays and state constraints ii verification and optimal feedbacks this paper which is the natural continuation of part i s federico b goldys and f gozzi siam j control optim 48 2010 pp 4910 4937 studies a class of optimal control problems with state constraints where the state equation is a differential equation with delays in part i the problem is embedded in a suitable hilbert space h and the regularity of the associated hamilton jacobi bellman equation is studied the goal of the present paper is to exploit the regularity result of part i to prove a verification theorem and find optimal feedback controls for the problem while it is easy to define a feedback control formally following the classical case the proof of its existence and optimality is hard due to lack of full regularity of v and to the infinite dimensionality of the problem the theory developed is applied to study economic problems of optimal growth for nonlinear time to build models in particular we show the existence and uniqueness of optimal controls and their characterization as feedbacks

generalized one sided estimates for solutions of hamilton jacobi equations and applications on etudie les proprietes locales des solutions de l equation de hamilton jacobi u t h t x u 0 et plus generalement de l equation u t h t x u u 0 ou t o t et x r n

optimization of convex functions with random pursuit we consider unconstrained randomized optimization of convex objective functions we analyze the random pursuit algorithm which iteratively computes an approximate solution to the optimization problem by repeated optimization over a randomly chosen one dimensional subspace this randomized method only uses zeroth order information about the objective function and does not need any problem specific parametrization we prove convergence and give convergence rates for smooth objectives assuming that the one dimensional optimization can be solved exactly or approximately by an oracle a convenient property of random pursuit is its invariance under strictly monotone transformations of the objective function it thus enjoys identical convergence behavior on a wider function class to support the theoretical results we present extensive numerical performance results of random pursuit two gradient free algorithms recently proposed by nesterov and a classical adaptive step size random search scheme we also pres

on the degree and separability of nonconvexity and applications to optimization problems we study qualitative indications for d c representations of closed sets in and functions on hilbert spaces the first indication is an index of nonconvexity which can be regarded as a measure for the degree of nonconvexity we show that a closed set is weakly closed if this indication is finite using this result we can prove the solvability of nonconvex minimization problems by duality a minimization problem on a feasible set in which this indication is low can be reduced to a quasi concave minimization over a convex set in a low dimensional space the second indication is the separability which can be incorporated in solving dual problems both the index of nonconvexity and the separability can be characteristics to good d c representations for practical computation we present a notion of clouds which enables us to obtain a good d c representation for a class of nonconvex sets using a generalized caratheodory s theorem we present various applications of clouds

optimality conditions for age structured control systems abstract we consider a fairly general model extension of the gurtin maccamy model of population dynamics of an age structured control system with nonlocal dynamics and nonlocal boundary conditions a necessary optimality condition is obtained in the form of pontryagin s maximum principle which is applicable to a number of practically meaningful models where the previously known results fail we discuss such models an epidemic control and a capital accumulation model as illustrations

complex jacobi matrices complex jacobi matrices play an important role in the study of asymptotics and zero distribution of formal orthogonal polynomials fops the latter are essential tools in several fields of numerical analysis for instance in the context of iterative methods for solving large systems of linear equations or in the study of pade approximation and jacobi continued fractions in this paper we present some known and some new results on fops in terms of spectral properties of the underlying infinite jacobi matrix with a special emphasis to unbounded recurrence coefficients here we recover several classical results for real jacobi matrices the inverse problem of characterizing properties of the jacobi operator in terms of fops and other solutions of a given three term recurrence is also investigated this enables us to give results on the approximation of the resolvent by inverses of finite sections with applications to the convergence of pade approximants

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

intrinsic cylindrical and spherical waves intrinsic waveforms associated with cylindrical and spherical bessel functions are obtained by eliminating the factors responsible for the inverse radius and inverse square radius laws of wave power per unit area of wavefront the resulting expressions are riccati bessel functions for both cases and these can be written in terms of amplitude and phase functions of order v and wave variable z when z is real it is shown that a spatial phase angle of the intrinsic wave can be defined and this together with its amplitude function is systematically investigated for a range of fixed orders and varying z the derivatives of riccati bessel functions are also examined all the component functions exhibit different behaviour in the near field depending on the order being less than equal to or greater than 1 2 plots of the phase angle can be used to display the locations of the zeros of the general riccati bessel functions and lead to new relations concerning the ordering of the real zeros of bessel functions and the occurrence of multiple zeros when the argument of the bessel function is fixed

the classical moment problem as a self adjoint finite difference operator abstract this is a comprehensive exposition of the classical moment problem using methods from the theory of finite difference operators among the advantages of this approach is that the nevanlinna functions appear as elements of a transfer matrix and convergence of pade approximants appears as the strong resolvent convergence of finite matrix approximations to a jacobi matrix as a bonus of this we obtain new results on the convergence of certain pade approximants for series of hamburger

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

optimal control with infinite horizon for distributed parameter systems with constrained controls optimal control problems for systems governed by linear partial differential state equations with constrained controls are studied if the dynamics are stabilizable with respect to the cost it is proved that the value function is a generalized solution for the associated stationary hamilton jacobi equation moreover the feedback formula and uniqueness are proved under suitable assumptions