quadratic integral games and causal synthesis the game problem for an input output system governed by a volterra integral equation with respect to a quadratic performance functional is an untouched open problem in this paper it is studied by a new approach called projection causality the main result is the causal synthesis which pro vides a causal feedback implementation of the optimal strategies in the saddle point sense the linear feedback operator is determined by the solution of a fredholm integral operator equation which is independent of data functions and control functions two application examples are included the first one is quadratic differential games of a linear system with arbitrary finite delays in the state variable and control variables the second is the standard linear quadratic differential games for which it is proved that the causal synthesis can be reduced to a known result where the feedback operator is determined by the solution of a differential riccati operator equation

laplace maps and constraints for a class of third order partial differential operators we explore the existence of a class of generalised laplace maps for third order partial differential operators of the form 1 2 3 a1 2 3 a2 1 3 a3 1 2 a12 3 a23 1 a13 2 a123 and related first order 3 systems and show that they require the satisfaction of constraints on the invariants for such operators

backward doubly stochastic volterra integral equations and their applications abstract in this paper we introduce a new class of equations called backward doubly stochastic volterra integral equations bdsvies for short first the well posedness of bdsvies in the sense of introduced m solutions is established second a comparison theorem of bdsvies is proved as an application of the comparison theorem we derive the existence of solutions of bdsvies with continuous coefficients third a duality principle between linear forward doubly stochastic volterra integral equations fdsvies for short and bdsvies is obtained moreover by virtue of the duality principle a maximum principle of pontryagin type is established for an optimal control problem of fdsvies

new trends towards lower energy estimates and optimality for nonlinearly damped vibrating systems abstract we present an approach based on comparison principles for energy and interpolation properties to derive lower energy estimates for nonlinearly either locally damped or boundary damped vibrating systems we show how the dissipation relation provides strong information on the asymptotic behavior of the energy of solutions the geometrical situations are either one dimensional or radial two dimensional or three dimensional for annulus domains we also consider the case of general domains but in this case for solutions with bounded velocities in time and space in all these cases the nonlinear damping function is assumed to have arbitrary strictly sublinear growth at the origin we give results for strong solutions and stronger lower estimates for smoother solutions the results are presented in two forms either on the side of energy comparison principles or through time pointwise lower estimates under additional geometric assumptions we give the resulting lower and upper estimates for four representative examples of damping functions we further give a weak lower estimate in the sense of a certain lim sup t and an upper estimate of the velocity for smoother solutions in case of general damping functions and for radial as well as multi dimensional domains we also discuss these estimates in the framework of optimality which is not proved here and indicate open problems raised by these results

backward stochastic volterra integral equations with additive perturbations the paper discusses a large class of stochastic volterra integral equations whose coefficients additively depend on small perturbations their solutions are compared in the l2 sense with the solutions of the appropriate unperturbed equations of the equal or simpler type more precisely we prove for an arbitrary 0 that there exists an interval t t 0 t on which the l2 difference between the solutions of the perturbed and unperturbed equations is less than

causal feedback optimal control for volterra integral equations the optimal control problem for volterra integral equations with respect to quadratic criteria is studied by a projection causality approach the work features a synthesis result where the optimal control is implemented via a linear causal feedback in which the feedback operator is determined by solving an independent fredholm integral operator equation

global existence and asymptotic behavior for a coupled hyperbolic system with localized damping abstract in this paper we investigate the global existence and energy decay rate for the solution of a coupled hyperbolic system the semi explicit energy decay rate is established by using piecewise multiplier techniques and weighted integral inequality we extend the energy decay result in alabau boussouira f alabau boussouira convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems appl math optim 51 2005 61 105 for a single equation to the coupled hyperbolic system

half riordan array sequences abstract we construct sequences of vertical and horizontal halves of various riordan arrays by using the vertical and horizontal halve riordan array transform operators the sequence characterizations of halves of riordan arrays are presented and applied to the halve riordan array transformation the conditions of transforming a riordan array to a pseudo involution riordan array by using the halve riordan array transform are given the condition of preserving the elements of a certain subgroup of the riordan group under the halve riordan array transformation are shown other properties of the halves of riordan arrays and their entries such as related recurrence relations double variable generating functions combinatorial interpretations are also studied

a unified approach via convexity for optimal energy decay rates of finite and infinite dimensional vibrating damped systems with applications to semi discretized vibrating damped systems the liapunov method is celebrated for its strength to establish strong decay of solutions of damped equations extensions to infinite dimensional settings have been studied by several authors see e g haraux 1991 11 and komornik and zuazua 1990 17 and references therein results on optimal energy decay rates under general conditions of the feedback is far from being complete the purpose of this paper is to show that general dissipative vibrating systems have structural properties due to dissipation we present a general approach based on convexity arguments to establish sharp optimal or quasi optimal upper energy decay rates for these systems and on comparison principles based on the dissipation property and interpolation inequalities in the infinite dimensional case for lower bounds of the energy we stress the fact that this method works for finite as well as infinite dimensional vibrating systems and as well as for applications to semi discretized nonlinear damped vibrating pde s a part of this approach has been introduced in alabau boussouira 2004 2005 1 2 in the present paper we identify a new simple and explicit criteria to select a class of nonlinear feedbacks for which we prove a simplified explicit energy decay formula comparatively to the more general but also more complex formula we give in alabau boussouira 2004 2005 1 2 moreover we prove optimality of the decay rates for this class in the finite dimensional case this class includes a wide range of feedbacks ranging from very weak nonlinear dissipation exponentially decaying in a neighborhood of zero to polynomial or polynomial logarithmic decaying feedbacks at the origin in the infinite dimensional case we establish a comparison principle on the energy of sufficiently smooth solutions through the dissipation relation this principle relies on suitable interpolation inequalities it allows us to give lower bounds for the energy of smooth initial data for the one dimensional wave equation with a distributed polynomial damping which improves haraux 1995 12 lower estimate of the energy for this case we also establish lower bounds in the multi dimensional case for sufficiently smooth solutions when such solutions exist we further mention applications of these various results to several classes of pde s namely the locally and boundary damped multi dimensional wave equation the locally damped plate equation and the globally damped coupled timoshenko beams system but it applies to several other examples furthermore we show that these optimal energy decay results apply to finite dimensional systems obtained from spatial discretization of infinite dimensional damped systems we illustrate these results on the one dimensional locally damped wave and plate equations discretized by finite differences and give the optimal energy decay rates for these two examples these optimal rates are not uniform with respect to the discretization parameter we also discuss and explain why optimality results have to be stated differently for feedbacks close to linear behavior at the origin

matrix characterizations of riordan arrays abstract here we discuss two matrix characterizations of riordan arrays p matrix characterization and a matrix characterization p matrix is an extension of the stieltjes matrix defined in 28 and the production matrix defined in 8 by modifying the marked succession rule introduced in 21 a combinatorial interpretation of the p matrix is given the p matrix characterizations of some subgroups of riordan group are presented which are used to find some algebraic structures of the subgroups we also give the p matrix characterizations of the inverse of a riordan array and the product of two riordan arrays a matrix characterization is defined in 20 and it is proved to be a useful tool for a riordan array while on the other side the a sequence characterization is very complex sometimes by using the fundamental theorem of riordan arrays a method of construction of a matrix characterizations from riordan arrays is given the converse process is also discussed several examples and applications of two matrix characterizations are presented

mean field linear quadratic gaussian lqg games for stochastic integral systems in this technical note we formulate and investigate a class of mean field linear quadratic gaussian lqg games for stochastic integral systems unlike other literature on mean field games where the individual states follow the controlled stochastic differential equations sdes the individual states in our large population system are characterized by a class of stochastic volterra type integral equations we obtain the nash certainty equivalence nce equation and hence derive the set of associated decentralized strategies the epsilon nash equilibrium properties are also verified due to the intrinsic integral structure the techniques and estimates applied here are significantly different from those existing results in mean field lqg games for stochastic differential systems for example some fredholm equation in the mean field setup is introduced for the first time as for applications two types of stochastic delayed systems are formulated as the special cases of our stochastic integral system and relevant mean field lqg games are discussed

schroder matrix as inverse of delannoy matrix abstract using riordan arrays we introduce a generalized delannoy matrix by weighted delannoy numbers it turns out that delannoy matrix pascal matrix and fibonacci matrix are all special cases of the generalized delannoy matrices meanwhile schroder matrix and catalan matrix also arise in involving inverses of the generalized delannoy matrices these connections are the focus of our paper the half of generalized delannoy matrix is also considered in addition we obtain a combinatorial interpretation for the generalized fibonacci numbers

invariants of hyperbolic partial differential operators we present a construction of a large class of laplace invariants for linear hyperbolic partial differential operators of fairly general form and arbitrary order

periodic output feedback stabilization of neutral systems a new feedback control technique called periodic output feedback is investigated in the context of infinite dimensional linear systems modeled by neutral functional differential equations in this method discrete output samples are multiplied by a periodic gain function to generate a continuous feedback control this work focuses on stabilization of neutral systems with delayed control modeled in the state space w sub 2 sup 1 r 0 r sup n spl times l sup 2 r 0 r sup p where w sub 2 sup 1 r 0 r sup n denotes the sobolev space of r sup n valued absolutely continuous functions with square integrable derivatives on r 0 we show that a class of these systems can be stabilized by periodic output feedback even though their input operators are unbounded we overcome this difficulty by representing the system state using an abstract integral variation of constants formula an algorithm is presented at the end of this paper to construct a periodic output feedback gain function an example is provided to illustrate the construction of the gain function

half of a riordan array and restricted lattice paths abstract for an infinite lower triangular matrix g g n k n k 0 we define the half of g to be the infinite lower triangular matrix h h n k n k 0 such that h n k g 2 n k n for all n k 0 in this paper we will show that if g g n k n k 0 is a riordan array then its half h h n k n k 0 is also a riordan array and we obtain new combinatorial interpretations for some riordan arrays in terms of weighted lattice paths

the linear algebra of the pascal matrix abstract pascal s triangle can be represented as a square matrix in two basically different ways as a lower triangular matrix pn or as a full symmetric matrix qn it has been found that the pn pnt is the cholesky factorization of qn pn can be factorized by special summation matrices it can be shown that the inverses of these matrices are the operators which perform the gaussian elimination steps for calculating cholesky s factorization by applying linear algebra we produce combinatorial identities and an existence theorem for diophantine equation systems finally an explicit formula for the sum of the kth powers is given

adapted solution of a backward stochastic differential equation abstract let wt t 0 1 be a standard k dimensional weiner process defined on a probability space f p and let ft denote its natural filtration given a f1 measurable d dimensional random vector x we look for an adapted pair of processes x t y t t 0 1 with values in rd and rd k respectively which solves an equation of the form x t t 1 f s x s y s ds t 1 g s x s y s dw s x a linearized version of that equation appears in stochastic control theory as the equation satisfied by the adjoint process we also generalize our results to the following equation x t t 1 f s x s y s ds t 1 g s x s y s dw s x under rather restrictive assumptions on g

a general stochastic maximum principle for optimal control problems the maximum principle for nonlinear stochastic optimal control problems in the general case is proved the control domain need not be convex and the diffusion coefficient can contain a control variable

palindromes and pseudo involution multiplication abstract a riordan array g f is called a pseudo involution if g f m or equivalently m g f where m 1 z is an involution this paper presents a palindromic property of pseudo involutions which seems both novel and useful if a and b are both pseudo involutions then so is the triple product aba with this it follows that if a b c are pseudo involutions so is any palindromic word using these symbols

a unified approach for the catalan matrices by using riordan arrays abstract we study generalized catalan matrices based on the riordan array and fuss catalan numbers a unified combinatorial interpretation for the entries of the generalized catalan matrices is presented by means of m dyck paths some properties and examples of the generalized catalan matrices are given including a new convolution formula for the generalized catalan numbers finally we present applications of generalized catalan matrices to the problems in counting the hill free and lower peak free m dyck paths

fuss catalan matrices their weighted sums and stabilizer subgroups of the riordan group abstract in this paper we present the riordan arrays called fuss catalan matrices which are constructed by the convolutions of the generating functions of the fuss catalan numbers we also discuss weighted sums of the fuss catalan matrices using such matrices as transformations of recursive sequences and their connection with stabilizer subgroups of the riordan group

on the halves of a riordan array and their antecedents abstract every riordan array has what we call a horizontal half and a vertical half these halves of a riordan array have been studied separately before here we place them in a common context showing that one may be obtained from the other using them we provide a canonical factorization of elements of the associated or lagrange subgroup of the riordan group the vertical half matrix is shown to be an element of the hitting time group we also ask and answer the question given a riordan array when is it the half either horizontal of vertical of a riordan array

obituary to a j pritchard on the 12th of august 2007 tony pritchard professor of mathematics at the university of warwick died suddenly at the age of 70 from a heart attack while on holiday with his family on the island of lanzarote both of the authors have had the privilege of working with him for many years we have lost both a colleague and a friend the following notes are dedicated to his memory

parametric catalan numbers and catalan triangles abstract here presented a generalization of catalan numbers and catalan triangles associated with two parameters based on the sequence characterization of bell type riordan arrays among the generalized catalan numbers a class of large generalized catalan numbers and a class of small generalized catalan numbers are defined which can be considered as an extension of large schroder numbers and small schroder numbers respectively using the characterization sequences of bell type riordan arrays some properties and expressions including the taylor expansions of generalized catalan numbers are given a few combinatorial interpretations of the generalized catalan numbers are also provided finally a generalized motzkin numbers and motzkin triangles are defined similarly an interrelationship among parametrical catalan triangle pascal triangle and motzkin triangle is presented based on the sequence characterization of bell type riordan arrays

identities induced by riordan arrays historically there exist two versions of the riordan array concept the older one better known as recursive matrix consists of bi infinite matrices d n k n k is an element of z k n implies d n k 0 deals with formal laurent series and has been mainly used to study algebraic properties of such matrices the more recent version consists of infinite lower triangular arrays d n k n k is an element of n k n implies d n k 0 deals with formal power series and has been used to study combinatorial problems here we show that every riordan array induces two characteristic combinatorial sums in three parameters n k m is an element of z these parameters can he specialized and generate an indefinite number of other combinatorial identities which are valid in the hi infinite realm of recursive matrices

symmetrical martingale solutions of backward doubly stochastic volterra integral equations abstract this paper aims to study a new class of integral equations called backward doubly stochastic volterra integral equations bdsvies for short the notion of symmetrical martingale solutions sm solutions for short is introduced for bdsvies and the existence and uniqueness theorem for bdsvies in the sense of sm solutions is established

optimal control of volterra equations with impulses we consider an optimal control problem for a system governed by a volterra integral equation with impulsive terms the impulses act on both the state and the control the control consists of switchings at discrete times the cost functional includes both an integrated cost rate continuous part and switching costs at the discrete impulse times discrete part we prove necessary optimality conditions of a form analogous to a discrete maximum principle for the particular case of a system governed by impulsive ordinary differential equations we obtain an impulsive maximum principle as a special case of the necessary optimality conditions for impulsive volterra equations

twisted laplace maps we consider general darboux maps arising from intertwining relations on second order linear partial differential operators as deformations of the classical laplace case we present lax pairs for the corresponding relations on invariants and discuss the conditions for a lattice structure analogous to 2d toda theory

notes on the total positivity of riordan arrays abstract let r d t h t be a riordan array we show that if both d t and h t are polya frequency formal power series then r is totally positive

numerical computation of the nonlinear feedback operators for the nonquadratic time variant optimal control problems in this paper a reliable numerical algorithm for the nonlinear feedback operators for the time variant optimal control with nonquadratic criteria is constructed our approach rests mainly on the modified adomian decomposition method the convergence of the decomposition method as applied to the synthesis equations is investigated the proposed scheme avoids the complexity provided by using other techniques

optimal processes governed by integral equations abstract we consider a control process governed by an integral equation with an integral constraint we derive optimality conditions using relaxed controls existence of optimal pairs is automatic furthermore for problems with linear and certain quadratic cost functions with linear integral equations and linear integral constraint the optimal controls are actually ordinary controls in the case of the linear problem considered if the control set is a convex polyhedron the control is piecewise continuous with values in the vertices of the polyhedron control processes governed by integral equations have been considered in the work of v r vinokurov siam 7 324 355 1969 however in op cit existence is not shown for the general problem and the method is rather complicated furthermore the work in op cit has some errors as pointed out in neustadt and warga s comments on vinokurov siam 8 572 1970

quadratic optimization for infinite dimensional linear differential difference type systems syntheses via the fredholm equation a closed loop solution for quadratic optimization is presented for control systems whose model is given by infinite dimensional linear differential difference equations frac dx t dt ax t sum i 1 n a i x left t h i right bu t sum j 1 m b j u left t r j right a nonsemigroup evolution formula for the fundamental solution is given first the system is then reduced to a volterra integral system by the semicausality approach a new optimality principle is established and the closed loop optimal control is given by real time state feedback plus retarded state integral feedback the feedback operator is determined by solving a linear fredholm integral equation the two crucial methods semicausal dynamical optimization for volterra integral systems and nonsemigroup evolution property are brought together to generalize the syntheses results to infinite dimensional cases

well posedness and regularity of backward stochastic volterra integral equations backward stochastic volterra integral equations bsvies for short are studied notion of adapted m solution is introduced well posedness of bsvies is established and some regularity results are proved for the adapted m solutions via malliavin calculus a pontryagin type maximum principle is presented for optimal controls of stochastic volterra integral equations

bijections and the riordan group one of the cornerstone ideas in mathematics is to take a problem and to look at it in a bigger space in this paper we examine combinatorial sequences in the context of the riordan group various subgroups of the riordan group each give us a different view of the original sequence in many cases this leads to both a combinatorial interpretation and to eco rewriting rules in this paper we will concentrate on just four of the subgroups of the riordan group to demonstrate some of the possibilities of this approach

volterra equations driven by semimartingales existence and uniqueness of solutions is established for stochastic volterra integral equations driven by right continuous semimartingales this resolves in the affirmative a conjecture of m berger and v mizel

darboux transformations for a twisted derivation and quasideterminant solutions to the super kdv equation this paper is concerned with a generalized type of darboux transformations defined in terms of a twisted derivation d satisfying d ab d a a b where is a homomorphism such twisted derivations include regular derivations difference and q difference operators and superderivatives as special cases remarkably the formulae for the iteration of darboux transformations are identical with those in the standard case of a regular derivation and are expressed in terms of quasideterminants as an example we revisit the darboux transformations for the manin radul super kdv equation studied in liu and manas liu manas 1997 a phys lett b 396 133 140 doi 10 1016 s0370 2693 97 00134 2 the new approach we take enables us to derive a unified expression for solution formulae in terms of quasideterminants covering all cases at once rather than using several subcases then by using a known relationship between quasideterminants and superdeterminants we obtain expressions for these solutions as ratios of superdeterminants this coincides with the results of liu and manas in all the cases they considered but also deals with the one subcase in which they did not obtain such an expression finally we obtain another type of quasideterminant solution to the manin radul super kdv equation constructed from its binary darboux transformations these can also be expressed as ratios of superdeterminants and are a substantial generalization of the solutions constructed using binary darboux transformations in earlier work on this topic

backward stochastic differential equations and partial differential equations with quadratic growth we provide existence comparison and stability results for one dimensional backward stochastic differential equations bsdes when the coefficient or generator f t y z is continuous and has a quadratic growth in z and the terminal condition is bounded we also give in this framework the links between the solutions of bsdes set on a diffusion and viscosity or sobolev solutions of the corresponding semilinear partial differential equations