optimization method for an evolutional type inverse heat conduction problem this paper deals with the determination of a pair q u in the heat conduction equation with initial and boundary conditions from the overspecified data u x t g x t by the time semi discrete scheme the problem is transformed into a sequence of inverse problems in which the unknown coefficients are purely space dependent based on the optimal control framework the existence uniqueness and stability of the solution q u are proved a necessary condition which is a couple system of a parabolic equation and parabolic variational inequality is deduced

on a generalized max type difference equation from automatic control theory abstract this paper studies the boundedness character of positive solutions of the following generalization of a max type difference equation from automatic control theory see mishkis 1977 18 and popov 1966 21 x n 1 max a x n p x n 1 q x n 2 r n n 0 where the parameters a p q and r are positive numbers in the study of the equation we also introduce a new method called oachkatzlschwoif

approximate controllability of a class of semilinear degenerate systems with boundary control abstract this paper concerns a class of control systems governed by semilinear degenerate equations with boundary control in one dimensional space the control is proposed on the degenerate part of the boundary the control systems are shown to be approximately controllable by kakutani s fixed point theorem

2d grushin type equations minimal time and null controllable data abstract we study internal null controllability for degenerate parabolic equations of grushin type g x x 2 x 2 y y 2 0 in the rectangle x y 1 1 0 1 previous works proved that null controllability holds for weak degeneracies small and fails for strong degeneracies large moreover in the transition regime and with strip shaped control domains a positive minimal time is required in this paper we work with controls acting on two strips symmetric with respect to the degeneracy we give the explicit value of the minimal time and we characterize some initial data that can be steered to zero in time t when the system is not null controllable their regularity depends on the control domain and the time t we also prove that with a control that acts on one strip touching the degeneracy line x 0 then grushin type equations are null controllable in any time t 0 and for any degeneracy 0 our approach is based on a precise study of the observability property for the one dimensional heat equations satisfied by the fourier coefficients in variable y this precise study is done through a transmutation process on the resulting one dimensional wave equations by lateral propagation of energy method

attractors for wave equations with degenerate memory abstract this paper is concerned with the long time dynamics of a semilinear wave equation with degenerate viscoelasticity u t t u 0 g s div a x u t s d s f u h x defined in a bounded domain of r 3 with dirichlet boundary condition and nonlinear forcing f u with critical growth the problem is degenerate in the sense that the function a x 0 in the memory term is allowed to vanish in a part of when a x does not degenerate and g decays exponentially it is well known that the corresponding dynamical system has a global attractor without any extra dissipation in the present work we consider the degenerate case by adding a complementary frictional damping b x u t which is in a certain sense arbitrarily small such that a b 0 in despite that the dissipation is given by two partial damping terms of different nature none of them necessarily satisfying a geometric control condition we establish the existence of a global attractor with finite fractal dimension

approximate controllability of a class of degenerate systems in this paper we consider the approximate controllability of a class of degenerate systems the equations may be weakly degenerate and strongly degenerate on the boundary we prove that the control systems are approximate controllable by constructing the controls via the conjugate problems

regional controllability of semilinear degenerate parabolic equations in bounded domains in this paper we study controllability properties for semilinear de generate parabolic equations with nonlinearities involving the first derivative in a bounded domain of r due to degeneracy classical null controllability results do not hold in general thus we investigate results of regional null controllability showing that we can drive the solution to rest at time t on a subset of the space domain contained in the set where the equation is nonde generate

on the global behavior of higher order recursive sequences our basic purpose in this paper is to establish some conditions for the global attractivity and convergence of the kth order recursive sequencex n f x n 1 x n 2 x n k n 1 2 3 we will show that if f is a real valued function with a convex basin w in r k which is differentiable with bounded differential then f has an equilibrium point our final comments are about informative examples

on a max type and a min type difference equation this note shows that every positive solution to the following third order non autonomous max type difference equationx n 1 maxa nx n x n 2 email protected n 0 when a n n n 0 is a three periodic sequence of positive numbers is periodic with period three the same result was proved for the following min type difference equationx n 1 mina nx n x n 2 email protected n 0

new periodic recurrences with applications abstract we develop two methods for constructing several new and explicit m periodic difference equations then we apply our results to two different problems firstly we show that two simple natural conditions appearing in the literature are not necessary conditions for the global periodicity of the difference equations secondly we present the first explicit non linear analytic potential differential system having a global isochronous center

a remark on null exact controllability of the heat equation it is well known that the heat equation u t delta u f chi omega in 0 t times omega with homogeneous dirichlet boundary conditions is null exactly controllable for any t 0 and any open nonempty subset omega of omega in this note we show that this property may be obtained as a singular limit of the exact controllability properties of singularly perturbed damped wave equations with a changing controller

identification of the unknown diffusion coefficient in a linear parabolic equation by the semigroup approach in this article we study the semigroup approach for the mathematical analysis of the inverse coefficient problems of identifying the unknown coefficient k x in the linear parabolic equation ut x t k x ux x t x with dirichlet boundary conditions u 0 t 0 u 1 t 1 main goal of this study is to investigate the distinguishability of the input output mappings k c1 0 t k c1 0 t via semigroup theory in this paper we show that if the null space of the semigroup t t consists of only zero function then the input output mappings and have the distinguishability property moreover the values k 0 and k 1 of the unknown diffusion coefficient k x at x 0 and x 1 respectively can be determined explicitly by making use of measured output data boundary observations f t k 0 ux 0 t or and h t k 1 ux 1 t in addition to these the values k 0 and k 1 of the unknown coefficient k x at x 0 and x 1 respectively are also determined via the input data furthermore it is shown that measured output data f t and h t can be determined analytically by an integral representation hence the input output mappings k c1 0 t k c1 0 t are given explicitly in terms of the semigroup finally by using all these results we construct the local representations of the unknown coefficient k x at the end points x 0 and x 1

null boundary controllability of a one dimensional heat equation with an internal point mass and variable coefficients in this paper we consider a linear hybrid system which is composed by two non homogeneous rods connected by a point mass with dirichlet boundary conditions on the left end and a boundary control acts on the right end we prove that this system is null controllable with dirichlet or neumann boundary controls our approach is mainly based on a detailed spectral analysis together with the moment method in particular we show that the associated spectral gap in both cases dirichlet or neumann boundary controls is positive without further conditions on the coefficients other than the regularities

exact boundary controllability for the linear korteweg de vries equation on the half line this paper is concerned with the controllability of the linear korteweg de vries equation on the domain o 0 infty the control being applied at the left endpoint x 0 it is shown that the em exact boundary controllability holds true in l 2 0 infty provided that the solutions are not required to be in l infty 0 t l 2 0 infty the proof rests on a carleman s estimate and an approximation theorem a similar result is obtained for the heat equation and for the schrodinger equation

dynamics of a higher order rational difference equation abstract our aim in this paper is to investigate the dynamics of difference equation y n 1 p qy n 1 y n k n 0 1 2 where the initial conditions y k y 1 y0 are non negative k 1 2 3 and the parameters p and q are non negative we study characteristics such as the global stability the boundedness of positive solutions and the character of semicycles of the above mentioned difference equation in particular our results solve the open problem introduced by kulenovic and ladas in their monograph m r s kulenovic g ladas dynamics of second order rational difference equations with open problems and conjectures chapman and hall crc boca raton 2002

on the controllability of parabolic systems with a nonlinear term involving the state and the gradient we present some results concerning the controllability of a quasi linear parabolic equation with linear principal part in a bounded domain of mathbb r n with dirichlet boundary conditions we analyze the controllability problem with distributed controls supported on a small open subset and boundary controls supported on a small part of the boundary we prove that the system is null and approximately controllable at any time if the nonlinear term f y nabla y grows slower than y log 3 2 1 y nabla y nabla y log 1 2 1 y nabla y at infinity generally in this case in the absence of control blow up occurs the proofs use global carleman estimates parabolic regularity and the fixed point method

a convergent quadratic time lattice algorithm for pricing european style asian options abstract asian options are strongly path dependent derivatives although efficient numerical methods and approximate closed form formulas are available most lack convergence guarantees asian options can also be priced on the lattice all efficient lattice algorithms keep only a polynomial number of states and use interpolation to compensate for the less than full representation of the states let the time to maturity be partitioned into n periods this paper presents the first o n 2 time convergent lattice algorithm for pricing european style asian options it is the most efficient lattice algorithm with convergence guarantees the algorithm relies on the lagrange multipliers to choose optimally the number of states for each node of the lattice the algorithm is also memory efficient extensive numerical experiments and comparison with existing pde analytical and lattice methods confirm the performance claims and the competitiveness of our algorithm this result places the problem of european style asian option pricing in the same complexity class as that of the vanilla option on the lattice

solving an inverse parabolic problem by optimization from final measurement data we consider an inverse problem of reconstructing the coefficient q in the parabolic equation ut u q x u 0 from the final measurement u x t where q is in some subset of l1 the optimization method combined with the finite element method is applied to get the numerical solution under some assumption on q the existence of minimizer as well as the convergence of approximate solution in finite dimensional space is proven the new ingredient in this paper is that we do not need uniformly a priori bounds of h1 norm on q numerical implementations are also presented

quenching of solutions to a class of semilinear parabolic equations with boundary degeneracy abstract this paper concerns the quenching phenomenon of solutions to a class of semilinear parabolic equations with boundary degeneracy in the case that the degeneracy is not strong it is shown that there exists a critical length which is positive such that the solution exists globally in time if the length of the spatial interval is less than it while quenches in a finite time if the length of the spatial interval is greater than it whereas in the case that the degeneracy is strong enough the solution must be quenching in a finite time no matter how long the spatial interval is furthermore for each quenching solution the set of quenching points is determined and it is proved that its derivative with respect to the time must blow up at the quenching time

a note on average rate options with discrete sampling the authors model the fair value of average rate financial options as the solution of partial differential equations when the average is sampled discretely the equation to be solved is the black scholes equation with the measure of the average to date being a parameter in the problem and a jump condition across sampling dates the authors show how to derive explicit results for the value of average rate options when the average is measured arithmetically when the average is measured geometrically the problem must be solved numerically

efficient pricing of discrete asian options asian options are popular path dependent financial derivatives this paper uses lattices to price fixed strike european style asian options that are discretely monitored the algorithm proposed can also be applied to floating strike asian options as well because fixed strike and floating strike asian options are related through an equation the discretely monitored version is usually found in practice instead of the continuously monitored version usually encountered in the literature this paper presents the first provably quadratic time convergent lattice algorithm for pricing fixed strike european style discretely monitored asian options it is the most efficient lattice algorithm with convergence guarantees the algorithm relies on the lagrange multipliers to choose the number of states for each node of the lattice extensive numerical experiments and comparisons with many existing numerical methods confirm the performance claims and the competitiveness of our algorithm this result places fixed strike european style discretely monitored asian options in the same complexity class as vanilla options

on the global periodicity of discrete dynamical systems and application to rational difference equations a new necessary condition for global periodicity of discrete dynamical systems and of difference equations is obtained here this condition will be applied to contribute to solving the problem of global periodicity for second order rational difference equations

carleman estimates for a class of degenerate parabolic operators given alpha in 0 2 and f in l 2 0 t times 0 1 we derive new carleman estimates for the degenerate parabolic problem w t x alpha w x x f where t x in 0 t times 0 1 associated to the boundary conditions w t 1 0 and w t 0 0 if 0 leq alpha 1 or x alpha w x t 0 0 if 1 leq alpha 2 the proof is based on the choice of suitable weighted functions and hardy type inequalities as a consequence for all 0 leq alpha 2 and omega subset subset 0 1 we deduce null controllability results for the degenerate one dimensional heat equation u t x alpha u x x h chi omega with the same boundary conditions as above

on the periodic cycle of xn 1 an bnxn cnxn 1 the behaviour of solutions of the difference equation x n 1 a n b nx n c nx n 1 n 0 1 2 are investigated where the sequence a n b n and c n are periodic

regional null controllability of a linearized crocco type equation we are interested in controllability problems of equations coming from a boundary layer model we simplify the problem by considering only equations with constant coefficients the problem is described by a degenerate parabolic equation a linearized crocco type equation where phenomena of diffusion and transport are coupled first we give a geometric characterization of the influence domain of a locally distributed con trol then we prove regional null controllability results on this domain the proof is based on an adequate observability inequality for the homogeneous adjoint problem this inequality is obtained by decomposition of the space time domain and carleman type estimates along characteristics in the second part of this paper we treat the case of a boundary control

finite difference scheme with a moving mesh for pricing asian options in this paper we propose a stable numerical method for pricing asian call options which is based on a central difference scheme with a moving mesh in the spatial discretization and the rannacher time stepping scheme in the time discretization at each time mesh point we make a piecewise uniform mesh to discretise the space interval which ensures that the matrix associated with the discrete operator is an m matrix hence the spatial discretization scheme is maximum norm stable for arbitrary volatility and arbitrary interest rate we show that the scheme is second order convergent with respect to both time and spatial variables numerical results support the theoretical results

approximate controllability for nonlinear degenerate parabolic problems with bilinear control abstract in this paper we study the global approximate multiplicative controllability for nonlinear degenerate parabolic cauchy neumann problems first we obtain embedding results for weighted sobolev spaces that have proved decisive in reaching well posedness for nonlinear degenerate problems then we show that the above systems can be steered in l 2 from any nonzero nonnegative initial state into any neighborhood of any desirable nonnegative target state by bilinear piecewise static controls moreover we extend the above result relaxing the sign constraint on the initial data

unique continuation for some evolution equations this work is concerned with unique continuation results for some evolution equations essentially of parabolic type to be more precise let l be an evolution operator acting on functions defined on some connected open set b of r r x r l is said to have the unique continuation property if every solution u of lu 0 which vanishes one some nonempty open set u i of fl vanishes in the horizontal component of o in 0 i e in x t e 0 3x x t e co the first general result of this type was established by mizohata 6 in the case where l is a second order parabolic operator with smooth coefficients however mizohata s proof is somewhat delicate it is based in particular on an extended class of calderon zygmund operators and does not lead obviously to a weakening of the smoothness of the coefficients on the other hand it is important for the applications to deal with nonsmooth coefficients since l is often obtained as the linearization of some nonlinear operator at a nonnecessarily smooth solution we refer to lions 4 and kernevez and lions 3 for applications to control theory and to 1 l for applications to transversality techniques let us now describe the content of this work in the first section we prove a unique continuation theorem when l is a second order parabolic equation our proof is simple and based on the derivation of a carleman estimate which is reminiscent of the classical carleman estimates for second order elliptic operators see 121 this carleman inequality allows the weakening of the 118 0022 0396187 3 00

a class of nonlinear degenerate integrodifferential control systems in this paper we study the approximate controllability and noncontrollability for a class of nonlinear degenerate integrodifferential control systems based on the property of finite propagation for the disturbances we prove noncontrollability we also establish sufficient conditions for approximate controllability of the system in l2 omega moreover we prove the global existence and uniqueness of generalized solutions

interior degenerate singular parabolic equations in nondivergence form well posedness and carleman estimates abstract we consider non smooth general degenerate singular parabolic equations in non divergence form with degeneracy and singularity occurring in the interior of the spatial domain in presence of dirichlet or neumann boundary conditions in particular we consider well posedness of the problem and then we prove carleman estimates for the associated adjoint problem

dynamics of the birational maps arising from f0 and dp3 quivers abstract the dynamics of the maps associated to f 0 and d p 3 quivers is studied in detail we show that the corresponding reduced symplectic maps are conjugate to globally periodic maps by providing explicit conjugations the dynamics in r n of the original maps is obtained by lifting the dynamics of these globally periodic maps and the solution of the discrete dynamical systems generated by each map is given a better understanding of the dynamics is achieved by considering first integrals the relationship between the complete integrability of the globally periodic maps and the dynamics of the original maps is explored

controllability of the semilinear parabolic equation governed by a multiplicative control in the reaction term a qualitative approach in this paper we are concerned with the global nonnegative approximate controllability property of a rather general semilinear heat equation with superlinear term governed in a bounded domain omega subset r n by a multiplicative bilinear control in the reaction term like v u x t where v is the control we show that any nonnegative target state in l 2 omega can approximately be reached from any nonnegative nonzero initial state by applying at most three static bilinear l infty omega controls subsequently in time this result is further applied to discuss the controllability properties of the nonhomogeneous version of this problem with bilinear term like v u x t theta x where theta is given our approach is based on an asymptotic technique allowing us to distinguish and make use of the pure diffusion and or pure reaction parts of the dynamics of the system at hand while suppressing the effect of a general nonlinear term

weak stability for coupled wave and or petrovsky systems with complementary frictional damping and infinite memory abstract in this paper we consider coupled wave wave petrovsky petrovsky and wave petrovsky systems in n dimensional open bounded domain with complementary frictional damping and infinite memory acting on the first equation we prove that these systems are well posed in the sense of semigroups theory and provide a weak stability estimate of solutions where the decay rate is given in terms of the general growth of the convolution kernel at infinity and the arbitrary regularity of the initial data we finish our paper by considering the uncoupled wave and petrovsky equations with complementary frictional damping and infinite memory and showing a strong stability estimate depending only on the general growth of the convolution kernel at infinity

heat equation on the heisenberg group observability and applications abstract we investigate observability and lipschitz stability for the heisenberg heat equation on the rectangular domain 1 1 t t taking as observation regions slices of the form a b t t or tubes a b y t with 1 a b 1 we prove that observability fails for an arbitrary time t 0 but both observability and lipschitz stability hold true after a positive minimal time which depends on the distance between and the boundary of t min 1 8 min 1 a 2 1 b 2 our proof follows a mixed strategy which combines the approach by lebeau and robbiano which relies on fourier decomposition with carleman inequalities for the heat equations that are solved by the fourier modes we extend the analysis to the unbounded domain 1 1 t r

global asymptotic stability in a rational recursive sequence in this paper we study the global stability of the difference equationx n a bx n 1 cx n 1 2d x n 2 n 1 2 where a b 0 and c d 0 we show that one nonnegative equilibrium point of the equation is a global attractor with a basin that is determined by the parameters and every positive solution of the equation in the basin exponentially converges to the attractor

controllability of a class of newtonian filtration equations with control and state constraints this paper addresses a study of the controllability of a class of newtonian filtration equations with nonnegative constraints on the control and state variables when the control enters the system through the whole domain where the equation evolves we characterize the set of nonnegative targets which are approximately controllable at any time t 0 the proof combines the fenchel rockafellar duality theory and a fixed point argument when the control is restricted to be active in a proper open subset of the whole domain we prove a negative controllability result by means of a localization technique which reflects the underlying obstruction phenomenon in the system

simultaneous control of a rod equation and a simple schro dinger equation abstract we show that a corollary of a local approximate controllability result for the bilinear rod equation in 1 is that the controls which steer the modes of the rod equation also move the modes of a controlled schrodinger equation specifying a target point for the schrodinger system restricts but does not determine the outcome of the controlled rod equation the local result is a special case of a general local result for hyperbolic systems which is used in 1 to obtain a global approximate controllability result after modifying the latter we obtain a global result for the two systems

null controllability and numerical method for crocco equation with incomplete data based on an exponential integrator and finite difference finite element method abstract we consider the linearized crocco equation in fluid dynamics with incomplete data and robin boundary conditions and address theoretical and numerical distributed null control the controllability problem is solved through a dual reformulation we first resolve some constrained extremal problems and apply appropriated duality techniques that lead to the formulation of equivalent unconstrained extremal problem in variational form novel numerical technique based on finite difference finite element space discretization and exponential integrator in time discretization is proposed for illustration numerical simulations are provided

null controllability of non autonomous ornstein uhlenbeck equations abstract we study the null controllability of parabolic equations associated to non autonomous ornstein uhlenbeck operators when a kalman type condition holds for some positive time t 0 these parabolic equations are shown to enjoy a gevrey regularizing effect at time t 0 thanks to this regularizing effect we prove by adapting the lebeau robbiano method that these parabolic equations are null controllable in time t 0 from control regions for which null controllability is classically known to hold in the case of the heat equation

multi parameters identification problem for a degenerate parabolic equation abstract this paper investigates an inverse problem of simultaneously reconstructing the initial value and source coefficient in a degenerate parabolic equation problems of this type have important applications in several fields of applied science being different from other inverse coefficient problems in classical parabolic equations non degenerate the principal coefficient in the mathematical model discussed in the paper may vanish on both extremities of the domain on the basis of carleman estimate the uniqueness and conditional stability of solution for the original problem are established then an iteration algorithm of landweber type is designed to obtain the numerical solution and some typical numerical experiments are also performed numerical results show that the proposed method is stable and the unknown coefficients are recovered quite well

an efficient convergent lattice algorithm for european asian options financial options whose payoff depends critically on historical prices are called path dependent options their prices are usually harder to calculate than options whose prices do not depend on past histories asian options are popular path dependent derivatives and it has been a long standing problem to price them efficiently and accurately no known exact pricing formulas are available to price them under the continuous time black scholes model although approximate pricing formulas exist they lack accuracy guarantees asian options can be priced numerically on the lattice a lattice divides the time to maturity into n equal length time steps the option price computed by the lattice converges to the option value under the black scholes model as n unfortunately only subexponential time algorithms are available if asian options are to be priced on the lattice without approximations efficient approximation algorithms are available for the lattice the fastest lattice algorithm published in the literature runs in o n 3 5 time whereas for the related pde method the fastest one runs in o n 3 time this paper presents a new lattice algorithm that runs in o n 2 5 time the best in the literature for such methods our algorithm exploits the method of lagrange multipliers to minimize the approximation error numerical results verify its accuracy and the excellent performance

global attractivity for a class of higher order nonlinear difference equations our aim in this paper is to investigate the global attractivity of all positive solutions of the higher order nonlinear difference equationx n 1 a bx email protected k i 0b ix n i n 0 1 where a b b email protected 0 a b 0 b k email protected 0 email protected 1 2 and the initial conditions x k x 0 are arbitrary real numbers we show that one positive equilibrium of the equation is a global attractor with a basin that depends on certain conditions posed on the coefficients

null controllability of one dimensional parabolic equations by the flatness approach we consider linear one dimensional parabolic equations with space dependent coefficients that are only measurable and that may be degenerate or singular considering generalized robin neumann boundary conditions at both extremities we prove the null controllability with one boundary control by following the flatness approach which provides explicitly the control and the associated trajectory as series both the control and the trajectory have a gevrey regularity in time related to the l p class of the coefficient in front of u t the approach applies in particular to the possibly degenerate or singular heat equation a x u x x u t 0 with a x 0 for a e x in 0 1 and a 1 a in l 1 0 1 or to the heat equation with inverse square potential u xx mu x 2 u u t 0 with mu ge 1 4

approximate controllability of the semilinear heat equation this article is concerned with the study of approximate controllability for the semilinear heat equation in a bounded domain when the control acts on any open and nonempty subset of or on a part of the boundary in the case of both an internal and a boundary control the approximate controllability in l p for 1 p l in the case of the interior control we also prove approximate controllability in c 0 the proof combines a variational approach to the controllability problem for linear equations and a fixed point method we also prove that the control can be taken to be of quasi bang bang form

local controllability of multidimensional quasi linear parabolic equations this paper studies the local controllability for a class of multidimensional quasi linear parabolic equations with homogenous dirichlet boundary conditions and an arbitrarily located internal controller unlike the known result in one spacial dimension controllability is analyzed in the frame of classical solutions the key of our approach is to seek a control function in the holder space for given data with certain regularity in order to give the cost estimate for the control function we establish an observability inequality for linear parabolic equations with an explicit estimate on the observability constant in terms of the c 1 coefficients of principal parts the later is based on a new global carleman estimate for general linear parabolic equations which has an independent interest

some results on controllability of a nonlinear degenerate parabolic system by bilinear control in this paper we discuss the controllability of a nonlinear degenerate parabolic system with bilinear control based on the shrinking property of the solutions we prove that the system is not globally approximately controllable furthermore we give an approximate null controllability result we also prove that the system is not globally exactly null controllable by a comparison principle

global steady state controllability of one dimensional semilinear heat equations we investigate the problem of exact boundary controllability of semilinear one dimensional heat equations we prove that it is possible to move from any steady state to any other by means of a boundary control provided that both are in the same connected component of the set of steady states the proof is based on an effective feedback stabilization procedure which is implemented

approximate controllability of a class of semilinear degenerate systems with convection term abstract this paper concerns a class of control systems governed by semilinear degenerate equations with convection term the equations may be weakly degenerate and strongly degenerate on a portion of the lateral boundary and the convection terms can be controlled by the diffusion terms the control systems are shown to be approximately controllable

carleman estimates and null controllability for a class of degenerate parabolic equations with convection terms this paper concerns a class of control systems governed by semilinear degenerate parabolic equations with convection terms which cannot be controlled by the diffusion terms the carleman estimates and the observability inequalities for the corresponding linear equations are established if the degeneracy is relatively weak subsequently it is proved that the control systems are null controllable

holder continuity of weak solutions of a class of linear equations with boundary degeneracy abstract this paper deals with a class of linear equations with boundary degeneracy according to the degenerate ratio the equations are divided into weakly degenerate ones and strongly degenerate ones which should be supplemented by different dirichlet boundary value conditions after establishing some necessary existence nonexistence and comparison principles we investigate the optimal holder continuity of weak solutions in these two cases utilizing the harnack inequality and the morrey theorem respectively

normal forms for rational difference equations with applications to the global periodicity problem we propose a classification and derive the associated normal forms for rational difference equations with complex coefficients as an application we study the global periodicity problem for second order rational difference equations with complex coefficients we find new necessary conditions as well as some new examples of globally periodic equations

backward uniqueness for parabolic equations it is shown that a function u satisfying t u m u u u x t mem x 2 in n br 0 t and u x 0 0 for x n br must vanish identically in n br 0 t

evolutionary weighted p laplacian with boundary degeneracy abstract the paper concerns the well posedness problem of an evolutionary weighted p laplacian with boundary degeneracy different from the classical theory for linear equations it is shown that the degenerate portion of the boundary should be decomposed into two parts the strongly degenerate boundary on which the equation exhibits hyperbolic characteristics and the weakly degenerate boundary on which the equation still exhibits parabolic characteristics we formulate reasonably the boundary value condition and establish the existence and uniqueness theorems

null controllability for the heat equation with singular inverse square potentials abstract we prove the null controllability of the heat equation perturbed by a singular inverse square potential arising in quantum mechanics and combustion theory this is done within the range of subcritical coefficients of the singular potential provided the control acts on an annular set around the singularity our proof uses a splitting argument on the domain decomposition in spherical harmonics new carleman inequalities and refined hardy inequalities

approximate controllability of a semilinear heat equation in bbb r n we prove the approximate controllability of the semilinear heat equation in bbb r n we introduce the weighted sobolev spaces of escobedo and kavian and in that functional setting we adapt the technique introduced by fabre puel and zuazua for the problem in bounded domains that is we first prove the approximate controllability of the linear equation and by a fixed point method obtain the main result

exact boundary controllability of 1 d parabolic and hyperbolic degenerate equations the boundary controllability of a class of one dimensional degenerate equations is studied in this paper the novelty of this work is that the control acts through the part of the boundary where degeneracy occurs first we consider a class of degenerate hyperbolic equations then we prove sharp observability estimates for these equations using nonharmonic fourier series the transmutation method yields a result for the corresponding class of degenerate parabolic equations

null controllability of a parabolic equation involving the grushin operator in some multi dimensional domains abstract we study the null controllability of the linear parabolic equation involving the grushin operator g s x x 2 s y s 0 in the domain 1 1 n 1 0 1 n 2 r n 1 r n 2 n 1 n 2 1 under an additive control supported in an open subset of we prove that the equation is null controllable in any positive time for s 1 and not null controllable for s 1 when s 1 a positive minimal time is required for null controllability

on rational recursive sequences our goal in this chapter is to gain some understanding of the dynamics of rational recursive sequences of the form x n 1 frac a sum limits i 0 k a i x n i b sum limits i 0 k b i x n i n 0 1 3 0 1 where k is a positive integer and the coefficients a b a 0 a k b 0 b k are nonnegative real numbers

controllability of the heat equation with an inverse square potential localized on the boundary this article is devoted to analyzing control properties for the heat equation with singular potential mu x 2 arising at the boundary of a smooth domain omega subset mathbb r n n geq 1 this problem was first studied by vancostenoble and zuazua j funct anal 254 2008 pp 1864 1902 and then generalized by ervedoza comm partial differential equations 33 2008 pp 1996 2019 in the context of interior singularity roughly speaking these results showed that for any value of parameters mu leq mu n n 2 2 4 the corresponding parabolic system can be controlled to zero with the control distributed in any open subset of the domain the critical value mu n stands for the best constant in the hardy inequality with interior singularity when considering the case of boundary singularity a better critical hardy constant is obtained namely mu n n 2 4 in this article we extend the previous results of vancostenoble and zuazua and of ervedoza to the case of boundary singula

controllability for distributed bilinear systems this paper studies controllability of systems of the form dw dt mathcal a w p t mathcal b w where mathcal a is the infinitesimal generator of a c 0 semigroup of bounded linear operators e mathcal a t on a banach space x mathcal b x to x is a c 1 map and p in l 1 0 t mathbb r is a control the paper i gives conditions for elements of x to be accessible from a given initial state w 0 and ii shows that controllability to a full neighborhood in x of w 0 is impossible for dim x infty examples of hyperbolic partial differential equations are provided

on the boundary controllability of a semilinear degenerate system with the convection term this paper concerns the approximate controllability of a semilinear degenerate equation with the convection term in one dimensional space the control acts on the degenerate part of the boundary by using kakutani fixed point theorem the control system is shown to be approximately controllable

on the lack of null controllability of the heat equation on the half line we consider the linear heat equation on the half line with a dirichlet boundary control we analyze the null controllability problem more precisely we study the class of initial data that may be driven to zero in finite time by means of an appropriate choice of the l2 boundary control we rewrite the system on the similarity variables that are a common tool when analyzing asymptotic problems next the control problem is reduced to a moment problem which turns out to be critical since it concerns the family of real exponentials eit j in which the usual summability condition on the inverses of the eigenvalues does not hold roughly speaking we prove that controllable data have fourier coefficients that grow exponentially for large frequencies this result is in contrast with the existing ones for bounded domains that guarantee that every initial datum belonging to a sobolev space of negative order may be driven to zero in an arbitrarily small time

on the max type equation xn 1 max axn xn 2 abstract we show that every well defined solution of the difference equation x n 1 max a x n x n 2 n n 0 where a r is eventually periodic with period three

accurate pricing formulas for asian options asian options have payoffs that depend on the average price of the underlying asset such as stocks commodities or financial indices as exact closed form formulas do not exist for these popular options how to price them numerically in an efficient and accurate manner has been extensively investigated there are two types of asian options fixed strike and floating strike asian options excellent lower bound formulas for both types of options have been derived by rogers and shi these formulas are extremely easy to calculate but they restrict the option s maturity to exactly 1 year this paper extends the rogers shi formulas to general maturities numerical experiments are performed to compare the formulas with many other numerical methods in the literature and under a wide variety of situations they confirm the extreme accuracy of the formulas

controllability for parabolic equations with nonlinear memory this paper is concerned with the controllability of a parabolic system with nonlinear memory based on the localized estimate of the solution we prove that the system with a superlinear growth memory is not controllable furthermore two controllability results for some initial data and targets are given as well

on global periodicity of difference equations abstract in this research we consider a difference equation of order k 2 of the following form y n 1 f y n y n k 1 n 0 1 2 we develop necessary and or sufficient conditions on f so that every solution of the above mentioned difference equation is periodic of the same period p