quadratic pencils and least squares piecewise polynomial approximation for a partition o o m denotes the space of piecewise polynomials of order k and of smoothness m 1 this space can be represented as the graph of the appropriate linear operator between two finite dimensional hilbert spaces it gives an approach to the c de boor problem 1972 on uniform boundedness with respect to in the l norm of the orthogonal projections onto stm and we give the detailed analysis of a quadratic pencil matrix valued polynomial of the second degree which appears in the case of a geometric mesh if 2m k the explicit calculations and estimates of zeros of the characteristic polynomial show that in the case sbx x the geometric mesh with the parameter x 0 m 0 and any partition ax n of the unit interval 0 1 01 0 1 0 1 n n i 1 we define the piecewise polynomial subspace 0 2 s t km n c m1 0 11 where p fel 01vfia k i pk e f ej l il f ia is a polnom1ial caxi of order k 0 s a s n each interior breakpoint a 1 a n generates m continuity conditions 0 3 ia c af ia ya i 0 in to make a functionf e pkt be an element of scm let q qs be the orthogonal projection onto s in l2 0 1 i e with respect to the inner product f g f x g x dx we are interested in q as a map in c 0 1 or l 0 1 and we would like to get the estimates of its norm iiqiioo sup iiqfli lflfloo f where ilf 11k0 esssupo x if x i f e l o 1 received september 29 1980 1980 mathematics subject classification primary 41a1 5 secondary 47a68 this paper was partly supported by nsf grant 7906079 1983 american mathematical society 0025 571 8 82 0000 0750 05 25

derivatives pricing with market impact and limit order book abstract this paper investigates the derivatives pricing under the existence of liquidity costs and market impact for the underlying asset in continuous time first we formulate the charge for the liquidity costs and the market impact on the derivatives prices through a stochastic control problem that aims to maximize the mark to market value of the portfolio less the quadratic variation multiplied by a risk aversion parameter during the hedging period and the liquidation cost at maturity then we obtain the derivatives price by reduction of this charge from the premium in the bachelier model second we consider a second order semilinear partial differential equation pde of parabolic type associated with the control problem which is analytically solved or approximated by an asymptotic expansion around a solution to an explicitly solvable nonlinear pde finally we present the numerical examples of the pricing for a variance option and a european call option and show comparative static analyses

a sparse collocation method for solving time dependent hjb equations using multivariate b splines this paper presents a sparse collocation method for solving the time dependent hamilton jacobi bellman hjb equation associated with the continuous time optimal control problem on a fixed finite time horizon with integral cost functional through casting the problem in a recursive framework using the value iteration procedure the value functions of every iteration step is approximated with a time varying multivariate simplex b spline on a certain state domain of interest in the collocation scheme the time dependent coefficients of the spline function are further approximated with ordinary univariate b splines to yield a discretization for the value function fully in terms of piece wise polynomials the b spline coefficients are determined by solving a sequence of highly sparse quadratic programming problems the proposed algorithm is demonstrated on a pair of benchmark example problems simulation results indicate that the method can yield increasingly more accurate approximations of the value function by refinement of the triangulation

a radial basis collocation method for hamilton jacobi bellman equations in this paper we propose a semi meshless discretization method for the approximation of viscosity solutions to a first order hamilton jacobi bellman hjb equation governing a class of nonlinear optimal feedback control problems in this method the spatial discretization is based on a collocation scheme using the global radial basis functions rbfs and the time variable is discretized by a standard two level time stepping scheme with a splitting parameter q a stability analysis is performed showing that even for the explicit scheme that q 0 the method is stable in time since the time discretization is consistent the method is also convergent in time numerical results performed to verify the usefulness of the method demonstrate that the method gives accurate approximations to both of the control and state variables

the absolute galois group acts faithfully on regular dessins and on beauville surfaces in this article we study the action of the absolute galois group gal q q on dessins d enfants and beauville surfaces a foundational result in grothendieck s theory of dessins d enfants is the fact that the absolute galois group gal q q acts faithfully on the set of all dessins however the question of whether this holds true when the action is restricted to the set of the more accessible regular dessins seems to be still an open question in the first part of this paper we give an affirmative answer to it in fact we prove the strongest result that the action is faithful on the set of quasiplatonic or triangle curves of any given hyperbolic type beauville surfaces are an important kind of algebraic surfaces introduced by catanese they are rigid surfaces of general type closely related to dessins d enfants here we prove that for any gal q q different from the identity and the complex conjugation there is a beauville surface s such that s and its galois conjugate s have non isomorphic fundamental groups this in turn easily implies that the action of gal q q on the set of beauville surfaces is faithful these results were conjectured by bauer catanese and grunewald and immediately imply that gal q q acts faithfully on the connected components of the moduli space of surfaces of general type a result due to the above mentioned authors

a fast semi discrete kansa method to solve the two dimensional spatiotemporal fractional diffusion equation abstract fractional order diffusion equations fdes extend classical diffusion equations by quantifying anomalous diffusion frequently observed in heterogeneous media real world diffusion can be multi dimensional requiring efficient numerical solvers that can handle long term memory embedded in mass transport to address this challenge a semi discrete kansa method is developed to approximate the two dimensional spatiotemporal fde where the kansa approach first discretizes the fde then the gauss jacobi quadrature rule solves the corresponding matrix and finally the mittag leffler function provides an analytical solution for the resultant time fractional ordinary differential equation numerical experiments are then conducted to check how the accuracy and convergence rate of the numerical solution are affected by the distribution mode and number of spatial discretization nodes applications further show that the numerical method can efficiently solve two dimensional spatiotemporal fde models with either a continuous or discrete mixing measure hence this study provides an efficient and fast computational method for modeling super diffusive sub diffusive and mixed diffusive processes in large two dimensional domains with irregular shapes

fixed final time constrained optimal control of nonlinear systems using neural network hjb approach in this paper fixed final time constrained optimal control laws using neural networks nns to solve hamilton jacobi bellman hjb equations for general affine in the constrained nonlinear systems are proposed an nn is used to approximate the time varying cost function using the method of least squares on a predefined region the result is an nn nearly constrained feedback controller that has time varying coefficients found by a priori offline tuning convergence results are shown the results of this paper are demonstrated in two examples including a nonholonomic system

controller design for markov jumping systems subject to actuator saturation in this paper the stochastic stabilization problem for a class of markov jumping linear systems mjls subject to actuator saturation is considered the concept of domain of attraction in mean square sense is used to analyze the closed loop stability when the jumping mode is available a mode dependent state feedback controller is developed otherwise we give a less conservative approach to design the mode independent state feedback controller both design procedures can be converted into a set of linear matrix inequalities lmis finally a numerical example is provided to show the effectiveness of the techniques

a new iterative method for discrete hjb equations a successive relaxation iterative algorithm for discrete hjb equations is proposed monotone convergence has been proved for the algorithm

pod based feedback control of the burgers equation by solving the evolutionary hjb equation a new approach for solving finite time horizon feedback control problems for distributed parameter systems is proposed it is based on model reduction by proper orthogonal decomposition combined with efficient numerical methods for solving the resulting low order evolutionary hamilton jacobi bellman hjb equation the feasibility of the proposed methodology is demonstrated by means of optimal feedback control for the burgers equation the method for solving the hjb equation is first tested on several 1 d problems and then successfully applied to the control of the reduced order burgers equation the effect of noise is investigated and parallelism is used for computational speedup

nearly optimal control laws for nonlinear systems with saturating actuators using a neural network hjb approach the hamilton jacobi bellman hjb equation corresponding to constrained control is formulated using a suitable nonquadratic functional it is shown that the constrained optimal control law has the largest region of asymptotic stability ras the value function of this hjb equation is solved for by solving for a sequence of cost functions satisfying a sequence of lyapunov equations le a neural network is used to approximate the cost function associated with each le using the method of least squares on a well defined region of attraction of an initial stabilizing controller as the order of the neural network is increased the least squares solution of the hjb equation converges uniformly to the exact solution of the inherently nonlinear hjb equation associated with the saturating control inputs the result is a nearly optimal constrained state feedback controller that has been tuned a priori off line

semismooth newton and newton iterative methods for hjb equation in this paper some semismooth methods are considered to solve a nonsmooth equation which can arise from a discrete version of the well known hamilton jacobi bellman equation by using the slant differentiability introduced by chen nashed and qi in 2000 a semismooth newton method is proposed the method is proved to have monotone convergence by suitably choosing the initial iterative point and local superlinear convergence rate moreover an inexact version of the proposed method is introduced which reduces the cost of computations and still preserves nice convergence properties some numerical results are also reported

non homeomorphic galois conjugate beauville structures on psl 2 p abstract catanese s rigidity results for surfaces isogenous to a product of curves indicate that beauville surfaces should provide a fertile source of examples of galois conjugate varieties that are not homeomorphic a phenomenon discovered by j p serre in the sixties in this paper we construct beauville surfaces s c 1 c 2 g with group g psl 2 p for p 7 and curves c 1 c 2 such that the orbit of s under the action of the absolute galois group gal q q contains non homeomorphic conjugate surfaces when p 7 the orbit consists exactly of two surfaces that have non isomorphic fundamental groups and the curves c 1 c 2 have genera 8 and 49 which is shown to be the minimum for which there is a pair of non homeomorphic galois conjugate beauville surfaces as p grows the orbits contain an arbitrarily large number of non homeomorphic surfaces along the way we prove a metric rigidity theorem for beauville surfaces which provides an elementary proof of the part of catanese s theory needed to prove our results

neural approximations for infinite horizon optimal control of nonlinear stochastic systems a feedback control law is proposed that drives the controlled vector v sub t of a discrete time dynamic system to track a reference v sub t over an infinite time horizon while minimizing a given cost function the behavior of v sub t over time is completely unpredictable random noises act on the dynamic system and the state observation channel which may be nonlinear it is assumed that all such random vectors are mutually independent and that their probability density functions are known so general a non lqg optimal control problem is very difficult to solve the proposed solution is based on three main approximating assumptions 1 the problem is stated in a receding horizon framework where v sub t is assumed to remain constant within a shifting time window 2 the control law is assigned a given structure that of a multilayer feedforward neural net in which a finite number of parameters have to be determined so as to minimize the cost function and 3 the control law is given a limited memory which prevents the amount of data to be stored from increasing over time errors resulting from the second and third assumptions are discussed due to the very general assumptions under which the control law is derived we are not able to report stability results however simulation results show that the proposed method may constitute an effective tool for solving to a sufficient degree of accuracy a wide class of control problems traditionally regarded as difficult ones an example of freeway traffic optimal control is given

inverses of infinite sign regular matrices let a be an infinite sign regular sr matrix which can be viewed as a bounded linear operator from lx to itself it is proved here that if the range of a contains the sequence 1 1 1 1 then a is onto if a exists then da d is also sr where d is the diagonal matrix with diagonal entries alternately 1 and 1 in case a is totally positive tp then da d is also tp under additional assumptions on a 0 introduction if the problem of spline interpolation is expressed in terms of b splines then the question of existence of a bounded spline interpolant to bounded data is seen to be equivalent to the question of whether a certain bounded band matrix has all bounded sequences in its range in 5 c a micchelli conjectured that there exists a unique bounded spline interpolant of a given order and a given knot sequence to any data sequence t y z in the plane with t strictly increasing and y bounded provided only that it is possible to interpolate the particular data sequence t l ez by sucn a sp ine there is apparently nothing special about the particular spline problem other than that it leads to a banded totally positive matrix therefore one of us quoted this conjecture in 3 p 319 prob lem 4 as a bi infinite banded totally positive matrix a is boundedly invert ible if and only if the linear system ax 1 has a bounded solution 2 micchelli gave a simple argument for the case when a is a toeplitz matrix cavaretta dahmen micchelli and smith 2 recently proved the conjecture in case a is a block toeplitz matrix this is all the more remarkable since it is easy to see in hindsight that the conjecture is faulty even in the original context of spline interpolation for example interpolation by bounded broken lines with breakpoint sequence z at the sequence t z 0 is possible to any bounded ordinate sequence y but not uniquely so since the value of the interpolant at 0 is freely choosable in matrix terms this corresponds to the matrix obtained from the bi infinite identity matrix by dropping one row but with the condition changed to has a unique bounded solution the conjecture was proved in 1

multi step heuristic dynamic programming for optimal control of nonlinear discrete time systems abstract policy iteration and value iteration are two main iterative adaptive dynamic programming frameworks for solving optimal control problems policy iteration converges fast while requiring an initial stabilizing control policy which is a strict constraint in practice value iteration avoids the requirement of initial admissible control policy while converging much slowly this paper tries to utilize the advantages of policy iteration and value iteration and avoids their drawbacks at the same time therefore a multi step heuristic dynamic programming mshdp method is developed for solving the optimal control problem of nonlinear discrete time systems mshdp speeds up value iteration and avoids the requirement of initial admissible control policy in policy iteration at the same time the convergence theory of mshdp is established by proving that it converges to the solution of the bellman equation for implementation purpose the actor critic neural network nn structure is developed the critic nn is employed to estimate the value function and its nn weight vector is computed with a least square scheme the actor nn is used to estimate the control policy and a gradient descent method is proposed for updating its nn weight vector according to the comparative simulation studies on two examples the effectiveness and advantages of mshdp are verified

meshless rbfs method for numerical solutions of two dimensional high order fractional sobolev equations abstract in this paper meshless rbfs method is proposed to solve two dimensional time fractional sobolev equations the proposed method uses rbfs for approximation of spatial operator finite difference formula of o t 2 0 1 is used for time fractional derivative approximation while rule 0 1 as time stepping scheme for the advancement of solution in time validation of the proposed method is made by considering various test examples from literature simulated results are found in very good agreement with available exact solutions a rigorous comparative analysis made with other methods testifies proposed method s superiority in higher dimensions efficiency and accuracy of rbfs method are examined by varying number of nodes n in the domain of influence time step size t as well as l 2 l and l rms error norms linearized stability analysis of the proposed method is thoroughly discussed and verified numerically to support the analysis

a general result on the stabilization of linear systems using bounded controls we present two constructions of controllers that globally stabilize linear systems subject to control saturation we allow essentially arbitrary saturation functions the only conditions imposed on the system are the obvious necessary ones namely that no eigenvalues of the uncontrolled system have positive real part and that the standard stabilizability rank condition hold one of the constructions is in terms of a neural network type one hidden layer architecture while the other one is in terms of cascades of linear maps and saturations

online adaptive optimal control for continuous time nonlinear systems with completely unknown dynamics an online adaptive optimal control is proposed for continuous time nonlinear systems with completely unknown dynamics which is achieved by developing a novel identifier critic based approximate dynamic programming algorithm with a dual neural network nn approximation structure first an adaptive nn identifier is designed to obviate the requirement of complete knowledge of system dynamics and a critic nn is employed to approximate the optimal value function then the optimal control law is computed based on the information from the identifier nn and the critic nn so that the actor nn is not needed in particular a novel adaptive law design method with the parameter estimation error is proposed to online update the weights of both identifier nn and critic nn simultaneously which converge to small neighbourhoods around their ideal values the closed loop system stability and the convergence to small vicinity around the optimal solution are all proved by means of the lyapunov theory the proposed ada

multi agent differential graphical games online adaptive learning solution for synchronization with optimality multi agent systems arise in several domains of engineering and can be used to solve problems which are difficult for an individual agent to solve strategies for team decision problems including optimal control n player games h infinity control and non zero sum and so on are normally solved for off line by solving associated matrix equations such as the coupled riccati equations or coupled hamilton jacobi equations however using that approach players cannot change their objectives online in real time without calling for a completely new off line solution for the new strategies therefore in this paper we bring together cooperative control reinforcement learning and game theory to present a multi agent formulation for the online solution of team games the notion of graphical games is developed for dynamical systems where the dynamics and performance indices for each node depend only on local neighbor information it is shown that standard definitions for nash equilibrium are not sufficient for graphical games and a new definition of interactive nash equilibrium is given we give a cooperative policy iteration algorithm for graphical games that converges to the best response when the neighbors of each agent do not update their policies and to the cooperative nash equilibrium when all agents update their policies simultaneously this is used to develop methods for online adaptive learning solutions of graphical games in real time along with proofs of stability and convergence

optimal nonlinear but continuous feedback control of systems with saturating actuators this paper presents a modification of the optimal saturating feedback control laws given by frankena and sivan 1979 and ryan 1982 a for asymptotically stable systems unlike their results which involve bang bang action and singular extremals the modified control law is continuous specifically the new control law is linear inside a cylinder set and saturated elsewhere using steady state hamilton jacobi bellman theory this control law is shown to be optimal for a modified performance functional with a discontinuous integrand

verification theorems within the framework of viscosity solutions abstract in optimal control theory the verification technique plays an important role in testing for the optimality of a given control and in constructing optimal feedback controls however the existing classical verification theorem is restrictive in that it requires the associated dynamic programming equation to have smooth solutions in this paper some verification theorems are presented within the framework of viscosity solutions under mild assumptions these theorems are shown to have wider applicability than the classical verification theorem as a relevant problem some differences and relationships between the viscosity solution and clarke s generalized gradient are also discussed

numerical methods for convection dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures finite element and finite difference methods are combined with the method of characteristics to treat a parabolic problem of the form cu t bu x au x x f optimal order error estimates in l 2 and w 1 2 are derived for the finite element procedure various error estimates are presented for a variety of finite difference methods the estimates show that for convection dominated problems b gg a these schemes have much smaller time truncation errors than those of standard methods extensions to n space variables and time dependent or nonlinear coefficients are indicated along with applications of the concepts to certain problems described by systems of differential equations

l sub 2 gain analysis of nonlinear systems and nonlinear state feedback h sub infinity control previously obtained results on l2 gain analysis of smooth nonlinear systems are unified and extended using an approach based on hamilton jacobi equations and inequalities and their relation to invariant manifolds of an associated hamiltonian vector field on the basis of these results a nonlinear analog is obtained of the simplest part of a state space approach to linear h sub infinity control namely the state feedback h sub infinity optimal control problem furthermore the relation with h sub infinity control of the linearized system is dealt with

conditional empirical processes consider a sequence i xi yi i 1 of independent d 1 random vectors with common d f h text h left text x y right mathbb p left text x x y y right text left text x y right in mathbb r text d 1

data driven zero sum neuro optimal control for a class of continuous time unknown nonlinear systems with disturbance using adp this paper is concerned with a new data driven zero sum neuro optimal control problem for continuous time unknown nonlinear systems with disturbance according to the input output data of the nonlinear system an effective recurrent neural network is introduced to reconstruct the dynamics of the nonlinear system considering the system disturbance as a control input a two player zero sum optimal control problem is established adaptive dynamic programming adp is developed to obtain the optimal control under the worst case of the disturbance three single layer neural networks including one critic and two action networks are employed to approximate the performance index function the optimal control law and the disturbance respectively for facilitating the implementation of the adp method convergence properties of the adp method are developed to show that the system state will converge to a finite neighborhood of the equilibrium the weight matrices of the critic and the two action networks are also convergent to finite neighborhoods of their optimal ones finally the simulation results will show the effectiveness of the developed data driven adp methods

a multivariate adaptive regression b spline algorithm bmars for solving a class of nonlinear optimal feedback control problems in this paper we present a novel method for solving a class of nonlinear optimal feedback control problems with moderately high dimensional state spaces based on an adapted version of the bmars algorithm numerical experiments were performed using problems with up to six state variables the numerical results clearly demonstrate the efficiency and potential of the method for solving high dimensional problems

reinforcement learning design based adaptive tracking control with less learning parameters for nonlinear discrete time mimo systems based on the neural network nn approximator an online reinforcement learning algorithm is proposed for a class of affine multiple input and multiple output mimo nonlinear discrete time systems with unknown functions and disturbances in the design procedure two networks are provided where one is an action network to generate an optimal control signal and the other is a critic network to approximate the cost function an optimal control signal and adaptation laws can be generated based on two nns in the previous approaches the weights of critic and action networks are updated based on the gradient descent rule and the estimations of optimal weight vectors are directly adjusted in the design consequently compared with the existing results the main contributions of this paper are 1 only two parameters are needed to be adjusted and thus the number of the adaptation laws is smaller than the previous results and 2 the updating parameters do not depend on the number of the subsystems for mimo systems and the tuning rules are replaced by adjusting the norms on optimal weight vectors in both action and critic networks it is proven that the tracking errors the adaptation laws and the control inputs are uniformly bounded using lyapunov analysis method the simulation examples are employed to illustrate the effectiveness of the proposed algorithm

convergence of meshfree collocation methods for fully nonlinear parabolic equations we prove the convergence of meshfree collocation methods for the terminal value problems of fully nonlinear parabolic partial differential equations in the framework of viscosity solutions provided that the basis function approximations of the terminal condition and the nonlinearities are successful at each time step a numerical experiment with a radial basis function demonstrates the convergence property

neural network based near optimal control for a class of discrete time affine nonlinear systems with control constraints in this paper the near optimal control problem for a class of nonlinear discrete time systems with control constraints is solved by iterative adaptive dynamic programming algorithm first a novel nonquadratic performance functional is introduced to overcome the control constraints and then an iterative adaptive dynamic programming algorithm is developed to solve the optimal feedback control problem of the original constrained system with convergence analysis in the present control scheme there are three neural networks used as parametric structures for facilitating the implementation of the iterative algorithm two examples are given to demonstrate the convergence and feasibility of the proposed optimal control scheme

neural network based finite horizon optimal control of uncertain affine nonlinear discrete time systems in this paper the finite horizon optimal control design for nonlinear discrete time systems in affine form is presented in contrast with the traditional approximate dynamic programming methodology which requires at least partial knowledge of the system dynamics in this paper the complete system dynamics are relaxed utilizing a neural network nn based identifier to learn the control coefficient matrix the identifier is then used together with the actor critic based scheme to learn the time varying solution referred to as the value function of the hamilton jacobi bellman hjb equation in an online and forward in time manner since the solution of hjb is time varying nns with constant weights and time varying activation functions are considered to properly satisfy the terminal constraint an additional error term is incorporated in the novel update law such that the terminal constraint error is also minimized over time policy and or value iterations are not needed and the nn weights are updated once a sampling instant the uniform ultimate boundedness of the closed loop system is verified by standard lyapunov stability theory under nonautonomous analysis numerical examples are provided to illustrate the effectiveness of the proposed method

selection of shape parameter in radial basis functions for solution of time fractional black scholes models abstract the current work aims to exploit two techniques namely residual power series method rpsm and collocation based meshfree method for the solution of time fractional black scholes models with constant and variable coefficients firstly using rpsm we obtain exact solutions of the considered models and then numerical solution by meshfree method computer simulations are performed for three test problems of european options pricing the simulations features excellent agreement with exact solutions accuracy and efficiency of the proposed numerical method is assessed via e2 e and erms error norms convergence of the proposed methods is also analyzed

compactly supported radial basis functions based collocation method for level set evolution in image segmentation the partial differential equation driving level set evolution in segmentation is usually solved using finite differences schemes in this paper we propose an alternative scheme based on radial basis functions rbfs collocation this approach provides a continuous representation of both the implicit function and its zero level set we show that compactly supported rbfs csrbfs are particularly well suited to collocation in the framework of segmentation in addition csrbfs allow us to reduce the computation cost using a kd tree based strategy for neighborhood representation moreover we show that the usual reinitialization step of the level set may be avoided by simply constraining the l1 norm of the csrbf parameters as a consequence the final solution is topologically more flexible and may develop new contours i e new zero level components which are difficult to obtain using reinitialization the behavior of this approach is evaluated from numerical simulations and from medical data of various kinds such as 3 d ct bone images and echocardiographic ultrasound images

neural network based online simultaneous policy update algorithm for solving the hji equation in nonlinear h_ infty control it is well known that the nonlinear h state feedback control problem relies on the solution of the hamilton jacobi isaacs hji equation which is a nonlinear partial differential equation that has proven to be impossible to solve analytically in this paper a neural network nn based online simultaneous policy update algorithm spua is developed to solve the hji equation in which knowledge of internal system dynamics is not required first we propose an online spua which can be viewed as a reinforcement learning technique for two players to learn their optimal actions in an unknown environment the proposed online spua updates control and disturbance policies simultaneously thus only one iterative loop is needed second the convergence of the online spua is established by proving that it is mathematically equivalent to newton s method for finding a fixed point in a banach space third we develop an actor critic structure for the implementation of the online spua in which only one critic nn is needed for approximating the cost function and a least square method is given for estimating the nn weight parameters finally simulation studies are provided to demonstrate the effectiveness of the proposed algorithm

error bounds of adaptive dynamic programming algorithms for solving undiscounted optimal control problems in this paper we establish error bounds of adaptive dynamic programming algorithms for solving undiscounted infinite horizon optimal control problems of discrete time deterministic nonlinear systems we consider approximation errors in the update equations of both value function and control policy we utilize a new assumption instead of the contraction assumption in discounted optimal control problems we establish the error bounds for approximate value iteration based on a new error condition furthermore we also establish the error bounds for approximate policy iteration and approximate optimistic policy iteration algorithms it is shown that the iterative approximate value function can converge to a finite neighborhood of the optimal value function under some conditions to implement the developed algorithms critic and action neural networks are used to approximate the value function and control policy respectively finally a simulation example is given to demonstrate the effectiveness of the developed algorithms

brief paper a neural network solution for fixed final time optimal control of nonlinear systems in this paper fixed final time optimal control laws using neural networks and hjb equations for general affine in the input nonlinear systems are proposed the method utilizes kronecker matrix methods along with neural network approximation over a compact set to solve a time varying hjb equation the result is a neural network feedback controller that has time varying coefficients found by a priori offline tuning convergence results are shown the results of this paper are demonstrated on an example

convergence of sums of mixing triangular arrays of random vectors with stationary rows this paper deals with the convergence in distribution to gaussian generalized poisson and infinitely divisible laws of the row sums of certain x or mixing triangular arrays of banach space valued random vectors with stationary rows necessary and sufficient conditions for convergence in terms of individual r v s are proved these include sufficient conditions for the convergence to a stable law of the normalized sums of certain 0 mixing stationary sequences an invariance principle for stationary 0 mixing triangular arrays is given

nonhomeomorphic conjugates of connected shimura varieties we show that conjugation by an automorphism of bbb c may change the topological fundamental group of a locally symmetric variety over bbb c as a consequence we obtain a large class of algebraic varieties defined over number fields with the property that different embeddings of the number field into bbb c give complex varieties with nonisomorphic fundamental groups

multi player non zero sum games online adaptive learning solution of coupled hamilton jacobi equations in this paper we present an online adaptive control algorithm based on policy iteration reinforcement learning techniques to solve the continuous time ct multi player non zero sum nzs game with infinite horizon for linear and nonlinear systems nzs games allow for players to have a cooperative team component and an individual selfish component of strategy the adaptive algorithm learns online the solution of coupled riccati equations and coupled hamilton jacobi equations for linear and nonlinear systems respectively this adaptive control method finds in real time approximations of the optimal value and the nzs nash equilibrium while also guaranteeing closed loop stability the optimal adaptive algorithm is implemented as a separate actor critic parametric network approximator structure for every player and involves simultaneous continuous time adaptation of the actor critic networks a persistence of excitation condition is shown to guarantee convergence of every critic to the actual optimal value function for that player a detailed mathematical analysis is done for 2 player nzs games novel tuning algorithms are given for the actor critic networks the convergence to the nash equilibrium is proven and stability of the system is also guaranteed this provides optimal adaptive control solutions for both non zero sum games and their special case the zero sum games simulation examples show the effectiveness of the new algorithm

optimal portfolios with regime switching and value at risk constraint we consider the optimal portfolio selection problem subject to a maximum value at risk mvar constraint when the price dynamics of the risky asset are governed by a markov modulated geometric brownian motion gbm here the market parameters including the market interest rate of a bank account the appreciation rate and the volatility of the risky asset switch over time according to a continuous time markov chain whose states are interpreted as the states of an economy the mvar is defined as the maximum value of the vars of the portfolio in a short time duration over different states of the chain we formulate the problem as a constrained utility maximization problem over a finite time horizon by utilizing the dynamic programming principle we shall first derive a regime switching hamilton jacobi bellman hjb equation and then a system of coupled hjb equations we shall employ an efficient numerical method to solve the system of coupled hjb equations for the optimal constrained portfolio we shall provide numerical results for the sensitivity analysis of the optimal portfolio the optimal consumption and the var level with respect to model parameters these results are also used to investigating the effect of the switching regimes

a novel actor critic identifier architecture for approximate optimal control of uncertain nonlinear systems an online adaptive reinforcement learning based solution is developed for the infinite horizon optimal control problem for continuous time uncertain nonlinear systems a novel actor critic identifier aci is proposed to approximate the hamilton jacobi bellman equation using three neural network nn structures actor and critic nns approximate the optimal control and the optimal value function respectively and a robust dynamic neural network identifier asymptotically approximates the uncertain system dynamics an advantage of using the aci architecture is that learning by the actor critic and identifier is continuous and simultaneous without requiring knowledge of system drift dynamics convergence of the algorithm is analyzed using lyapunov based adaptive control methods a persistence of excitation condition is required to guarantee exponential convergence to a bounded region in the neighborhood of the optimal control and uniformly ultimately bounded uub stability of the closed loop system simulation results demonstrate the performance of the actor critic identifier method for approximate optimal control

an iterative algorithm for a quasivariational inequality system related to hjb equation this paper presents a new iterative algorithm for a quasivariational inequality system related to hjb equation a domain decomposition method based on this algorithm is proposed the convergence theorems have been established

green forms and the arithmetic siegel weil formula we construct natural green forms for special cycles in orthogonal and unitary shimura varieties in all codimensions and for compact shimura varieties of type mathrm o p 2 and mathrm u p 1 we show that the resulting local archimedean height pairings are related to special values of derivatives of siegel eisentein series a conjecture put forward by kudla relates these derivatives to arithmetic intersections of special cycles and our results settle the part of his conjecture involving local archimedean heights

infinite horizon self learning optimal control of nonaffine discrete time nonlinear systems in this paper a novel iterative adaptive dynamic programming adp based infinite horizon self learning optimal control algorithm called generalized policy iteration algorithm is developed for nonaffine discrete time dt nonlinear systems generalized policy iteration algorithm is a general idea of interacting policy and value iteration algorithms of adp the developed generalized policy iteration algorithm permits an arbitrary positive semidefinite function to initialize the algorithm where two iteration indices are used for policy improvement and policy evaluation respectively it is the first time that the convergence admissibility and optimality properties of the generalized policy iteration algorithm for dt nonlinear systems are analyzed neural networks are used to implement the developed algorithm finally numerical examples are presented to illustrate the performance of the developed algorithm

distributed cooperative h optimal tracking control of mimo nonlinear multi agent systems in strict feedback form via adaptive dynamic programming the design of distributed cooperative h optimal controllers for multi agent systems is a major challenge when the agents models are uncertain multi input and multi output nonlinear systems in str

conditional empirical processes defined by φ mixing sequences for a random vector x y in r 2 let m y x p y y x x be the conditional distribution function it is assumed that x 1 y 1 x 2 y 2 form a strictly stationary mixing sequence of random vectors with the same distribution as x y considering an estimate mn x of m x it is shown that nan 1 2 mn x 0 m x 0 converges to a certain gaussian process depending on x 0 where an is a sequence of bandwiths

iterative adaptive dynamic programming for solving unknown nonlinear zero sum game based on online data h infty control is a powerful method to solve the disturbance attenuation problems that occur in some control systems the design of such controllers relies on solving the zero sum game zsg but in practical applications the exact dynamics is mostly unknown identification of dynamics also produces errors that are detrimental to the control performance to overcome this problem an iterative adaptive dynamic programming algorithm is proposed in this paper to solve the continuous time unknown nonlinear zsg with only online data a model free approach to the hamilton jacobi isaacs equation is developed based on the policy iteration method control and disturbance policies and value are approximated by neural networks nns under the critic actor disturber structure the nn weights are solved by the least squares method according to the theoretical analysis our algorithm is equivalent to a gauss newton method solving an optimization problem and it converges uniformly to the optimal solution the online data can also be used repeatedly which is highly efficient simulation results demonstrate its feasibility to solve the unknown nonlinear zsg when compared with other algorithms it saves a significant amount of online measurement time

on application of an alternating direction method to hamilton jacobin bellman equations this paper presents a numerical method for the approximation of viscosity solutions to a hamilton jacobi bellman hjb equation governing a class of optimal feedback control problems the first order hjb equation is first perturbed by adding a diffusion term with a singular perturbation parameter the time and spatial variables in the resulting equation are then discretized respectively by an implicit modified method of characteristics and the alternating direction ad scheme we show that the ad procedure s perturbation error is virtually negligible due to the small perturbation parameter and the efficient ad scheme can be applied to our hjb equation without generating splitting error numerical results performed to verify the usefulness of the method show that the method gives accurate approximate solutions to both of the control and the state variables

multi agent zero sum differential graphical games for disturbance rejection in distributed control this paper addresses distributed optimal tracking control of multi agent linear systems subject to external disturbances the concept of differential game theory is utilized to formulate this distributed control problem into a multi player zero sum differential graphical game which provides a new perspective on distributed tracking of multiple agents influenced by disturbances in the presented differential graphical game the dynamics and performance indices for each node depend on local neighbor information and disturbances it is shown that the solution to the multi agent differential graphical games in the presence of disturbances requires the solution to coupled hamilton jacobi isaacs hji equations multi agent learning policy iteration pi algorithm is provided to find the solution to these coupled hji equations and its convergence is proven it is also shown that l 2 bounded synchronization errors can be guaranteed using this technique an online pi algorithm is given to solve the zero sum game in real time a simulation example is provided to show the effectiveness of the online approach

policy iterations on the hamilton jacobi isaacs equation for h_ infty state feedback control with input saturation an hinfin suboptimal state feedback controller for constrained input systems is derived using the hamilton jacobi isaacs hji equation of a corresponding zero sum game that uses a special quasi norm to encode the constraints on the input the unique saddle point in feedback strategy form is derived using policy iterations on both players the hji equation is broken into a sequence of differential equations linear in the cost for which closed form solutions are easier to obtain policy iterations on the disturbance are shown to converge to the available storage function of the associated l2 gain dissipative dynamics the resulting constrained optimal control feedback strategy has the largest domain of validity within which l2 performance for a given gamma is guaranteed

l boundedness of l 2 projections on splines for a multiple geometric mesh this paper concerns the l2 projectors from loo to the normed linear space of polynomial splines it is shown that for the multiple geometric meshes the loo norms of the corresponding l2 projectors are bounded independently of the mesh ratio 0 introduction let us begin with some notation let k be a positive integer and x x i z a real nondecreasing knot sequence with xi xi k all i set x 00 lim xi i x o lim xji i x oo x00 i 0 i o00o fori e z and x e i define mi k x k xi xi k x i ni k x xi k xi mi k x k where po pr f denotes the rth divided difference of the function f at the points po 0 pr then ni k is called the i th b spline of order k on the knot sequence x for a e rz the rule f x a i niak x defines a function on i if we take the sum to be pointwise every such function is called a polynomial spline of order k with the knot sequence x and their collection is denoted by skx further let denote the normed linear space of bounded polynomial splines of order k with the knot sequence x and norm fll ii sup i f x ix xei we shall be concerned with p the orthogonal projector onto with respect to the ordinary inner product f g ff x g x dx but restricted to l i we want to bound its norm 11 p ii sup iipsf ii 1 iii o fel j received march 26 1984 revised december 16 1985 1980 mathematics subject classification primary 41a15 15a60 this is a part of the author s thesis which was submitted to the graduate school university of wisconsin madison in partial fullfillment of the requirement for ph d degree present address department of mathematics zhejiang university hangzhou zhejiang the people s republic of china 1987 american mathematical society 0025 5718 87 1 00 25 per page 675 this content downloaded from 207 46 13 178 on wed 15 jun 2016 07 00 20 utc all use subject to http about jstor org terms

a meshfree hermite type radial point interpolation method for kirchhoff plate problems a meshfree computational method is proposed in this paper to solve kirchhoff plate problems of various geometries the deflection of the thin plate is approximated by using a hermite type radial basis function approximation technique the standard galerkin method is adopted to discretize the governing partial differential equations which were derived from using the kirchhoff s plate theory the degrees of freedom for the slopes are included in the approximation to make the proposed method effective in enforcing essential boundary conditions numerical examples with different geometric shapes and various boundary conditions are given to verify the efficiency accuracy and robustness of the method copyright 2005 john wiley sons ltd

meshless spectral method for solution of time fractional coupled kdv equations abstract in this article an efficient and accurate meshless spectral interpolation method is formulated for the numerical solution of time fractional coupled kdv equations that govern shallow water waves meshless shape functions constructed via radial basis functions rbfs and point interpolation are used for discretization of the spatial operator approximation of fractional temporal derivative is obtained via finite differences of order o 2 and a quadrature formula the formulated method is applied to various test problems available in the literature for its validation approximation quality and efficiency of the method is measured via discrete error norms e2 e and erms convergence analysis of the proposed method in space and time is numerically determined by varying nodal points m and time step size respectively stability of the proposed method is discussed and affirmed computationally which is an important ingredient of the current study

finite horizon approximate optimal guaranteed cost control of uncertain nonlinear systems with application to mars entry guidance this paper studies the finite horizon optimal guaranteed cost control gcc problem for a class of time varying uncertain nonlinear systems the aim of this problem is to find a robust state feedback controller such that the closed loop system has not only a bounded response in a finite duration of time for all admissible uncertainties but also a minimal guaranteed cost a neural network nn based approximate optimal gcc design is developed initially by modifying the cost function to account for the nonlinear perturbation of system the optimal gcc problem is transformed into a finite horizon optimal control problem of the nominal system subsequently with the help of the modified cost function together with a parametrized bounding function for all admissible uncertainties the solution to the optimal gcc problem is given in terms of a parametrized hamilton jacobi bellman phjb equation then a nn method is developed to solve offline the phjb equation approximately and thus obtain the nearly optimal gcc policy furthermore the convergence of approximate phjb equation and the robust admissibility of nearly optimal gcc policy are also analyzed finally by applying the proposed design method to the entry guidance problem of the mars lander the achieved simulation results show the effectiveness of the proposed controller

norms of projectors onto spaces with riesz bases abstract many important linear operators p x s of a linear space x onto a subspace s of x are defined by the minimum problem f p f min f u u s f x where the norm on x is induced by an inner product we study the l norm of such operators p when s has a riesz basis b then we apply the results in detail to the best l2 and l2 approximations for univariate polynomial splines spaces and to box spline series

some properties of viscosity solutions of hamilton jacobi equations abstract recently m g crandall and p l lions introduced the notion of viscosity solutions of scalar nonlinear first order partial differential equations viscosity solutions need not be differentiable anywhere and thus are not sensitive to the classical problem of the crossing of characteristics the value of this concept is established by the fact that very general existence uniqueness and continuous dependence results hold for viscosity solutions of many problems arising in fields of application the notion of a viscosity solution admits several equivalent formulations here we look more closely at two of these equivalent criteria and exhibit their virtues by both proving several new facts and reproving various known results in a simpler manner moreover by forsaking technical generality we hereby provide a more congenial introduction to this subject than the original paper author

brief intelligent optimal control of robotic manipulators using neural networks the paper is concerned with the application of quadratic optimization for motion control to feedback control of robotic systems using neural networks explicit solutions to the hamilton jacobi bellman h j b equation for optimal control of robotic systems are found by solving an algebraic riccati equation it is shown how neural networks can cope with nonlinearities through optimization with no preliminary off line learning phase required the adaptive learning algorithm is derived from lyapunov stability analysis so that both system tracking stability and error convergence can be guaranteed in the closed loop system the filtered tracking error or critic gain and the lyapunov function for the nonlinear analysis are derived from the user input in terms of a specified quadratic performance index simulation results on a two link robot manipulator show the satisfactory performance of the proposed control schemes even in the presence of large modeling uncertainties and external disturbances

neural network based robust optimal control design for a class of uncertain nonlinear systems via adaptive dynamic programming in this paper the neural network based robust optimal control design for a class of uncertain nonlinear systems via adaptive dynamic programming approach is investigated first the robust controller of the original uncertain system is derived by adding a feedback gain to the optimal controller of the nominal system it is also shown that this robust controller can achieve optimality under a specified cost function which serves as the basic idea of the robust optimal control design then a critic network is constructed to solve the hamilton jacobi bellman equation corresponding to the nominal system where an additional stabilizing term is introduced to verify the stability the uniform ultimate boundedness of the closed loop system is also proved by using the lyapunov approach moreover the obtained results are extended to solve decentralized optimal control problem of continuous time nonlinear interconnected large scale systems finally two simulation examples are presented to illustrate the effectiveness of the established control scheme

fractional diffusion equations by the kansa method this study makes the first attempt to apply the kansa method in the solution of the time fractional diffusion equations in which the multiquadrics and thin plate spline serve as the radial basis function in the discretization formulation the finite difference scheme and the kansa method are respectively used to discretize time fractional derivative and spatial derivative terms the numerical solutions of one and two dimensional cases are presented and discussed which agree well with the corresponding analytical solution

policy iteration adaptive dynamic programming algorithm for discrete time nonlinear systems this paper is concerned with a new discrete time policy iteration adaptive dynamic programming adp method for solving the infinite horizon optimal control problem of nonlinear systems the idea is to use an iterative adp technique to obtain the iterative control law which optimizes the iterative performance index function the main contribution of this paper is to analyze the convergence and stability properties of policy iteration method for discrete time nonlinear systems for the first time it shows that the iterative performance index function is nonincreasingly convergent to the optimal solution of the hamilton jacobi bellman equation it is also proven that any of the iterative control laws can stabilize the nonlinear systems neural networks are used to approximate the performance index function and compute the optimal control law respectively for facilitating the implementation of the iterative adp algorithm where the convergence of the weight matrices is analyzed finally the numerical results and analysis are presented to illustrate the performance of the developed method

galerkin approximations of the generalized hamilton jacobi bellman equation in this paper we study the convergence of the galerkin approximation method applied to the generalized hamilton jacobi bellman ghjb equation over a compact set containing the origin the ghjb equation gives the cost of an arbitrary control law and can be used to improve the performance of this control the ghjb equation can also be used to successively approximate the hamilton jacobi bellman equation we state sufficient conditions that guarantee that the galerkin approximation converges to the solution of the ghjb equation and that the resulting approximate control is stabilizing on the same region as the initial control the method is demonstrated on a simple nonlinear system and is compared to a result obtained by using exact feedback linearization in conjunction with the lqr design method