identification of mixture models using support variations abstract we consider the issue of identifying nonparametrically continuous mixture models in these models all observed variables depend on a common and unobserved component but are mutually independent conditional on it such a structure applies for instance to measurement error matching and auction models traditional approaches rely on parametric assumptions or strong functional restrictions we show that these models are actually identified nonparametrically if the supports of the observed variables move with the value of the unobserved component moreover this assumption is testable nonparametrically using tools from extreme value theory we develop an appropriate test and derive its asymptotic properties

continuous time stochastic model reference adaptive control the authors establish global convergence and asymptotic properties of a direct adaptive controller for continuous time stochastic linear systems by presenting a direct adaptive control algorithm and an associated proof of convergence this result is comprehensive and covers many other existing results as special cases it has practical implications for the discrete time case since it reveals how the existing discrete time results must be modified so that they have meaningful limits as the sampling period decreases

the backward euler method for numerical solution of the hodgkin huxley equations of nerve conduction a widely accepted model of nerve conduction is based on nonlinear parabolic partial differential equations pdes derived for the giant axon of the squid loligo by hodgkin and huxley when consider

stability and asymptotic behavior of a numerical solution corresponding to a diffusion reaction equation solved by a finite difference scheme crank nicolson abstract in this paper some partial differential equations arising in biochemistry are studied from the numerical point of view these nonlinear equations based on diffusion reaction processes are solved by using a crank nicolson technique and convergence is proved by using lower and upper solutions stability and asymptotic behavior are also examined a numerical algorithm has been deduced and is presently used by chemists

stochastic linear quadratic adaptive control for continuous time first order systems in this paper we discuss the linear quadratic lq adaptive control problem for the following continuous time first order scalar stochastic system dxt axt dt but dt c dwt with cost function min lim sup jt u u u t where jt u 1t 0t q1x2s q2u2s ds q1 0 0 based on the self convergence property of continuous time weighted least squares cwls algorithm 9 and the similar parameter modification method 13 the stability of the closed loop system is achieved without invoking any excitations here self convergence means that the convergence is automatic in the sense that no excitation conditions on the signals or measurements is needed it is a terminology from 10 the strong consistency of cwls and the optimality of the adaptive control are established by incorporating with diminishing excitations

a linearly implicit predictor corrector method for reaction diffusion equations abstract a novel linearly implicit predictor corrector scheme is developed for the numerical solution of reaction diffusion equations iterative processes are avoided by treating the nonlinear reaction terms explicitly while maintaining superior accuracy and stability properties compared to the well known methods and linearly implicit runge kutta methods the proposed method allows the opportunity of solving large systems of reaction diffusion equations by alleviating the necessity of solving the accompanying large linear systems of algebraic equations due to the natural parallelism which surfaces across the system numerical results confirm the enhanced stability accuracy and efficiency of the method when applied to reaction diffusion equations arising in biochemistry and population ecology

mean field stochastic adaptive control for noncooperative games the mean field mf methodology provides decentralized strategies which yield nash equilibria for large population systems in the asymptotic limit of an infinite mass population the mf control laws use only the local information of each agent on its own state and own dynamical parameters while the mass effect is calculated offline using the distribution function of i the population s dynamical parameters and ii the population s cost function parameters for the infinite population case these laws yield approximate equilibria when applied in the finite population in this paper these a priori information conditions are relaxed and incrementally the cases are considered where first the agents estimate their own dynamical parameters and second estimate the distribution parameter in i and ii above an mf stochastic adaptive control sac law in which each agent observes a random subset of the population of agents is specified where the ratio of the cardinality of the observed set to that of the number of agents decays to zero as the population size tends to infinity each agent estimates its own dynamical parameters via the recursive weighted least squares rwls algorithm and the distribution of the population s dynamical parameters via maximum likelihood estimation mle under reasonable conditions on the population dynamical parameter distribution the mf sac law applied by each agent results in i the strong consistency of the self parameter estimates and the strong consistency of the population distribution function parameters ii the long run average l2 stability of all agent systems iii a strong nash equilibrium for the population of agents for all 0 and iv the a s equality of the long run average cost and the non adaptive cost in the population limit

moment reconstruction and moment adjusted imputation when exposure is generated by a complex nonlinear random effects modeling process for the classical homoscedastic measurement error model moment reconstruction freedman et al 2004 2008 and moment adjusted imputation thomas et al 2011 are appealing computationally simple imputation like methods for general model fitting like classical regression calibration the idea is to replace the unobserved variable subject to measurement error with a proxy that can be used in a variety of analyses moment reconstruction and moment adjusted imputation differ from regression calibration in that they attempt to match multiple features of the latent variable and also to match some of the latent variable s relationships with the response and additional covariates in this note we consider a problem where true exposure is generated by a complex nonlinear random effects modeling process and develop analogues of moment reconstruction and moment adjusted imputation for this case this general model includes classical measurement errors berkson measurement errors mixtures of berkson and classical errors and problems that are not measurement error problems but also cases where the data generating process for true exposure is a complex nonlinear random effects modeling process the methods are illustrated using the national institutes of health aarp diet and health study where the latent variable is a dietary pattern score called the healthy eating index 2005 we also show how our general model includes methods used in radiation epidemiology as a special case simulations are used to illustrate the methods

l0 stable split linear multistep formulas for parabolic pdes abstract new types of split linear multistep methods slmms are applied to the numerical solution of parabolic differential equations the formulas derived achieve up to fourth order in time and unlike the crank nicolson method are l0 stable

continuous time stochastic adaptive control this paper establishes global boundedness for a continuous time stochastic adaptive control algorithm it is shownn that with probability one the system inputs and outputs satisfy a sample mean square boundedness property the algorithm and method of analysis are not directly analogous to the discrete time case since special features are necessary to handle the continuous time problem

semiparametric regression modeling with mixtures of berkson and classical error with application to fallout from the nevada test site nous construisons des methodes bayesiennes pour une modelisation semi parametrique d une fonction de regression monotone quand les predicteurs sont mesures avec une erreur classique une erreur de berkson ou les deux de telles methodes demandent une distribution pour le predicteur non observe variable latente distribution que nous modelisons aussi de facon semi parametrique de telles combinaisons de methodes semi parametriques pour la reponse a des doses aussi bien que pour la distribution de la variable latente n ont pas ete etudiees dans la litterature sur les erreurs de mesure quelle que soit la forme de l erreur de mesure de plus nos methodes proposent une nouvelle approche des problemes ou l erreur de mesure combine une composante de berkson et une composante classique ces methodes sont generales mais nous les developpons autour d une application particuliere qui est l etude des maladies de la thyroide en relation avec les radiations venant du site de test du nevada nous utilisons ces donnees pour illustrer nos methodes lesquelles suggerent une estimation ponctuelle moyenne a posteriori du risque relatif a des doses pres de deux fois plus elevee que celle obtenue par les analyses precedentes mais suggere aussi une bien plus grande incertitude sur le risque relatif

regressions with berkson errors in covariates a nonparametric approach this paper establishes that so called instrumental variables enable the identification and the estimation of a fully nonparametric regression model with berkson type measurement error in the regressors an estimator is proposed and proven to be consistent its practical performance and feasibility are investigated via monte carlo simulations as well as through an epidemiological application investigating the effect of particulate air pollution on respiratory health these examples illustrate that berkson errors can clearly not be neglected in nonlinear regression models and that the proposed method represents an effective remedy

predictor corrector methods for parabolic partial differential equations in this paper we extend predictor corrector methods commonly used for the numerical solution of ordinary differential equations o d e s to parabolic partial differential equations p d e s typically of the form ut auxx u ux x t we describe linear multistep methods for p d e s the nonlinear algebraic equations arising from implicit formulae being solved using a corrector analogous to those used for o d e s a sufficient condition for convergence of the iteration is then derived and is found in most cases to be far less restrictive than that obtained from the usual method of lines approach numerical results are presented to investigate the necessity of this condition they also indicate that we can accelerate convergence by reducing the time increment this allows us to achieve convergence within a prescribed number of iterations and so to construct pcm methods corresponding to p ec m methods for o d e s numerical results are also given to test the absolute stability of the crank nicolson corrector for various predictors p and iterations m

a real distinct poles rational approximation of generalized mittag leffler functions and their inverses applications to fractional calculus abstract the generalized mittag leffler function and its inverse is very useful in solving fractional differential equations and structural derivatives respectively however their computational complexities have made them difficult to deal with numerically we propose a real distinct pole rational approximation of the generalized mittag leffler function under some mild conditions this approximation is proven and empirically shown to be l acceptable due to the complete monotonicity property of the mittag leffler function we derive a rational approximation for the inverse generalized mittag leffler function these approximations are especially useful in developing efficient and accurate numerical schemes for partial differential equations of fractional order several applications are presented such as complementary error function solution of fractional differential equations and the ultraslow diffusion model using the structural derivative

recent progress in extrapolation methods for ordinary differential equations the paper is intended to give a survey on the state of the art of extrapolation methods for initial value problems in ordinary differential equation systems all basic discretizations which are su

convergence results for continuous time adaptive stochastic filtering algorithms abstract the adaptive stochastic filtering problem for gaussian processes is considered the self tuning synthesis procedure is used to derive two algorithms for this problem almost sure convergence for the parameter estimate and the filtering error will be established the convergence analysis is based on an almost supermartingale convergence lemma that allows a stochastic lyapunov like approach

continuous time stochastic adaptive tracking robustness and asymptotic properties adaptive estimation and control problems are considered for continuous time stochastic systems containing both modeled and unmodeled dynamics the least squares method is used to estimate unknown parameters included in the modeled part which are used to update an adaptive control law it is shown that both the estimation error and the tracking error are bounded and that the bounds are proportional to constants dominating the unmodeled dynamics moreover convergence rates of the tracking errrors are established in the case where no unmodeled dynamics exist

split linear multistep methods for the numerical integration of stiff differential systems a new approach to the problem of numerically integrating stiff differential systems is described in this approach a linear multistep method the basic method is split into a kind of predictor corrector scheme where the predictor is also implicit if this splitting is done in an appropriate manner the modified method has considerably better stability properties than the basic method as a result splitting methods are particularly useful for problems where conventional integration methods experience stability difficulties in particular some highly stable split linear multistep methods based on backward differentiation formulae are derived and a highly stable variable step implementation is proposed

bootstrapping general empirical measures it is proved that the bootstrapped central limit theorem for empirical processes indexed by a class of functions mathscr f and based on a probability measure p holds a s if and only if mathscr f in mathrm clt p and int f 2 dp infty where f sup f in mathscr f f and it holds in probability if and only if mathscr f in mathrm clt p thus for a large class of statistics no local uniformity of the clt about p is needed for the bootstrap to work consistency of the bootstrap the bootstrapped law of large numbers is also characterized these results are proved under certain weak measurability assumptions on mathscr f

convergence rates of continuous time stochastic els parameter estimation discrete time convergence rates for extended least squares els algorithms are generalized to the continuous time case an essential difference in the estimation is the appropriate prefiltering while in the theory the existence of solutions of the stochastic equations is a concern

persistence of excitation in linear systems abstract this paper develops output reachability characterizations of linear finite dimensional multivariate systems so as to translate excitation properties of system inputs to excitation properties of system outputs states or associated regression vectors such properties are of fundamental concern for convergence of algorithms involving on line identification adaptive state estimation prediction and control persistence of excitation guarantees convergence without a priori stability assumptions and ensures robustness properties

an ordinary differential equation technique for continuous time parameter estimation an ordinary differential equation technique is developed via averaging theory and weak convergence theory to analyze the asymptotic behavior of continuous time recursive stochastic parameter estimators this technique is an extension of l ljung s 1977 work in discrete time using this technique the following results are obtained for various continuous time parameter estimators the recursive prediction error method with probability one converges to a minimum of the likelihood function the same is true of the gradient method the extended kalman filter fails with probability one to converge to the true values of the parameters in a system whose state noise covariance is unknown an example of the extended least squares algorithm is analyzed in detail analytic bounds are obtained for the asymptotic rate of convergence of all three estimators applied to this example

nonlinear errors in variables estimation of some engel curves abstract the most common solution to the errors in variables problem for the linear regression model is the use of instrumental variable estimation however this methodology cannot be applied in the nonlinear regression framework in this paper we develop consistent estimators for nonlinear regression specifications when errors in variables are present we apply our methodology to estimation of engel curves on household data first we find that the lesser working specification of budget shares regressed on the log of income or expenditure should be generalized to higher order terms in log income also we find that errors in variables in either reported income or expenditure should be accounted for lastly and perhaps most interesting we find rather strong support for the gorman rank restriction on the matrix of coefficients for the polynomial terms in income

control theory methods for consistency in some least squares identification problems the consistency of a family of least squares estimates of some unknown parameters of a continuous time linear stochastic system is verified the unknown parameters appear affinely in the linear transformations of the state and the control the method of verification of consistency associates a family of control problems to the identification problem and the asymptotic behavior of the solutions of a family of algebraic riccati equations from the control problems implies a persistent excitation property for the identification problem

continuous time stochastic adaptive control non explosion e consistency and stability abstract for completely observed continuous time constant parameter stochastic linear systems an indirect adaptive control law is presented which subject principally to a weak location hypothesis concerning the true parameter and a persistent excitation hypothesis generates consistent recursive least squares parameter estimates and ensures the system is mean square sample path stable the adaptive control algorith mentails i recursively calculating the least squares estimate of the system parameter and ii recursively generating the lqr feedback gain matrix lying in a set of matrix gains known to contain a stabilizing gain the a s non explosion of the system is a direct consequence of this construction

non parametric regression estimation from data contaminated by a mixture of berkson and classical errors estimation of a regression function is a well known problem in the context of errors in variables where the explanatory variable is observed with random noise this noise can be of two types which are known as classical or berkson and it is common to assume that the error is purely of one of these two types in practice however there are many situations where the explanatory variable is contaminated by a mixture of the two errors in such instances the berkson component typically arises because the variable of interest is not directly available and can only be assessed through a proxy whereas the inaccuracy that is related to the observation of the latter causes an error of classical type we propose a non parametric estimator of a regression function from data that are contaminated by a mixture of the two errors we prove consistency of our estimator derive rates of convergence and suggest a data driven implementation finite sample performance is illustrated via simulated and real data examples

convergence of adaptive minimum variance algorithms via weighting coefficient selection weighted least squares and related stochastic approximation algorithms are studied for parameter estimation adaptive state estimation adaptive n step ahead prediction and adaptive control in both white and colored noise environments for the fundamental algorithm which is the basis for the various applications the step size in the stochastic approximation versions and the weighting coefficient in the weighted least squares schemes are selected according to a readily calculated stability measure associated with the estimator the selection is guided by the convergence theory in this way strong global convergence of the parameter estimates state estimates and prediction or tracking errors is not only guaranteed under the appropriate noise passivity and stability or minimum phase conditions but the convergence is also as fast as it appears reasonable to achieve given the simplicity of the adaptive scheme

a universality advantage of stochastic excitation signals for adaptive control abstract adaptive control schemes usually depend on estimation of system parameters which in turn depends excitation of modes associated with these parameters such excitation is supplied by plant noise the adaptive control signal itself and any external excitation such as a reference signal in this note it is shown that there is a universality advantage for any externally applied signal to be stochastic rather than deterministic the crucial property of a stochastic signal exploited in this note is its unpredictability by any causal system such as an adaptive control scheme when such unpredictable signals excite an adaptive control scheme there is no need to deliberately constrain the adaptation to be slow or excitation maintaining to ensure adequate identification

persistence of excitation in extended least squares in least squares parameter estimation schemes persistency of excitation conditions on the plant states are required for consistent estimation in the case of extended least squares the persistency conditions are on the state estimates here these persistency of excitation conditions are translated into sufficiently rich conditions on the plant noise and inputs in the case of adaptive minimum variance control schemes the sufficiently rich conditions are on the noise and specified output trajectory with sufficiently rich input signals guaranteed convergence rates of prediction errors improve and it is conjectured that the algorithms are consequently more robust

robust and consistent estimation of nonlinear errors in variables models this paper considers the consistent estimation of nonlinear errors in variables models it adopts the functional modeling approach by assuming that the true but unobserved regressors are random variables but making no parametric assumption on the distribution from which the latent variables are drawn this paper shows how the information extracted from the replicate measurements can be used to identify and consistently estimate a general nonlinear errors in variables model the identification is established through characteristic functions the estimation procedure involves nonparametric estimation of the conditional density of the latent variables given the measurements using the identification results at the first stage and at the second stage a semiparametric nonlinear least squares estimator is proposed the consistency of the proposed estimator is also established finite sample performance of the estimator is investigated through a monte carlo study

a note on continuous time els abstract the parameter estimation of continuous time finite dimensional linear stochastic systems is a problem of long standing interest the method usually used is the extended least squares els algorithm described by a nonlinear stochastic differential equation sld with the existence of the global strong solution assumed this paper shows that the els estimate does exist in 0 and at the same time presents a number of convergence results paralleling those for the discrete time case

the use of positive matrices for the analysis of the large time behavior of the numerical solution of reaction diffusion systems in this paper we study the numerical solution of nonlinear reaction diffusion systems with homogeneous neumann boundary conditions via the known o method we show that if conditions for the positivity of solutions are imposed then the study of the asymptotic behavior of the numerical solution can be done by means of the theory of stochastic matrices in this way it is shown that the numerical solution reproduces the asymptotic behavior of the corresponding theoretical one in particular we obtain the decay of the solution to its mean value an analysis of the asymptotic stability of the equilibrium points and the convergence of the numerical scheme is given based on the use of m matrices finally we consider the case in which the nonlinear term satisfies a condition of quasi monotonicity

linearly implicit methods for nonlinear pdes with linear dispersion and dissipation the linear stability of imex implicit explicit methods and exponential integrators for stiff systems of odes arising in the discrete solution of pdes is examined for nonlinear pdes with both linear dispersion and dissipation and a clear method of visualization of the linear stability regions is proposed predictions are made based on these visualizations and are supported by a series of experiments on five pdes including quasigeostrophic equations and stratified boussinesq equations the experiments involving 24 imex and exponential methods of third and fourth order confirm the predictions of the linear stability analysis that the methods are typically limited by small eigenvalues of the linear term and by eigenvalues on or near the imaginary axis rather than by large eigenvalues near the negative real axis the experiments also demonstrate that imex methods achieve comparable stability to exponential methods and that exponential methods are significantly more accurate only when the problem is nearly linear novel imex predictor corrector methods are also derived

on persistent excitation for linear systems with stochastic coefficients a wiener input process is shown to be persistently exciting pe for linear stochastic systems with time varying convergent random coefficients provided asymptotic noise controllability holds a s the pe result is in the sense that the minimum eigenvalue of the integrated outer product of the state process is of o t t being the upper time limit of the integral

nonparametric regression with errors in variables the effect of errors in variables in nonparametric regression estimation is examined to account for errors in covariates deconvolution is involved in the construction of a new class of kernel estimators it is shown that optimal local and global rates of convergence of these kernel estimators can be characterized by the tail behavior of the characteristic function of the error distribution in fact there are two types of rates of convergence according to whether the error is ordinary smooth or super smooth it is also shown that these results hold uniformly over a class of joint distributions of the response and the covariate which is rich enough for many practical applications furthermore to achieve optimality we show that the convergence rates of all possible estimators have a lower bound possessed by the kernel estimators

nonparametric methods for solving the berkson errors in variables problem summary it is common in errors in variables problems in regression to assume that the errors are incurred after the experiment in that the observed value of the explanatory variable is an independent perturbation of its true value however if the errors are incurred before the experiment then the true value of the explanatory variable equals a perturbation of its observed value this is the context of the berkson model which is increasingly attracting attention in parametric and semiparametric settings we introduce and discuss nonparametric techniques for analysing data that are generated by the berkson model our approach permits both random and regularly spaced values of the target doses in the absence of data on dosage error it is necessary to propose a distribution for the latter but we show numerically that our method is robust against that assumption the case of dosage error data is also discussed a practical method for smoothing parameter choice is suggested our techniques for errors in variables regression are shown to achieve theoretically optimal convergence rates

semi nonparametric maximum likelihood estimation on developpe une approche basee sur une innovation due a phillips 1983 sur les approximations des fonctions de densite

on methods of sieves and penalization we develop a general theory which provides a unified treatment for the asymptotic normality and efficiency of the maximum likelihood estimates mle s in parametric semiparametric and nonparametric models we find that the asymptotic behavior of substitution estimates for estimating smooth functionals are essentially governed by two indices the degree of smoothness of the functional and the local size of the underlying parameter space we show that when the local size of the parameter space is not very large the substitution standard nonsieve substitution sieve and substitution penalized mle s are asymptotically efficient in the fisher sense under certain stochastic equicontinuity conditions of the loglikelihood moreover when the convergence rate of the estimate is slow the degree of smoothness of the functional needs to compensate for the slowness of the rate in order to achieve efficiency when the size of the parameter space is very large the standard and penalized maximum likelihood procedures may be inefficient whereas the method of sieves may be able to overcome this difficulty this phenomenon is particularly manifested when the functional of interest is very smooth especially in the semiparametric case

o h 4 locally overconvergent semidiscrete scheme for the equation u t u xx f t x u abstract in the equation ut uxx f t x u the second derivative uxx is approximated by the finite difference operator lh which has o h2 local truncation error at the points h and 1 h next to the ends of the interval 0 1 and o h4 at the interior mesh points 2h 3h 1 2h it is proved that the obtained semidiscrete scheme is o h4 globally convergent the semidiscrete scheme is solved by the pure implicit finite difference scheme combined with the method of iteration and the gauss elimination method for tridiagonal matrices the nonstationary liouville s equation ut uxx e u has been solved by the proposed algorithm

estimation of nonlinear models with berkson measurement errors this paper is concerned with general nonlinear regression models where the predictor variables are subject to berkson type measurement errors the measurement errors are assumed to have a general parametric distribution which is not necessarily normal in addition the distribution of the random error in the regression equation is nonparametric a minimum distance estimator is proposed which is based on the first two conditional moments of the response variable given the observed predictor variables to overcome the possible computational difficulty of minimizing an objective function which involves multiple integrals a simulation based estimator is constructed consistency and asymptotic normality for both estimators are derived under fairly general regularity conditions

estimation of nonlinear models with measurement error this paper presents a solution to an important econometric problem namely the root n consistent estimation of nonlinear models with measurement errors in the explanatory variables when one repeated observation of each mismeasured regressor is available while a root n consistent estimator has been derived for polynomial specifications see hausman ichimura newey and powell 1991 such an estimator for general nonlinear specifications has so far not been available using the additional information provided by the repeated observation the suggested estimator separates the measurement error from the true value of the regressors thanks to a useful property of the fourier transform the fourier transform converts the integral equations that relate the distribution of the unobserved true variables to the observed variables measured with error into algebraic equations the solution to these equations yields enough information to identify arbitrary moments of the true unobserved variables the value of these moments can then be used to construct any estimator that can be written in terms of moments including traditional linear and nonlinear least squares estimators or general extremum estimators the proposed estimator is shown to admit a representation in terms of an influence function thus establishing its root n consistency and asymptotic normality monte carlo evidence and an application to engel curve estimation illustrate the usefulness of this new approach

a second order l stable time discretisation of the semiconductor device equations abstract the application of a second order and l stable predictor corrector method to the time discretisation of the semiconductor device equations is investigated the semidiscrete system is linearised at an equilibrium point and the resulting method is shown to be stable it is also shown that the method conserves the total current balance at each time level

adaptive control via a simple switching algorithm in this paper we present an adaptive stabilization control for systems with unknown constant parameters and stochastic disturbances which may be neither open loop stable nor minimum phase the ideas come from previous works j f zhang and h f chen adaptive stabilization under the weakest condition proc 31st control and design conference december 14 18 1992 pp 3620 3621 and h f chen continuous time stochastic adaptive control stabilizing the system and minimizing the quadratic loss function tech report institute of systems science academia sinica beijing 1992 but here we not only simplify the construction procedure of an adaptive control but also reduce the computational load significantly so that the adaptive control in this paper is more practical furthermore parameter estimation is carried out in only a finite time period and unlike previous work the parameter estimates are generated by ordinary differential equations rather than stochastic differential equations

adaptive continuous time linear quadratic gaussian control the adaptive linear quadratic gaussian control problem where the linear transformation of the state a and the linear transformation of the control b are unknown is solved assuming only that a b is controllable and a q sub 1 sup 1 2 is observable where q sub 1 determines the quadratic form for the state in the integrand of the cost functional a weighted least squares algorithm is modified by using a random regularization to ensure that the family of estimated models is uniformly controllable and observable a diminishing excitation is used with the adaptive control to ensure that the family of estimates is strongly consistent a lagged certainty equivalence control using this family of estimates is shown to be self optimizing for an ergodic quadratic cost functional

uplink power adjustment in wireless communication systems a stochastic control analysis this paper considers mobile to base station power control for lognormal fading channels in wireless communication systems within a centralized information stochastic optimal control framework under a bounded power rate of change constraint the stochastic control problem and its associated hamilton jacobi bellman hjb equation are analyzed by the viscosity solution method then the degenerate hjb equation is perturbed to admit a classical solution and a suboptimal control law is designed based on the perturbed hjb equation when a quadratic type cost is used without a bound constraint on the control the value function is a classical solution to the degenerate hjb equation and the feedback control is affine in the system power in addition in this case we develop approximate but highly scalable solutions to the hjb equation in terms of a local polynomial expansion of the exact solution when the channel parameters are not known a priori one can obtain on line estimates of the parameters and get adaptive versions of the control laws in numerical experiments with both of the above cost functions the following phenomenon is observed whenever the users have different initial conditions there is an initial convergence of the power levels to a common level and then subsequent approximately equal behavior which converges toward a stochastically varying optimum

linearization methods for reaction diffusion equations 1 d problems three types of linearized finite difference methods are presented the first type is based on a formulation which provides discrete solutions in both space and time it is shown that this type is a linearly implicit rosenbrock s or a w method depending on whether the full or an appropriate jacobian is employed the second one requires the solution of two point linear ordinary differential equations and provides either piecewise continuous or piecewise differentiable solutions in space and discrete in time the third type is based on the discretization of the spatial coordinate and provides a system of linear ordinary differential equations in time which can be integrated analytically therefore the third type of linearized finite difference methods provides continuous solutions in time and discrete in space and may be transformed into one of the first type by discretizing the time variable