probabilistic metrization of fuzzy uniformities abstract modifying the structure of fuzzy uniform spaces introduced by lowen 5 we propose a further concept of fuzzy uniformities so called fuzzy t

construction of aggregation operators with noble reinforcement this paper examines disjunctive aggregation operators used in various recommender systems a specific requirement in these systems is the property of noble reinforcement allowing a collection of high valued arguments to reinforce each other while avoiding reinforcement of low valued arguments we present a new construction of lipschitz continuous aggregation operators with noble reinforcement property and its refinements

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

owa trees and their role in security modeling using attack trees we introduce the idea of an owa node as an extension of the and or node and use them to generalize and or trees to owa trees we provide a semantics for an owa node specifically while an or node requires only one of the children to be satisfied and an and node requires all the children be satisfied the owa node allows us to model situations in which there is some probabilistic uncertainty in the number of children that need be satisfied we then use these owa nodes in the security related problem of constructing attack trees techniques for the evaluation of an owa attack tree for the overall probability of success and cost of an attack are provided a method is provided for describing different types of attackers

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

consistency property of extended least squares parameter estimation for stochastic diffusion equation the extended least squares parameter estimate for stochastic heat diffusion equations is considered the unknown parameter is a heat diffusion coefficient which is a function of a spatial variable almost sure convergence for the estimated parameter is proved a numerical example is demonstrated for supporting the theoretical results developed here

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

discrete time stochastic adaptive control this paper establishes global convergence of a stochastic adaptive control algorithm for discrete time linear systems it is shown that with probability one the algorithm will ensure the system inputs and outputs are sample mean square bounded and the conditional mean square output tracking error achieves its global minimum possible value for linear feedback control thus asymptotically the adaptive control algorithm achieves the same performance as could be achieved if the system parameters were known

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

on the issue of obtaining owa operator weights we first investigate the issue of obtaining the weights associated with the owa aggregation in the situation when we have observed data on the arguments and the aggregated value we next introduce a family of owa operators called exponential owa operators finally we look at a simple procedure for generating the weights given a required degree of orness

a convergent approximation of the continuous time optimal parameter estimator continuous time linear stochastic systems that are bilinear in the state and parameters are considered a specific approximation to the optimal nonlinear filter used as a recursive parameter estimator is derived by retaining third order moments and using a gaussian approximation for higher order moments with probability one the specific approximation is proved to converge to a minimum of the likelihood function the proof uses the ordinary differential equation technique and requires that the trajectories of the slow system be bounded on finite time intervals and that the fixed parameter fast system by asymptotically stable the fixed parameter fast system is proved to be asymptotically stable if the parameter update gain is small enough essentially the specific approximation is asympotically equivalent to the recursive prediction error method thus inheriting its asymptotic rate of convergence a numerical simulation for a simple example indicates that the specific approximation has better transient response than other commonly used convergent parameter estimators

adaptive stabilization of unstable and nonminimum phase stochastic systems abstract the system under consideration is the discrete time stochastic system a z yn zb z un wn driven by a martingale difference sequence wn where a z are polynomilas in backward shift operator z with unknown coefficients and both a z and b z may be unstable with the purpose of demonstrating theoretical possibility rather than designing a practically applicable control law this paper constructs an adaptive control that stabilizes the system and simultaneously guarantees strong consistency of the least squares estimates for unknown coefficients

refining aggregation operator based orderings in multifactorial evaluation part i continuous scales aggregation operators are often needed when building preference relations in multicriteria decision making problems most existing approaches have limitations due to incomparability between decisions or ties due to the use of some aggregation operations that produce a ranking the natural way of overcoming the lack of discrimination power is to refine the obtained ranking we bring an overview of methods that enable aggregation based rankings to be refined generalizing concepts like discrimin max leximin max and lorentz orderings that refine such aggregation operations like the minimum the maximum and the sum

representation similarity measures and aggregation methods using fuzzy sets for content based recommender systems representation of features of items and user feedback and reasoning about their relationships are major problems in recommender systems this is because item features and user feedback are subjective imprecise and vague the paper presents a fuzzy set theoretic method ftm for recommender systems that handles the non stochastic uncertainty induced from subjectivity vagueness and imprecision in the data and the domain knowledge and the task under consideration the research further advances the application of fuzzy modeling for content based recommender systems initially presented by ronald yager the paper defines a representation method similarity measures and aggregation methods as well as empirically evaluates the methods performance through simulation using a benchmark movie data ftm consist of a representation method for items features and user feedback using fuzzy sets and a content based algorithm based on various fuzzy set theoretic similarity measures the fuzzy set extensions of the jaccard index cosine proximity or correlation similarity measures and aggregation methods for computing recommendation confidence scores the maximum minimum or weighted sum fuzzy set theoretic aggregation methods compared to the baseline crisp set based method csm presented the empirical evaluation of the ftm using the movie data and simulation shows an improvement in precision without loss of recall moreover the paper provides a guideline for recommender systems designers that will help in choosing from a combination of one of the fuzzy set theoretic aggregation methods and similarity measures

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

a suboptimal deadlock control policy for designing non blocking supervisors in flexible manufacturing systems this work aims to resolve deadlock problems in flexible manufacturing systems a novel deadlock control policy in the frame of petri nets formalism is presented for a class of generalized petri nets namely g systems which can usually model assembly and disassembly operations of multiple resource acquisition based on the concept of elementary siphons all strict minimal siphons sms in an uncontrolled net system are first divided into elementary siphons and dependent ones after that a set of linear inequality constraints expressed by state vectors can be formalized for elementary siphons after being modified by utilizing the conjunctive disjunctive resources upstream neighborhood policy a set of generalized mutual exclusion constraints expressed by marking vectors can be obtained then the additional monitors are only designed for elementary siphons whose control depth variables can be obtained by solving a linear programming problem as a result the controllability of dependent ones can be ensured by properly supervising its elementary siphons finally all strict minimal siphons are max controlled and no insufficiently marked siphon is generated which indicates that the sequential resources are allocated reasonably to guarantee the absence of deadlock states the proposed method can usually lead to a near optimal non blocking supervisor with simple structure a g system example prone to deadlocks is used to illustrate the applicability and the effectiveness of the proposed method

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

toward a theory of fuzzy information granulation and its centrality in human reasoning and fuzzy logic there are three basic concepts that underlie human cognition granulation organization and causation informally granulation involves decomposition of whole into parts organization involves integration of parts into whole and causation involves association of causes with effects granulation of an object a leads to a collection of granules of a with a granule being a clump of points objects drawn together by indistinguishability similarity proximity or functionality for example the granules of a human head are the forehead nose cheeks ears eyes etc in general granulation is hierarchical in nature a familiar example is the granulation of time into years months days hours minutes etc modes of information granulation ig in which the granules are crisp c granular play important roles in a wide variety of methods approaches and techniques crisp ig however does not reflect the fact that in almost all of human reasoning and concept formation the granules are fuzzy f granular the granules of a human head for example are fuzzy in the sense that the boundaries between cheeks nose forehead ears etc are not sharply defined furthermore the attributes of fuzzy granules e g length of nose are fuzzy as are their values long short very long etc the fuzziness of granules their attributes and their values is characteristic of ways in which humans granulate and manipulate information

synthesis of petri net supervisors for fms via redundant constraint elimination the minimal number of control places problem mcpp which is formulated to obtain optimal and structurally minimal supervisors needs extensive computation the current methods to reduce the computational burden have mainly focused on revision of the original formulation of mcpp instead this paper presents methods to accelerate its solution by eliminating its redundant reachability constraints the optimization problem scale required for supervisor synthesis is thus drastically reduced first a sufficient and necessary condition for a reachability constraint to be redundant is established in the form of an integer linear program ilp based on a newly proposed concept called feasible region of supervisors then two kinds of redundancy elimination methods are proposed an ilp one and a non ilp one most of the redundant reachability constraints can be eliminated by our methods in a short time the computational time to solve mcpp is greatly reduced after the elimination especially for large scale systems the obtained supervisors are still optimal and structurally minimal finally numerical tests are conducted to show the efficiency and effectiveness of the proposed methods

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

practical aggregation operators for gradual trust and distrust trust and distrust are two increasingly important metrics in social networks reflecting users attitudes and relationships towards each other in this paper we study the indirect derivation of these metrics values for users that do not know each other but are connected through the network in particular we study bilattice based aggregation approaches and investigate how they can be improved by using ordered weighted averaging techniques or through the incorporation of knowledge defects experiments on a real world data set from couchsurfing org demonstrate that the best operators from a theoretical perspective are not always the most suitable ones in practice and that the sophisticated aggregation methods can outperform the more obvious bilattice based approaches

fuzzy logic methods in recommender systems here we consider methodologies for constructing recommender systems the approaches studied here differ from collaborative filtering they are based solely on the preferences of the single individual for whom we are providing the recommendation and make no use of the preferences of other collaborators we have called these reclusive methods another important feature distinguishing these reclusive methods from collaborative methods is that they require a representation of the objects considerable use is made of fuzzy set methods for the representation and subsequent construction of justifications and recommendation rules it is pointed out these reclusive methods rather than being competitive with collaborative methods are complementary

brief paper an ekf based nonlinear observer with a prescribed degree of stability the topic of this article is a nonlinear observer based on a slight modification of the extended kalman filter the purpose of this modification is twofold first the degree of stability can be assigned in advance and secondly this modification allows an effective treatment of the nonlinearities using the second method of lyapunov it is proved that the proposed observer is an exponential observer to examine the practical usefulness the proposed observer is applied to the highly nonlinear flux and angular velocity estimation problem for induction machines

on continuous time parameter identification by using observers this paper describes a continuous time parameter estimation technique this technique consists of designing two observers and comparing the observation bias due to parameter uncertainties the comparison of both biases is used to estimate the parameter values it comes out that under some conditions the system of equations to solve reduces to a division of output signals of the observers optimization of the design to reject disturbances is also discussed

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

stability of a class of hybrid linear stochastic systems the objective of the paper is to establish l q stability results for the state processes of a new class of slowly time varying hybrid continuous time linear stochastic systems the systems are hybrid in the sense that jumps both in the dynamics and the state may occur in both cases a jump amounts to resetting to an initial value jump times are defined by a more or less arbitrary point process most of the analysis is based on a carefully organized stochastic lyapunov function argument we motivate our study by a brief summary of some basic problems in continuous time recursive estimation of linear stochastic systems within a framework analogous to the one developed by benveniste et al

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

on structural minimality of optimal supervisors for flexible manufacturing systems this paper develops a place invariant based deadlock prevention method to obtain an optimal i e maximally permissive liveness enforcing petri net supervisor with a minimal supervisory structure that means the minimal number of control places maximal permissiveness can be achieved by designing place invariants that make all legal markings reachable while all first met bad markings unreachable an integer linear programming problem is formulated to compute all place invariants and its objective function minimizes the number of place invariants aiming to yield a minimal supervisory structure importantly we develop a technique to greatly improve the efficiency of the proposed method by reducing the number of constraints and variables in the integer linear programming problem under consideration a number of examples from the literature are used to illustrate the proposed approaches

optimal adaptive control of linear quadratic gaussian systems we consider the problem of adaptively controlling an unknown linear gaussian system with a standard quadratic cost criterion including a control cost by means of a counterexample it is shown that a commonly mentioned adapative control scheme can lead to severe problems to overcome this a new adaptive control law based on biasing the usual least squares parameter estimation criterion with a term favoring parameters associated with lower optimal costs is introduced a salient feature of this adaptive control scheme is its imperviousness to the closed loop identification problem properties such as closed loop system identification convergence of the adaptive control law to an optimal control law overall stability of the controlled system and optimality with respect to the long term average cost of the adaptive controller are proved

refinements of the maximin approach to decision making in a fuzzy environment abstract the most popular approach to decision making in the setting of fuzzy sets is the maximin ranking of solutions this method is natural when interpreting the fuzzy sets as flexible constraints that cannot compensate with one another however the obtained ranking of solutions is very coarse two kinds of refinements to this ordering are introduced a partial ordering according to the least satisfied discriminating constraint and a lexicographical ranking the latter refines the former and combines utilitarist and egalitarist points of view on the aggregation of feasibility degrees these orderings are characterized in several ways and their representation by means of two place numerical functions is studied dual refinements of the maximax ranking are provided

an optimization approach to petri net monitor design this note addresses the problem of enforcing generalized mutual exclusion constraints on a petri net plant first we replace the classical partition of the event set into controllable and uncontrollable events from supervisory control theory by associating a control and observation cost to each event this leads naturally to formulate the supervisory control problem as an optimal control problem monitor places which enforce the constraint are devised as a solution of an integer linear programming problem whose objective function is expressed in terms of the introduced costs second we consider timed models for which the monitor choice may lead to performance optimization if the plant net belongs to the class of mono t semiflow nets we present an integer linear fractional programming approach to synthesize the optimal monitor so as to minimize the cycle time lower bound of the closed loop net for strongly connected marked graphs the cycle time of the closed loop net can be minimized

generalizing leximin to t norms and t conorms the lexit and lexis orderings the leximin procedure introduced by dubois and prade in 1996 provides a methodology for adjudicating ties in aggregated values when using the min t norm here with the introduction of the lexit procedure we generalize these ideas to provide a tool for adjudicating ties when aggregating data using any t norm we then provide the lexis procedure for adjudicating ties when aggregating arguments using t conorms

reduction of constraints for controller synthesis based on safe petri nets in this paper we present an efficient method based on safe petri nets to construct a controller a set of linear constraints allows forbidding the reachability of specific states the number of these so called forbidden states and consequently the number of constraints are large and lead to a large number of control places a systematic method to reduce the size and the number of constraints for safe petri nets is offered by using a method based on petri net invariants maximal permissive controllers are determined

a collaborative recommender system based on probabilistic inference from fuzzy observations the problem of building recommender systems has attracted considerable attention in recent years the objective of this paper is to automatically suggest and rank a list of new items to a user based on the past voting patterns of other users with similar tastes the proposed model can be considered as a soft computing based collaborative recommender system the combination of bayesian networks which enables an intuitive representation of the mechanisms that govern the relationships between the users and the fuzzy set theory enabling us to represent ambiguity or vagueness in the description of the ratings improves the accuracy of the system

on the use of aggregation operations in information fusion processes this position paper discusses the role of the existing body of fuzzy set aggregation operations in various kinds of problems where the process of fusion of items coming from several sources is central several kinds of membership functions can be useful according to the nature of the information to be merged numerical vs ordinal inputs preferences vs uncertain data observations vs constraints in each case some aggregation operations look more plausible or feasible than others the aim of this discussion is to suggest directions for putting at work the results of recent mathematical investigations in the structure of aggregation operations

brief paper suboptimal supervisory control of petri nets in presence of uncontrollable transitions via monitor places this paper deals with the problem of enforcing generalized mutual exclusion constraints gmec on place transition nets with uncontrollable transitions an efficient control synthesis technique which has been proposed in the literature enforces gmec constraints by introducing monitor places to create suitable place invariants the method has been shown to be maximally permissive and to give a unique control structure in the case that the set of legal markings is controllable this paper investigates on and formally shows that the class of controllers obtained by this technique may not have a supremal element for uncontrollable specifications moreover it is shown that the family of monitor places enforcing an uncontrollable specification can be parameterized with respect to the solution of a linear system of equation an algorithm to obtain such parameterization is presented here

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

reduced order controllers for continuous and discrete time singular h control problems based on lmi this paper describes a method for constructing a petri net controller for a discrete event system modeled by a petri net the controller consists only of places and arcs and is computed based on the concept of petri net place invariants the size of the controller is proportional to the number of constraints that must be satisfied this method is computationally efficient and can accommodate constraints written as boolean logic formulas in the conjunctive normal form or algebraic inequalities that contain elements of the marking and or the firing vectors

noble reinforcement in disjunctive aggregation operators we discuss the role of disjunctive aggregation operators in prototype based reasoning systems such as recommender systems we indicate a need for these aggregation operators to have the additional property of noble reinforcement allowing a collection of high valued arguments to reinforce each other to give complete satisfaction while avoiding the situation in which a collection of low valued arguments act in the same way we describe the difficulty that most t conorms have in realizing this feature of noble reinforcement we show that by relaxing the requirement of associativity we can provide disjunctive type aggregation operators that manifest noble reinforcement

leximin optimality and fuzzy set theoretic operations abstract the leximin ranking of vectors of values taken from a totally ordered set is sometimes encountered in fields like operational research social choice or numerical analysis but has seldom been studied in connexion with fuzzy optimization in this paper we prove that a leximin optimal solution to a vector ranking problem on the unit hypercube can be obtained as the limit of optimal solutions to a problem of fuzzy multiple criteria optimization where fuzzy sets are aggregated either using a triangular norm or a generalized mean or an ordered weighted average owa operation

brief paper design of a maximally permissive liveness enforcing supervisor with a compressed supervisory structure for flexible manufacturing systems in this paper a deadlock prevention policy for flexible manufacturing systems fms is proposed which can obtain a maximally permissive liveness enforcing petri net supervisor while the number of control places is compressed by using a vector covering approach the sets of legal markings and first met bad markings fbm are reduced to two small ones i e the minimal covering set of legal markings and the minimal covered set of fbm a maximally permissive control purpose can be achieved by designing control places such that all markings in the minimal covered set of fbm are forbidden and no marking in the minimal covering set of legal markings is forbidden an integer linear programming problem is designed to minimize the number of control places under an assumption that a control place is associated with a p semiflow the resulting net has the minimal number of control places on the premise that the assumption holds and possesses all permissive states of a plant the only problem of the proposed method is its computational complexity that makes it inapplicable to large scale petri net models an fms example from the literature is presented to illustrate the proposed method

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