semi global practical asymptotic stability and averaging abstract we prove a generalized liapunov theorem which guarantees practical asymptotic stability based on this theorem we show that if the averaged system x f av x corresponding to x f x t is globally asymptotically stable then starting from an arbitrarily large set of initial conditions the trajectories of x f x t e converge uniformly to an arbitrarily small residual set around the origin when e 0 is taken sufficiently small in other words the origin is semi globally practically asymptotically stable

almost sure convergence of extremum seeking algorithm using stochastic perturbation abstract this paper presents an extremum seeking es algorithm where the perturbation signal is a martingale difference sequence m d s with a vanishing variance the measurement noise at the plant output is modeled by a superposition of deterministic component and a non stationary colored noise signal the optimizing set point of the uncertain reference to output equilibrium map is estimated by a stochastic approximation sa type algorithm the algorithm has a vanishing gain sequence dependent on the set point estimates by utilizing powerful tools of the martingale convergence theory it is proved that with probability one the set point estimates converge to the optimizing equilibrium point in spite of the presence of a measurement noise this result is derived without requiring boundedness or any prior condition on the set point estimates

quadratic functional equations in a set of lebesgue measure zero abstract let r be the set of real numbers y a banach space and f r y we prove the hyers ulam stability theorem for the quadratic functional inequality f x y f x y 2 f x 2 f y for all x y where r 2 is of lebesgue measure 0 using the same method we dealt with the stability of two more functional equations in a set of lebesgue measure 0

stochastic processes originating in deterministic microscopic dynamics we investigate the probability distribution of the scaled trajectory of a test particle moving in an equilibrium fluid according to the laws of classical mechanics i e ifq t is the displacement of the test particle we letqa t q at a and consider the distribution of the trajectory qa t in the limit a the randomness of the motion is due entirely to the randomness of the initial state of the fluid test particle or both and the process is generally non markovian nevertheless it can be proven in some cases and we expect it to be true in many more that qa t looks like brownian motion in the limit a some results for simple model systems are presented

brief stability of extremum seeking feedback for general nonlinear dynamic systems while the mainstream methods of adaptive control both linear and nonlinear deal only with regulation to known set points or reference trajectories in many applications the set point should be selected to achieve a maximum of an uncertain reference to output equilibrium map the techniques of the so called extremum control or self optimizing control developed for this problem in the 1950 1960s have long gone out of fashion in the theoretical control literature because of the difficulties that arise in a rigorous analytical treatment in this paper we provide the first proof of stability of an extremum seeking feedback scheme by employing the tools of averaging and singular perturbation analysis our scheme is much more general that the existing extremum control results which represent the plant as a static nonlinear map possibly cascaded with a linear dynamic block we allow the plant to be a general nonlinear dynamic system possibly non affine in control and open loop unstable whose reference to output equilibrium map has a maximum and whose equilibria are locally exponentially stabilizable

newton based stochastic extremum seeking in this paper we introduce a newton based approach to stochastic extremum seeking and prove local stability of newton based stochastic extremum seeking algorithm in the sense of both almost sure convergence and convergence in probability the convergence of the newton algorithm is proved to be independent of the hessian matrix and can be arbitrarily assigned which is an advantage over the standard gradient based stochastic extremum seeking simulation shows the effectiveness and advantage of the proposed algorithm over gradient based stochastic extremum seeking

maximizing map sensitivity and higher derivatives via extremum seeking we present a generalization to the scalar newton based extremum seeking algorithm which through perturbation induced measurements of an unknown steady state input to output map maximizes the map s higher derivatives as with other extremum seeking es problems the scheme relies on derivative estimators and learning dynamics operating in different time scales we provide analysis for the typical deterministic sinusoidal perturbation of the optimal parameter estimate also we alternatively include a stochastic method where the map is perturbed via the sinusoid of brownian motion about the boundary of a circle by properly demodulating the map output corresponding to the manner in which it is perturbed the es algorithm maximizes the n rm th derivative only through measurements of the map the newton based es approach removes the dependence of the convergence rate on the unknown hessian of the higher derivative an effort to improve performance over standard gradient based es our design stems from the existing multivariable newton based es algorithm where a differential riccati equation estimates the inverse hessian of the function to be maximized algebraically computing a direct estimate of the inverse hessian is susceptible to singularity whereas employing the riccati filter removes that potential we prove local stability of the algorithm for general nonlinear equilibrium profiles of dynamic maps and compare the newton based method against the gradient based method we also extend this abstraction to multiplayer noncooperative games but limit our attention to a two player game for notational simplicity and length considerations

addendum to on the stability of functional equations on square symmetric groupoid abstract let x be a square symmetric groupoid and y d a complete metric divisible square symmetric groupoid in this paper we obtain the hyers ulam stability problem of functional inequality d g x y g x g y e x y for approximate mapping g x y of functional equation f x y f x f y differing by e x 2 0 in particular we have investigated the case of f x f y h f x 1 t f y 1 t on some set y

general regular variation popa groups and quantifier weakening abstract we introduce general regular variation a theory of regular variation containing the existing karamata bojanic karamata de haan and beurling theories as special cases the unifying theme is the popa groups of our title viewed as locally compact abelian ordered topological groups together with their haar measure and fourier theory the power of this unified approach is shown by the simplification it brings to the whole area of quantifier weakening so important in this field

decomposition of mappings approximately inner product preserving abstract for hilbert spaces e and f we consider the class of mappings f e f preserving approximately inner product in the following sense f x f y x y x y x y e for suitable function we show that the orthogonal projection of f onto some closed linear subspace h of f is a linear isometry while the projection onto h is bounded by x x several consequences of such a decomposition are given in particular stability and superstability of inner product preserving mappings are considered

the fractal properties of generalised random walks in one dimension the properties of a simple random walk and random walks with persistence of velocity in one dimension are reported the finite fractal properties of the walks are obtained with exact expressions for indefinitely long walks remarkably simple expressions are found for the length l epsilon at scale epsilon computer simulations of these walks are reported for finite trajectories which help to explain the properties of atomic trajectories in realistic liquids determined by molecular dynamics simulation some new molecular dynamics simulations are reported for a typical liquid its coexisting vapour and the crystal all at the same temperature the general form of the fractal properties of the three states of matter are well represented by the present one dimensional model indeed even the quantitative nature is quite well represented which suggests that results for random walks in three dimensions will be substantially the same

learning generalized nash equilibria in a class of convex games we consider multiagent decision making where each agent optimizes its convex cost function subject to individual and coupling constraints the constraint sets are compact convex subsets of a euclidean space to learn nash equilibria we propose a novel distributed payoff based algorithm where each agent uses information only about its cost value and the constraint value with its associated dual multiplier we prove convergence of this algorithm to a nash equilibrium under the assumption that the game admits a strictly convex potential function in the absence of coupling constraints we prove convergence to nash equilibria under significantly weaker assumptions not requiring a potential function namely strict monotonicity of the game mapping is sufficient for convergence we also derive the convergence rate of the algorithm for strongly monotone game maps

hyers ulam stability of functional equations with a square symmetric operation abstract the stability of the functional equation f x y h f x f y x y s is investigated where h is a homogeneous function and is a square symmetric operation on the set s the results presented include and generalize the classical theorem of hyers obtained in 1941 on the stability of the cauchy functional equation

stability of multi jensen mappings in non archimedean normed spaces in this paper we study the stability of the multi jensen mappings in non archimedean normed spaces the results improve and extend some recent results

shahshahani gradient like extremum seeking in this paper we present novel extremum seeking controllers designed to achieve online constrained optimization of uncertain input to output maps of a class of nonlinear dynamical systems the algorithms inspired by a class of evolutionary dynamics that emerge in the context of population games generate on line shahshahani gradient like systems able to achieve extremum seeking under simplex like constraints on the positive orthant the method attains semi global practical convergence to the optimal point even when the initial conditions are not in the feasible set and it can be naturally adapted to address distributed extremum seeking problems in multi agent systems where an all to all communication structure is not available potential applications include problems on distributed dynamic resource allocation congestion and flow control in networked systems as well as portfolio optimization in financial markets via simulations we illustrate our results under different scenarios

distributed robust adaptive equilibrium computation for generalized convex games this paper considers a class of generalized convex games where each player is associated with a convex objective function a convex inequality constraint and a convex constraint set the players aim to compute a nash equilibrium through communicating with neighboring players the particular challenge we consider is that the component functions are unknown a priori to associated players we study two distributed computation algorithms and analyze their convergence properties in the presence of data transmission delays and dynamic changes of network topologies the algorithm performance is verified through demand response on the ieee 30 bus test system our technical tools integrate convex analysis variational inequalities and simultaneous perturbation stochastic approximation

lie bracket approximation of extremum seeking systems extremum seeking feedback is a powerful method to steer a dynamical system to an extremum of a partially or completely unknown map it often requires advanced system theoretic tools to understand the qualitative behavior of extremum seeking systems in this paper a novel interpretation of extremum seeking is introduced we show that the trajectories of an extremum seeking system can be approximated by the trajectories of a system which involves certain lie brackets of the vector fields of the extremum seeking system it turns out that the lie bracket system directly reveals the optimizing behavior of the extremum seeking system furthermore we establish a theoretical foundation and prove that uniform asymptotic stability of the lie bracket system implies practical uniform asymptotic stability of the corresponding extremum seeking system we use the established results in order to prove local and semi global practical uniform asymptotic stability of the extrema of a certain map for multi agent extremum seeking systems

continuous on rays solutions of an equation of the golab schinzel type abstract let x be a real linear space we characterize continuous on rays solutions f g x r of the equation f x g x y f x f y our result refers to papers of j chudziak 2006 14 and j brzd k 2003 11

multivariable newton based extremum seeking we present a newton based extremum seeking algorithm for the multivariable case the design extends the recent newton based extremum seeking algorithms for the scalar case and introduces a dynamic estimator of the inverse of the hessian matrix that removes the difficulty with the possible singularity of a possible direct estimate of the hessian matrix the estimator of the inverse of the hessian has the form of a differential riccati equation we prove local stability of the new algorithm for general nonlinear dynamic systems using averaging and singular perturbations in comparison with the standard gradient based multivariable extremum seeking the proposed algorithm removes the dependence of the convergence rate on the unknown hessian matrix and makes the convergence rate of both the parameter estimates and of the estimates of the hessian inverse user assignable in particular the new algorithm allows all the parameters to converge with the same speed yielding straight trajectories to the extremum even with maps that have highly elongated level sets in contrast to curved steepest descent trajectories of the gradient algorithm simulation results show the advantage of the proposed approach over gradient based extremum seeking by assigning equal desired convergence rates to all the parameters using newton s approach

on the quotient stability of a family of functional equations abstract we show some stability results for a family of functional equations that contains the exponential and the golab schinzel functional equations

asymptotic stability of perturbation based extremum seeking control for nonlinear plants we introduce a perturbation based extremum seeking controller for general nonlinear dynamical plants with an arbitrary number of tunable plant parameters the controller ensures asymptotic convergence of the plant parameters to their performance optimizing values for any initial plant condition under the assumptions in this work the key to this result is that the amplitude and the frequencies of the perturbations as well as other tuning parameters of the controller are time varying remarkably the time varying tuning parameters can be chosen such that asymptotic convergence is achieved for all plants that satisfy the assumptions thereby guaranteeing stability of the resulting closed loop system of plant and controller regardless of tuning

stochastic averaging in continuous time and its applications to extremum seeking we investigate stochastic averaging theory in continuous time for locally lipschitz systems and the applications of this theory to stability analysis of stochastic extremum seeking algorithms first we establish a general stochastic averaging principle and some related stability theorems for a class of continuous time nonlinear systems with stochastic perturbations and remove or weaken several significant restrictions present in existing results global lipschitzness of the nonlinear vector field equilibrium preservation under the stochastic perturbation global exponential stability of the average system and compactness of the state space of the perturbation process then we propose a continuous time extremum seeking algorithm with stochastic excitation signals instead of deterministic periodic signals we analyze the stability of stochastic extremum seeking for static maps and for general nonlinear dynamic systems

batch to batch finite horizon lq control for unknown discrete time linear systems via stochastic extremum seeking we employ our recent discrete time stochastic averaging theorems and stochastic extremum seeking to iteratively batch to batch optimize open loop control sequences for unknown but reachable discrete time linear systems with a scalar input and without known system dimension for a cost that is quadratic in the measurable output and the input first for a multivariable gradient based stochastic extremum seeking algorithm we prove local exponential convergence to the optimal open loop control sequence second to remove the convergence rate s dependence on the hessian matrix of the cost function which is unknown since the system s model the system matrices a b c is unknown we develop a multivariable discrete time newton based stochastic extremum seeking method design the newton based algorithm for the iteration of the input sequence and prove local exponential convergence to the optimal input sequence finally two simulation examples are given to illustrate the effectiveness of the two methods

extremum seeking under stochastic noise and applications to mobile sensors in this paper the extremum seeking algorithm with sinusoidal perturbations has been extended and modified in two ways a the output of the system is corrupted with measurement noise b the amplitudes of the perturbation signals as well as the gain of the integrator block are time varying and tend to zero at a pre specified rate convergence to the extremal point with probability one has been proved also as a consequence of being able to cope with a stochastic environment it has been shown how the proposed algorithm can be applied to mobile sensors as a tool for achieving the optimal observation positions the proposed algorithm has been illustrated through several simulations

on the hyers ulam stability of the linear differential equation abstract we obtain some results on generalized hyers ulam stability of the linear differential equation in a banach space as a consequence we improve some known estimates of the difference between the perturbed and the exact solutions

distributed algorithms for the computation of noncooperative equilibria abstract in this paper a general class of nonquadratic convex nash games is studied from the points of view of existence stability and iterative computation of noncooperative equilibria conditions for contraction of general nonlinear operators are obtained which are then used in the stability study of such games these lead to existence and uniqueness conditions for stable nash equilibrium solutions under both global and local analysis also convergence of an algorithm which employs inaccurate search techniques is verified it is shown in the context of a fish war example that the algorithm given is in some aspects superior to various algorithms found in the literature and is furthermore more meaningful for real world implementation

extremum seeking with stochastic perturbations extremum seeking es using deterministic periodic perturbations has been an effective method for non model based real time optimization when only limited plant knowledge is available however periodicity can naturally lead to predictability which is undesirable in some tracking applications and unrepresentative of biological optimization processes such as bacterial chemotaxis with this in mind it is useful to investigate employing stochastic perturbations in the context of a typical es architecture and to compare the approach with existing stochastic optimization techniques in this work we show that convergence towards the extremum of a static map can be guaranteed with a stochastic es algorithm and quantify the behavior of a system with gaussian distributed perturbations at the extremum in terms of the es constants and map parameters we then examine the closed loop system when actuator dynamics are included as the separation of time scales between the perturbation signal and plant dynamics recommended in periodic es schemes cannot be guaranteed with stochastic perturbations consequently we investigate how actuator dynamics influence the allowable range of es parameters and necessitate changes in the closed loop structure finally simulation results are presented to demonstrate convergence and to validate predicted behavior about the extremum for the sake of analogy with the classical methods of stochastic approximation stochastic es in this technical note is pursued in discrete time

distributed asynchronous computation of fixed points we present an algorithmic model for distributed computation of fixed points whereby several processors participate simultaneously in the calculations while exchanging information via communication links we place essentially no assumptions on the ordering of computation and communication between processors thereby allowing for completely uncoordinated execution we provide a general convergence theorem for algorithms of this type and demonstrate its applicability to several classes of problems including the calculation of fixed points of contraction and monotone mappings arising in linear and nonlinear systems of equations optimization problems shortest path problems and dynamic programming

distributed convergence to nash equilibria in two network zero sum games this paper considers a class of strategic scenarios in which two networks of agents have opposing objectives with regard to the optimization of a common objective function in the resulting zero sum game individual agents collaborate with neighbors in their respective network and have only partial knowledge of the state of the agents in the other network for the case when the interaction topology of each network is undirected we synthesize a distributed saddle point strategy and establish its convergence to the nash equilibrium for the class of strictly concave convex and locally lipschitz objective functions we also show that this dynamics does not converge in general if the topologies are directed this justifies the introduction in the directed case of a generalization of this distributed dynamics which we show converges to the nash equilibrium for the class of strictly concave convex differentiable functions with globally lipschitz gradients the technical approach combines tools from algebraic graph theory nonsmooth analysis set valued dynamical systems and game theory

stability problem for the golab schinzel type functional equations let x be a vector space over a field k of real or complex numbers n n and k 0 we study the stability problem for the golab schinzel type functional equations f x f x ny f x f y in the class of functions f x k such that the set x x f x 0 has an algebraically interior point

the mimo iterative waterfilling algorithm this paper considers the noncooperative maximization of mutual information in the vector gaussian interference channel in a fully distributed fashion via game theory this problem has been widely studied in a number of works during the past decade for frequency selective channels and recently for the more general multiple input multiple output mimo case for which the state of the art results are valid only for nonsingular square channel matrices surprisingly these results do not hold true when the channel matrices are rectangular and or rank deficient matrices the goal of this paper is to provide a complete characterization of the mimo game for arbitrary channel matrices in terms of conditions guaranteeing both the uniqueness of the nash equilibrium and the convergence of asynchronous distributed iterative waterfilling algorithms our analysis hinges on new technical intermediate results such as a new expression for the mimo waterfilling projection valid also for singular matrices a mean value theorem for complex matrix valued functions and a general contraction theorem for the multiuser mimo watefilling mapping valid for arbitrary channel matrices the quite surprising result is that uniqueness convergence conditions in the case of tall possibly singular channel matrices are more restrictive than those required in the case of full rank fat channel matrices we also propose a modified game and algorithm with milder conditions for the uniqueness of the equilibrium and convergence and virtually the same performance in terms of nash equilibria of the original game

distributed seeking of time varying nash equilibrium for non cooperative games in this note we address a nash equilibrium seeking problem for non cooperative games in contrast to previous works on nash equilibrium seeking the nash equilibrium under consideration can be time varying a non model based seeking scheme is proposed to achieve time varying nash equilibrium seeking where each player updates its strategy by employing an extremum seeking method the proposed nash seeking scheme consists of a gradient estimation algorithm and a gradient search algorithm which can be designed in a modular fashion for symmetric quadratic games the proposed nash equilibrium seeking method enables the estimated strategy to globally asymptotically converge to the nash equilibrium for general quadratic games that are not necessarily symmetric the estimated strategy converges to a neighborhood of the nash equilibrium for more general non quadratic games that may admit multiple equilibria local convergence to the nash equilibrium is proven

on the characterization of local nash equilibria in continuous games we present a unified framework for characterizing local nash equilibria in continuous games on either infinite dimensional or finite dimensional non convex strategy spaces we provide intrinsic necessary and sufficient first and second order conditions ensuring strategies constitute local nash equilibria we term points satisfying the sufficient conditions differential nash equilibria further we provide a sufficient condition non degeneracy guaranteeing differential nash equilibria are isolated and show that such equilibria are structurally stable we present tutorial examples to illustrate our results and highlight degeneracies that can arise in continuous games

a distributed simultaneous perturbation approach for large scale dynamic optimization problems we consider distributed optimization problems of large scale dynamic systems where the global cost is the sum of all individual costs of subsystems which are only known to the associated agent to solve this problem a distributed simultaneous perturbation approach d spa is proposed based on simultaneous perturbation techniques as well as consensus theory the proposed method is model free so long as all individual costs can be measured and requires little knowledge on the coupling structure of the problem to be optimized the convergence of the proposed scheme is proved using singular perturbation and averaging theory in particular with proper choice of the parameters under design we show that the proposed scheme is able to converge to the neighborhood of the pareto optimum of the problem so long as the energy of perturbation signals is sufficiently small moreover the proposed approach is applied to a simulated offshore wind farm for energy maximization and a comprehensive comparison with the existing state of the art technique is made to illustrate its effectiveness

continuous time gradient like descent algorithm for constrained convex unknown functions penalty method application abstract this paper suggests a novel continuous time gradient descent algorithm of a constrained convex unknown function with a stochastic noise in the observed data the penalty function approach is employed to introduce the restrictions of the system and to provide a successful optimization process the solution is restricted to a static scheme dealing with the class of strongly convex functions subject to a set of constraints to estimate the stochastic gradient we employ a modified version of the synchronous detection method all the parameters of the proposed approach are decreasing in time to both compensate noise effect in the observations and to provide the mean square convergence of the suggested approach we present two different numerical examples to validate the contributions of the paper

the space of psi gamma additive mappings on semigroups in this paper we introduce the concept of pseudoadditive mappings from a semigroup into a banach space and we provide a generalized solution of ulam s problem for approximately additive mappings

on solutions of a common generalization of the golab schinzel equation and of the addition formulae abstract under some additional assumptions we determine solutions of the equation f x m f x y f x f y where f r r is lebesgue measurable or baire measurable m r r and r 2 r all three functions are unknown

extremum seeking for limit cycle minimization in many physical problems equilibrium stabilization is not possible and the controlled system is in a limit cycle if the size of the limit cycle depends on some of the control parameters then a reasonable objective would be to tune this parameter to minimize the size of the limit cycle we propose a method for achieving this this method is an extension of our earlier result kristic and wang 2000 on extremum seeking for equilibria we illustrate the method with a van der pol oscillator example and present analysis for it using averaging and singular perturbations

on non local stability properties of extremum seeking control in this paper we consider several extremum seeking schemes and show under appropriate conditions that these schemes achieve extremum seeking from an arbitrarily large domain of initial conditions if the parameters in the controller are appropriately adjusted this non local stability result is proved by showing semi global practical stability of the closed loop system with respect to the design parameters we show that reducing the size of the parameters typically slows down the convergence rate of the extremum seeking controllers and enlarges the domain of the attraction our results provide guidelines on how to tune the controller parameters in order to achieve extremum seeking simulation examples illustrate our results

measure zero stability problem of a new quadratic functional equation abstract let r be the set of real numbers y a banach space and f r y we prove the ulam hyers stability theorem for the new quadratic functional equation j 2 k f x 1 x j f x 1 x j 2 k 1 f x 1 2 j 2 k f x j for all x 1 x k where r k is of lebesgue measure 0 using the result we obtain an asymptotic behavior of the equation

iterative techniques for the nash solution in quadratic games with unknown parameters we study adaptive schemes for repeated quadratic nash games in a deterministic and a stochastic framework the convergence of the schemes is demonstrated under certain conditions

an equilibrium theory for multiperson decision making with multiple probabilistic models this paper develops an equilibrium theory for two person two criteria stochastic decision problems with static information patterns wherein the decision makers dm s have different probabilistic models of the underlying process the objective functionals are quadratic and the decision spaces are general inner product spaces under two different modes of decision making viz symmetric and asymmetric sufficient conditions are obtained for the existence and uniqueness of equilibrium solutions stable in the former case and in each case a uniformly convergent iterative scheme is developed whereby the equilibrium policies of the dm s can be obtained by evaluating a number of conditional expectations when the probability measures are gaussian the equilibrium solution is linear under the symmetric mode of decision making whereas it is generically nonlinear in the asymmetric case with the linear structure prevailing only in some special cases which are delineated in the paper

extremum seeking of dynamical systems via gradient descent and stochastic approximation methods this paper examines the use of gradient based methods for extremum seeking control of possibly infinite dimensional dynamic nonlinear systems with general attractors within a periodic sampled data framework first discrete time gradient descent method is considered and semi global practical asymptotic stability with respect to an ultimate bound is shown next under the more complicated setting where the sampled measurements of the plant s output are corrupted by an additive noise three basic stochastic approximation methods are analysed namely finite difference random directions and simultaneous perturbation semi global convergence to an optimum with probability one is established a tuning parameter within the sampled data framework is the period of the synchronised sampler and hold device which is also the waiting time during which the system dynamics settle to within a controllable neighbourhood of the steady state input output behaviour

fractal properties of model trajectories with exponential velocity autocorrelation functions abstract the length l at scale has been determined for four different models for atomic motion in a fluid three distinct types of random walk with constant speed and the langevin model this has been done for trajectories in one two three and sometimes four dimensions all these motions have exactly exponential velocity autocorrelation functions however the functional form of l is different for the different models contrary to a recent assertion s toxvaed phys lett a 114 1986 159 a theory for l is lacking except for the limits of small and very large our results can be used as experimental results for the testing of such theories which are quite different from the conventional analysis of molecular motion in fluids a comparison is made of these model results for l with those for simulated trajectories in realistic models of liquids and for the actual trajectories of suspended particles in solution

stochastic adaptive stabilization via extremum seeking in case of unknown control directions this technical note presents a novel extremum seeking es based algorithm for global stabilization of unstable first order linear discrete time ldt stochastic systems with unknown control directions the probing signal is a martingale difference sequence with a vanishing variance the controller parameter estimate converges almost surely a s to the optimizing value the sample mean square value of the output converges a s to the minimal variance

hyers ulam stability of the linear differential operator with nonconstant coefficients abstract we obtain some stability results for the linear differential operator of order one in banach spaces as a consequence we derive a result on hyers ulam stability for the linear differential operator of higher order with nonconstant coefficients

dither re use in nash equilibrium seeking nash equilibrium seeking nes scheme consists of a number of decentralized extremum seeking es agents each controlling an input such that an associated selfish cost is regulated to its steady state nash equilibrium a non local stability result for the nes scheme is provided which allows two agents to use the same dither signal if the effect of each agent on the other s steady state cost function is sufficiently weak its application to plants with quadratic cost functions is presented as an example it is then demonstrated in simulation that by reducing the number of distinct dither signals the proposed scheme still has acceptable convergence properties while design effort is reduced

on the stability of a pexiderized gol ab schinzel equation in this paper we establish the stability of the pexiderized email protected b schinzel functional equation f x yf x g x h y under the condition that lim t 0h tx exists and is non zero

minimum seeking for clfs universal semiglobally stabilizing feedback under unknown control directions employing extremum seeking es for seeking minima of control lyapunov function clf candidates we develop 1 the first systematic design of es controllers for unstable plants 2 a simple non model based universal feedback law that emulates in an average sense the l g v controllers for stabilization with inverse optimality and 3 a new strategy for stabilization of systems with unknown control directions as an alternative to nussbaum gain controllers that lack exponential stability lack transient performance guarantees and lack robustness to changes in the control direction the stability analysis that underlies our designs is inspired by an analysis approach synthesized in a recent work by durr stankovic and johansson which combines a lie bracket averaging result of gurvits and li with a semiglobal practical stability result under small parametric perturbations by moreau and aeyels

general stability of functional equations of linear type making use of a dynamical systems notion called shadowing we prove a stability result for linear functional equations in metric groups as a corollary we obtain stability of the quadratic functional equation in the case when the target space is a metric group satisfying some local 2 divisibility condition

exponential type functional equation and its hyers ulam stability abstract we give the solution of the functional equation f x y f x f y x y under some conditions also we show its hyers ulam stability

fast extremum seeking on hammerstein plants partial plant knowledge may be used to develop model based extremum seekers however existing results rely on a type of time scale separation which leads to slow optimization relative to the plant dynamics in this work a fast model based extremum seeking scheme is proposed for a hammerstein plant and semi global stability results are provided the structure of a hammerstein plant is used to advantage in designing filters that enable the extremum seeker to act on a faster time scale than the plant dynamics this leads to fast convergence while maintaining semi global stability

games with incomplete information played by bayesian players i iii this article originally appeared in management science november 1967 volume 14 number 3 pp 159 182 published by the institute of management sciences the paper develops a new theory for the analysis of games with incomplete information where the players are uncertain about some important parameters of the game situation such as the payoff functions the strategies available to various players the information other players have about the game etc however each player has a subjective probability distribution over the alternative possibilities in most of the paper it is assumed that these probability distributions entertained by the different players are mutually consistent in the sense that they can be regarded as conditional probability distributions derived from a certain basic probability distribution over the parameters unknown to the various players but later the theory is extended also to cases where the different players subjective probability distributions fail to satisfy this consistency assumption in cases where the consistency assumption holds the original game can be replaced by a game where nature first conducts a lottery in accordance with the basic probability distribution and the outcome of this lottery will decide which particular subgame will be played i e what the actual values of the relevant parameters will be in the game yet each player will receive only partial information about the outcome of the lottery and about the values of these parameters however every player will know the basic probability distribution governing the lottery thus technically the resulting game will be a game with complete information it is called the bayes equivalent of the original game part i of the paper describes the basic model and discusses various intuitive interpretations for the latter part ii shows that the nash equilibrium points of the bayes equivalent game yield bayesian equilibrium points for the original game finally part iii considers the main properties of the basic probability distribution

on approximate solutions of the linear functional equation of higher order abstract we show that under some assumptions every approximate solution of the linear functional equation of higher order in single variable generates a solution of the equation that is close to it we also give a description of a procedure that yields such a solution estimate the distance between those approximate and exact solutions to the equation and discuss the problem of uniqueness moreover as a consequence we obtain some results concerning the hyers ulam stability of the equation

christensen measurable solutions of some functional equation abstract let x be a separable f space over the field k of reals or complex numbers we characterize solutions of the equation f x m f x y f x f y in the class of functions m k k and christensen measurable functions f x k

comments on the core of the direct method for proving hyers ulam stability of functional equations in this short paper the core of the direct method for proving stability of functional equations is described in a clear way and in a quite general form

distributed nash equilibrium seeking this paper presents an asynchronous gossip based algorithm for finding a nash equilibrium ne of a game in a distributed multi player network the algorithm is designed in such a way that players make decisions based on estimates of the other players actions obtained from local neighbors using a set of standard assumptions on the cost functions and communication graph the paper proves almost sure convergence to a ne for diminishing step sizes for constant step sizes an error bound on expected distance from a ne is established the effectiveness of the proposed algorithm is demonstrated via simulation for both diminishing and constant step sizes

stochastic averaging in discrete time and its applications to extremum seeking we investigate stochastic averaging theory for locally lipschitz discrete time nonlinear systems with stochastic perturbation and its applications to convergence analysis of discrete time stochastic extremum seeking algorithms firstly by defining two average systems one is continuous time the other is discrete time we develop discrete time stochastic averaging theorem for locally lipschitz nonlinear systems with stochastic perturbation our results only need some simple and applicable conditions which are easy to verify and remove a significant restriction present in existing results global lipschitzness of the nonlinear vector field secondly we provide a discrete time stochastic extremum seeking algorithm for a static map in which measurement noise is considered and an ergodic discrete time stochastic process is used as the excitation signal finally for discrete time nonlinear dynamical systems in which the output equilibrium map has an extremum we present a discrete time stochastic extremum seeking scheme and with a singular perturbation reduction we prove the stability of the reduced system compared with classical stochastic approximation methods while the convergence that we prove is in a weaker sense the conditions of the algorithm are easy to verify and no requirements e g boundedness are imposed on the algorithm itself

nash equilibrium computation in subnetwork zero sum games with switching communications in this paper we investigate a distributed nash equilibrium computation problem for a time varying multi agent network consisting of two subnetworks where the two subnetworks share the same objective function we first propose a subgradient based distributed algorithm with heterogeneous stepsizes to compute a nash equilibrium of a zero sum game we then prove that the proposed algorithm can achieve a nash equilibrium under uniformly jointly strongly connected ujsc weight balanced digraphs with homogenous stepsizes moreover we demonstrate that for weighted unbalanced graphs a nash equilibrium may not be achieved with homogenous stepsizes unless certain conditions on the objective function hold we show that there always exist heterogeneous stepsizes for the proposed algorithm to guarantee that a nash equilibrium can be achieved for ujsc digraphs finally in two standard weight unbalanced cases we verify the convergence to a nash equilibrium by adaptively updating the stepsizes along with the arc weights in the proposed algorithm

the fractal nature of molecular trajectories in fluids the scaled lengths of molecular trajectories obtained by molecular dynamics simulation of a hard sphere fluid are shown to have the same fractal dimensiond 2 as the random walk self similarity first appears on length scales typically a factor of 25 greater than the mean free path length whereas for the simple random walk with constant step size the onset occurs after only six steps the reason for the slow convergence is shown to be the near exponential distribution of intercollision path lengths of the fluid molecules the influence of density on the scaled path lengths is also discussed

a switching based extremum seeking control scheme abstractin this paper a novel sliding mode analysis method is proposed for the traditional sliding mode based extremum seeking control esc scheme based on the method a more rigorous and more concise stability proof of the scheme is presented furthermore a novel switching based esc scheme is designed by combining traditional sliding mode based esc and a perturbation based esc without steady state oscillation this proposed scheme can overcome the steady state oscillation of sliding mode based esc scheme as well as keep rapid robust convergence the stability analysis of the proposed scheme is also provided the simulation examples show the effectiveness and advantages of the scheme

a framework for a class of hybrid extremum seeking controllers with dynamic inclusions abstract this paper presents a prescriptive framework for the design and analysis of a class of deterministic extremum seeking controllers escs based on hybrid dynamic inclusions this type of escs combines continuous and discrete time dynamics during the seeking process and its evolution in time is characterized by differential and difference inclusions rather than standard difference and differential equations examples of hybrid escs include but are not limited to purely continuous esc with optimizers described by set valued mappings esc with arbitrarily fast and slow switching modes esc with weakly jumping parameters as well as distributed escs for multi agent systems with time varying graphs since solutions of systems described by set valued mappings are usually not unique we do not insist on this property but rather we characterize the behavior of all possible solutions generated by the closed loop system making use of recent results in singular perturbations averaging and limit sets of sets for hybrid dynamical systems we establish a practical asymptotic stability result some examples of common classes of seeking dynamics described by hybrid systems are presented as well as some numerical simulations illustrating their application in standard maximization problems and nash seeking problems in game theoretical scenarios

stability of functional equations in single variable abstract this paper discusses hyers ulam stability for functional equations in single variable including the forms of linear functional equation nonlinear functional equation and iterative equation surveying many known and related results we clarify the relations between hyers ulam stability and other senses of stability such as iterative stability continuous dependence and robust stability which are used for functional equations applying results of nonlinear functional equations we give the hyers ulam stability of bottcher s equation we also prove a general result of hyers ulam stability for iterative equations

game theory approach for the integrated design of structures and controls the problem of design of actively controlled structures subject to constraints on the damping parameters of the closed loop system is formulated as a multiobjeetive optimization problem the structural weight and the controlled system energy are considered as objective functions for minimization with cross sectional areas of members as design variables a computational procedure is developed for solving the multiobjeetive optimization problem using cooperative game theory the feasibility of the procedure is demonstrated through the design of two truss structures

stability of the multi jensen equation abstract in this paper we study the stability in the sense of th m rassias of multi jensen equation

nash equilibrium seeking in noncooperative games we introduce a non model based approach for locally stable convergence to nash equilibria in static noncooperative games with n players in classical game theory algorithms each player employs the knowledge of the functional form of his payoff and the knowledge of the other players actions whereas in the proposed algorithm the players need to measure only their own payoff values this strategy is based on the extremum seeking approach which has previously been developed for standard optimization problems and employs sinusoidal perturbations to estimate the gradient we consider static games with quadratic payoff functions before generalizing our results to games with non quadratic payoff functions that are the output of a dynamic system specifically we consider general nonlinear differential equations with n inputs and n outputs where in the steady state the output signals represent the payoff functions of a noncooperative game played by the steady state values of the input signals we employ the standard local averaging theory and obtain local convergence results for both quadratic payoffs where the actual convergence is semi global and non quadratic payoffs where the potential existence of multiple nash equilibria precludes semi global convergence our convergence conditions coincide with conditions that arise in model based nash equilibrium seeking however in our framework the user is not meant to check these conditions because the payoff functions are presumed to be unknown for non quadratic payoffs convergence to a nash equilibrium is not perfect but is biased in proportion to the perturbation amplitudes and the higher derivatives of the payoff functions we quantify the size of these residual biases and confirm their existence numerically in an example noncooperative game in this example we present the first application of extremum seeking with projection to ensure that the players actions remain in a given closed and bounded action set

the stability of a cubic type functional equation with the fixed point alternative abstract in this note we investigate the generalized hyers ulam rassias stability for the new cubic type functional equation f x y 2 z f x y 2 z f 2 x f 2 y 2 f x y 2 f x z 2 f x z 2 f y z 2 f y z by using the fixed point alternative the first systematic study of fixed point theorems in nonlinear analysis is due to g isac and th m rassias internat j math math sci 19 1996 219 228

on the stability of the golab schinzel functional equation abstract motivated by the problem of r ger we consider the stability of the golab schinzel functional equation we show that in the class of functions satisfying some weak regularity assumption the equation is superstable

extremum seeking control for discrete time systems we present an extremum seeking control algorithm for discrete time systems applied to a class of plants that are represented as a series combination of a linear input dynamics a static nonlinearity with an extremum and a linear output dynamics by using the two time scale averaging theory we derive a mild sufficient condition under which the plant output exponentially converges to an o spl alpha sup 2 neighborhood of the extremum value where spl alpha is the magnitude of modulation signal the sufficient condition is related to positive realness of linear parts of the plant but only at the modulation frequency the algorithm is illustrated with a brief simulation study

the pexiderized golab schinzel functional equation let x be a linear space over a commutative field k we characterize a general solution f g h k x k of the pexiderized golab schinzel equation f x g x y h x k y as well as real continuous solutions of the equation

distributed seeking of nash equilibria with applications to mobile sensor networks we consider the problem of distributed convergence to a nash equilibrium in a noncooperative game where the players generate their actions based only on online measurements of their individual cost functions corrupted with additive measurement noise exact analytical forms and or parameters of the cost functions as well as the current actions of the players may be unknown additionally the players actions are subject to linear dynamic constraints we propose an algorithm based on discrete time stochastic extremum seeking using sinusoidal perturbations and prove its almost sure convergence to a nash equilibrium we show how the proposed algorithm can be applied to solving coordination problems in mobile sensor networks where motion dynamics of the players can be modeled as 1 single integrators velocity actuated vehicles 2 double integrators force actuated vehicles and 3 unicycles a kinematic model with nonholonomic constraints examples are given in which the cost functions are selected such that the problems of connectivity control formation control rendezvous and coverage control are solved in an adaptive and distributed way the methodology is illustrated through simulations

dynamic fictitious play dynamic gradient play and distributed convergence to nash equilibria we consider a continuous time form of repeated matrix games in which player strategies evolve in reaction to opponent actions players observe each other s actions but do not have access to other player utilities strategy evolution may be of the best response sort as in fictitious play or a gradient update such mechanisms are known to not necessarily converge we introduce a form of dynamic fictitious and gradient play strategy update mechanisms these mechanisms use derivative action in processing opponent actions and in some cases can lead to behavior converging to nash equilibria in previously nonconvergent situations we analyze convergence in the case of exact and approximate derivative measurements of the dynamic update mechanisms in the ideal case of exact derivative measurements we show that convergence to nash equilibrium can always be achieved in the case of approximate derivative measurements we derive a characterization of local convergence that shows how the dynamic update mechanisms can converge if the traditional static counterparts do not we primarily discuss two player games but also outline extensions to multiplayer games we illustrate these methods with convergent simulations of the well known shapley and jordan counterexamples