Extremum Seeking Control (ESC) control is a non-model based real-time optimization method that is used to find an extremum of an unknown map between inputs and outputs of a stable dynamic system . The first scheme of ESC control uses a deterministic sinusoidal signal for estimating gradient of a given map which results optimizing the map by the proper filtering . Despite few applications of ltr" Despite the theoretical developments and applications , uncertainty of convergence rate is a serious problem which exists in all ESC schemes; this is roughly due to the fact that the target function is unknown . Bounded E xtremum Seeking Control (BESC) is a recent ESC scheme in which the uncertainty is confined to the argument of a sine/cosine function , resulting in guaranteed bounds on update rate in extremum seeking . The aim of this research is to achieve nash equilibrium in a non-cooperative game using BESC . Although the practical stability is important, i n some application it requires to expotential stability . O nly the practical stability of BESC has been proved . In this thesis , we first prove the expotential stability for both static and dynamic maps using averaging theory and singular perturbation theory . It is proposed the Newton-based bounded extremum seeking control for a single-valued mapping . Then , the application of the bounded extremum seeking is investigated in two sets of non-cooperative static games. The first category are static games with quadratic payoff. The second category is a static noncooperative game with nonquadratic payoff . Stability analysis shows that in both games to the nash equilibrium seeking based bounded extremum seeking control. The simulation results for both game categories indicate that the convergence of the bounded extremum seeking control is faster than the classic extremum s eeking control . Key Words: Extremum seeking control, Singular Perturbation Theory, Averaging Theory, Quadratic game, Nash Equilibrium.
