a comparison of some methods for the evaluation of highly oscillatory integrals abstract in part i the extended clenshaw curtis method for finite fourier integrals is discussed and a number of timed comparisons are made between the various implementations which appear in the literature part ii deals with irregular oscillatory integrals and outlines the various methods which have been proposed for their evaluation their effectiveness is discussed in the light of a set of test examples

a closed form feedback controller for stabilization of the linearized 2 d navier stokes poiseuille system we present a formula for a boundary control law which stabilizes the parabolic profile of an infinite channel flow which is linearly unstable for high reynolds numbers also known as the poiseuille flow this problem is frequently cited as a paradigm for transition to turbulence whose stabilization for arbitrary reynolds numbers without using discretization has so far been an open problem our result achieves exponential stability in the l2 h1 and h2 norms for the linearized navier stokes equations explicit solutions are obtained for the closed loop system this is the first time explicit formulae are produced for solutions of the linearized navier stokes equations in a channel flow with feedback in the boundary conditions used to make this possible the result is presented for the 2d case for clarity of exposition an extension to 3d is available and will be presented in a future publication

closed form boundary state feedbacks for a class of 1 d partial integro differential equations in this paper a problem of boundary stabilization of a class of linear parabolic partial integro differential equations p i des in one dimension is considered using the method of backstepping avoiding spatial discretization required in previous efforts the problem is formulated as a design of an integral operator whose kernel is required to satisfy a hyperbolic p i de the kernel p i de is then converted into an equivalent integral equation and by applying the method of successive approximations the equation s well posedness and the kernel s smoothness are established it is shown how to extend this approach to design optimally stabilizing controllers an adaptation mechanism is developed to reduce the conservativeness of the inverse optimal controller and the performance bounds are derived for a broad range of physically motivated special cases feedback laws are constructed explicitly and the closed loop solutions are found in closed form a numerical scheme for the kernel p i de is proposed its numerical effort compares favorably with that associated with operator riccati equations

stabilization of a plane channel flow by wall normal controllers abstract the steady state flows in a two dimensional channel with periodic conditions along the axis can be stabilized by boundary controllers with vertical velocity observation which acts on the normal component of velocity only

fuzzy differential game of guarding a movable territory in most earlier works of differential game problems which are based solely on crisp mathematics it is generally assumed that all information obtained by each player is perfect thus the games are treated with full information in many practical situations the information obtained by any player may contain some sort of uncertainties which are usually difficult to characterize either deterministically or stochastically we treat such uncertain information as fuzzy information and consider a differential game problem with fuzzy information as a fuzzy differential game problem in this paper we consider the fuzzy differential game of guarding a movable territory in a general problem of guarding a territory key concepts are the favorable region of the invader and the boundary curve for simplicity the movable territory is assumed to be a circle in this paper we will show that when the center of the territory is located outside the favorable region of the invader the value of the game exists otherwise the value does not exist a fuzzy control strategy is proposed and some computer simulations are provided in this paper

extended exponentially fitted interpolation formulas for oscillatory functions we consider exponentially fitted interpolation formulas that use not only the pointwise values of an w dependent function f but also of its derivatives up to the nth order at the two boundary nodes of a closed interval a b the function f is of the form f x f 1 x cos wx f 2 x sin wx email protected a b where the functions f 1 and f 2 are smooth on the interval as n increases numerical results about the errors for the formulas are illustrated and compared some properties of the coefficients of the formulas at the two nodes are presented and extended to the case with three nodes we investigate if the properties of the coefficients can be generalised to the cases with more than three nodes finally some other numerical results regarding the errors are given

a continuation gmres method for fast computation of nonlinear receding horizon control in this paper a fast numerical algorithm for nonlinear receding horizon control is proposed the control input is updated by a differential equation to trace the solution of an associated state dependent two point boundary value problem a linear equation involved in the differential equation is solved by the generalized minimum residual method one of the krylov subspace methods with jacobians approximated by forward differences the error in the entire algorithm is analyzed and is shown to be bounded under some conditions the proposed algorithm is applied to a two link arm whose dynamics is highly nonlinear simulation results show that the proposed algorithm is faster than the conventional algorithms

polytopic approximation of explicit model predictive controllers a model predictive control law mpc is given by the solution to a parametric optimization problem that can be precomputed offline which provides an explicit map from state to input that can be rapidly evaluated online however the primary limitations of these optimal explicit solutions are that they are applicable to only a restricted set of systems and that the complexity can grow quickly with problem size in this paper we compute approximate explicit control laws that trade off complexity against approximation error for mpc controllers that give rise to convex parametric optimization problems the algorithm is based on the classic double description method and returns a polyhedral approximation to the optimal cost function the proposed method has three main advantages from a control point of view it is an incremental approach meaning that an approximation of any specified complexity can be produced it operates on implicitly defined convex sets meaning that the prohibitively complex optimal explicit solution is not required and finally it can be applied to any convex parametric optimization problem a sub optimal controller based on barycentric interpolation is then generated from this approximate polyhedral cost function that is feasible and stabilizing the resulting control law is continuous although non linear and defined over a non simplical polytopic partition of the state space the non simplical nature of the partition generates significantly simpler approximate control laws which is demonstrated on several examples

frequency dependent interpolation rules using first derivatives for oscillatory functions we construct frequency dependent rules to interpolate oscillatory functions y x with frequency w of the form y x f 1 x cos wx f 2 x sin wx at equidistant nodes on the interval of interest where the functions f 1 and f 2 are smooth error analysis of the rules is investigated and numerical results are discussed we provide numerical illustrations to compare the accuracy of classical hermite polynomials and newly constructed frequency dependent rules

pseudospectra of the orr sommerfeld operator this paper investigates the pseudospectra and the numerical range of the orr sommerfeld operator for plane poiseuille flow a number z in bf c is in the epsilon pseudospectrum of a matrix or operator a if zi a 1 geq epsilon 1 or equivalently if z is in the spectrum of a e for some perturbation e satisfying e leq epsilon the numerical range of a is the set of numbers of the form au u where cdot cdot is the inner product and u is a vector or function with huh u 1 the spectrum of the orr sommerfeld operator consists of three branches it is shown that the eigenvalues at the intersection of the branches are highly sensitive to perturbations and that the sensitivity increases dramatically with the reynolds number the associated eigenfunctions are nearly linearly dependent even though they form a complete set to understand the high sensitivity of the eigenvalues a model operator is considered related to the airy equation tha

boundary control of the linearized ginzburg landau model of vortex shedding in this paper we continue the development of state feedback boundary control laws based on the backstepping methodology for the stabilization of unstable parabolic partial differential equations we consider the linearized ginzburg landau equation which models for instance vortex shedding in bluff body flows asymptotic stabilization is achieved by means of boundary control via state feedback in the form of an integral operator the kernel of the operator is shown to be twice continuously differentiable and a series approximation for its solution is given under certain conditions on the parameters of the ginzburg landau equation compatible with vortex shedding modelling on a semi infinite domain the kernel is shown to have compact support resulting in partial state feedback simulations are provided in order to demonstrate the performance of the controller in summary the paper extends previous work in two ways 1 it deals with two coupled partial differential equations and 2 under certain circumstances handles equations defined on a semi infinite domain

error analysis for frequency dependent interpolation formulas using first derivatives abstract we call attention to the entire errors which result when frequency dependent interpolation formulas are utilized to approximate oscillatory functions f x at some x on the domain of interest with a frequency of the form f x f 1 x cos x f 2 x sin x where the functions f 1 and f 2 are smooth enough to be approximated by polynomials the interpolation formulas to be considered utilize not only the pointwise values of the function f but also of its derivative f at two or three nodes on a closed and bounded interval in particular investigations about the interpolation formulas i or i using three equally spaced nodes or three unequally spaced nodes enable us to construct i or i related composite formulas which are obtained from applying the formulas i or i onto subintervals where the union of all the subintervals is the domain of interest numerical results show that newly constructed composite formulas are superior in their accuracy to other approximations to interpolate the oscillatory functions finally the entire errors with respect to the interpolation formulas using the derivative information at two or three nodes are obtained

two frequency dependent gauss quadrature rules we construct two frequency dependent gauss quadrature rules which can be applied for approximating the integration of the product of two oscillatory functions with different frequencies 1 and 1 of the forms yi x fi 1 x cos ix fi 2 x sin ix i 1 2 where the functions fi j x are smooth a regularization procedure is presented to avoid the singularity of the jacobian matrix of nonlinear system of equations which is induced as one frequency approaches the other frequency we provide numerical results to compare the accuracy of the classical gauss rule and one and two frequency dependent rules

receding horizon control with numerical solution for nonlinear parabolic partial differential equations the optimal control of nonlinear partial differential equations pdes is an open problem with applications that include fluid thermal biological and chemical systems receding horizon control is a kind of optimal feedback control and its performance index has a moving initial time and a moving terminal time in this study we develop a design method of receding horizon control for systems described by nonlinear parabolic pdes the objective of this study is to develop a novel algorithm for numerically solving the receding horizon control problem for nonlinear parabolic pdes the effectiveness of the proposed method is verified by numerical simulations

a dynamic games approach to controller design disturbance rejection in discrete time it is shown that the discrete time disturbance rejection problem formulated in finite and infinite horizons and under perfect state measurements can be solved by making direct use of some results on linear quadratic zero sum dynamic games for the finite horizon problem an optimal minimax controller exists and can be expressed in terms of a generalized time varying discrete time riccati equation the existence of an optimum also holds in the infinite horizon case under an appropriate observability condition with the optimal control given in terms of a generalized algebraic riccati equation also being stabilizing in both cases the corresponding worst case disturbances turn out to be correlated random sequences with discrete distributions which means that the problem viewed as a dynamic game between the controller and the disturbance does not admit a pure strategy saddle point results for the delayed state measurement and the nonzero initial state cases are presented

exponentially fitted gauss laguerre quadrature rule for integrals over an unbounded interval new quadrature formulae are introduced for the computation of integrals over the whole positive semiaxis when the integrand has an oscillatory behaviour with decaying envelope the new formulae are derived by exponential fitting and they represent a generalization of the usual gauss laguerre formulae their weights and nodes depend on the frequency of oscillation in the integrand and thus the accuracy is massively increased rules with one up to six nodes are treated with details numerical illustrations are also presented

a strategy for a payoff switching differential game based on fuzzy reasoning in this paper a new concept of a payoff switching differential game is introduced in this new game any one player at any time may have several choices of payoffs for the future moreover the payoff switching process including the time of payoff switching and the outcome payoff of any one player is unknown to the other indeed the overall payoff which is a sequence of several payoffs is unknown until the game ends an algorithm for determining a reasoning strategy based on fuzzy reasoning is proposed in this algorithm the fuzzy theory is used to estimate the behavior of one player during a past time interval by deriving two fuzzy matrices gsm game similarity matrix and vgsm variation of gsm the behavior of the player can be quantified two weighting vectors are selected to weight the relative importance of the player s behavior at each past time instant finally a simple fuzzy inference rule is adopted to generate a linear reasoning strategy the advantage of this algorithm is that it provides a flexible way for differential game specialists to convert their knowledge into a reasonable strategy a practical example of guarding three territories is given to illustrate our main ideas

quadrature rules using first derivatives for oscillatory integrands we consider the integral of a function y x i y x 11 y x dx and its approximation by a quadrature rule of the form qn y x k 1n wky xk k 1n ky xk i e by a rule which uses the values of both y and its derivative at nodes of the quadrature rule we examine the cases when the integrand is either a smooth function or an dependent function of the form y x f1 x sin x f2 x cos x with smoothly varying f1 and f2 in the latter case the weights wk and k are dependent we establish some general properties of the weights and present some numerical illustrations

continuation methods theory and applications this paper surveys in a tutorial fashion theoretical and applications aspects of the confirmation method for the solution of large scale system engineering problems the continuation method is motivated and defined the existence of the method is formulated in terms of degree theory a la algebraic topology examples are given throughout the paper and applications to engineering and economic problems are cited and described

nonlinear control of incompressible fluid flow application to burgers equation and 2d channel flow abstract this paper proposes a methodology for the synthesis of nonlinear finite dimensional feedback controllers for incompressible newtonian fluid flows described by two dimensional navier stokes equations combination of galerkin s method with approximate inertial manifolds is employed for the derivation of low order ordinary differential equation ode systems that accurately describe the dominant dynamics of the flow these ode systems are subsequently used as the basis for the synthesis of nonlinear output feedback controllers that guarantee stability and enforce the output of the closed loop system to follow the reference input asymptotically the method is successfully used to synthesize nonlinear finite dimensional output feedback controllers for the burgers equation and the two dimensional channel flow that enhance the convergence rate to the spatially uniform steady state and the parabolic velocity profile respectively the performance of the proposed controllers is successfully tested through simulations and is shown to be superior to the one of linear controllers

h infty feedback boundary stabilization of the two dimensional navier stokes equations we study the robust or h infty exponential stabilization of the linearized navier stokes equations around an unstable stationary solution in a two dimensional domain omega the disturbance is an unknown perturbation in the boundary condition of the fluid flow we determine a feedback boundary control law robust with respect to boundary perturbations by solving a max min linear quadratic control problem next we show that this feedback law locally stabilizes the navier stokes system similar problems have been studied in the literature in the case of distributed controls and disturbances to the authors knowledge it is the first time that the robust stabilization of the navier stokes equations is studied for boundary controls and boundary disturbances

stability enhancement by boundary control in 2 d channel flow in this paper we stabilize the parabolic equilibrium profile in a two dimensional 2 d channel flow using actuators and sensors only at the wall the control of channel flow was previously considered by speyer and coworkers and bewley and coworkers who derived feedback laws based on linear optimal control and implemented by wall normal actuation with an objective to achieve global lyapunov stabilization we arrive at a feedback law using tangential actuation using teamed pairs of synthetic jets or rotating disks and only local measurements of wall shear stress allowing to embed the feedback in microelectromechanical systems mems hardware without need for wiring this feedback is shown to guarantee global stability in at least h sup 2 norm which by sobolev s embedding theorem implies continuity in space and time of both the flow field and the control as well as their convergence to the desired steady state the theoretical results are limited to low values of reynolds number however we present simulations that demonstrate the effectiveness of the proposed feedback for values five order of magnitude higher

a first approach to fuzzy differential game problem guarding a territory abstract in this paper the fuzzy differential game problem is proposed and the problem of guarding a territory is considered we reformulate this differential game problem by considering that the locations of the territory and or of one s opponent are imprecisely known to the players thus treating them as fuzzy information a fuzzy control strategy is proposed for this fuzzy differential game problem it will be seen that in the limiting case the result of the fuzzy differential game problem approaches that of the unfuzzified case some computer simulation results are provided in this paper