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

a multiplicity result for a nonlinear fractional schrodinger equation in rn without the ambrosetti rabinowitz condition abstract in this paper we study the existence and the multiplicity of positive solutions for the following class of fractional schrodinger equations 2 s s u v x u f u in r n where 0 is a parameter s 0 1 n 2 s v r n r is a continuous positive potential and f r r is a c 1 superlinear nonlinearity which does not satisfy the ambrosetti rabinowitz condition the main result is established by using minimax methods and ljusternik schnirelmann theory of critical points

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

ground state of kirchhoff type fractional schrodinger equations with critical growth abstract in this paper we study the critical kirchhoff type fractional schrodinger equation 0 1 1 r 3 r 3 u x u y 2 x y 3 2 s d x d y s u u f u u 2 s 1 in r 3 where s 0 1 and 2 s 6 3 2 s we establish the pohoz ev type identity of 0 1 when s 3 4 1 under some conditions on and f u we obtain some results on the existence of ground state solutions when s 0 3 4 we also prove the non existence result in particular when 0 we obtain an existence result

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

bounded extremum seeking with discontinuous dithers the analysis of discontinuous extremum seeking es controllers e g those applicable to digital systems has historically been more complicated than that of continuous controllers we establish a simple and general extension of a recently developed bounded form of es to a general class of oscillatory functions including functions discontinuous with respect to time such as triangle or square waves with dead time we establish our main results by combining a novel idea for oscillatory control with an extension of functional analytic techniques originally utilized by kurzweil jarnik sussmann and liu in the late 80s and early 90s and recently studied by durr et al we demonstrate the value of the result with an application to inverter switching control

accuracy improvement of extremum seeking control in this work we present a modification for the classic and phasor extremum seeking control algorithms in order to improve the accuracy by removing or reducing the convergence error the modulation signals were replaced by a sum of sinusoids in order to remove the equilibrium shift in the controlled variable of the averaged system the convergence error is calculated as a function of the number of sinusoids used in the modulation signal a simulation example is presented to illustrate the improvement

a concave convex elliptic problem involving the fractional laplacian we study a nonlinear elliptic problem defined in a bounded domain involving fractional powers of the laplacian operator together with a concave convex term we completely characterize the range of parameters for which solutions of the problem exist and prove a multiplicity result we also prove an associated trace inequality and some liouville type results

estimation based approach for real time optimisation of uncertain nonlinear systems abstractin this paper the minimisation of an unknown but measurable cost function with uncertain dynamics is considered the drift term of the uncertain dynamical system and the gradient of the objective function are treated as unknown time varying parameters a novel estimation scheme based on almost invariant manifolds is proposed to estimate the time varying parameters a direct gradient based adaptive extremum seeking controller is designed to solve the uncertain optimisation problem this approach is shown to improve the transient performance of real time optimisation control systems

on a class of generating vector fields for the extremum seeking problem lie bracket approximation and stability properties abstract in this paper we describe a broad class of control functions for extremum seeking problems we show that it unifies and generalizes existing extremum seeking strategies which are based on lie bracket approximations and allows to design new controls with favorable properties in extremum seeking and vibrational stabilization tasks the second result of this paper is a novel approach for studying the asymptotic behavior of extremum seeking systems it provides a constructive procedure for defining frequencies of control functions to ensure the practical asymptotic and exponential stability in contrast to many known results we also prove asymptotic and exponential stability in the sense of lyapunov for the proposed class of extremum seeking systems under appropriate assumptions on the vector fields

on some critical problems for the fractional laplacian operator abstract we study the effect of lower order perturbations in the existence of positive solutions to the following critical elliptic problem involving the fractional laplacian 2 u u q u n n u 0 in u 0 on where r n is a smooth bounded domain n 1 0 0 q n n 0 min n 2 for suitable conditions on depending on q we prove in the case q 1 there exist at least two solutions for every 0 and some 0 at least one if no solution if for q 1 we show existence of at least one solution for 0 1 and nonexistence for 1 when q 1 the existence is shown for every 0 also we prove that the solutions are bounded and regular

a note on the error in gaussian quadrature the error in gaussian quadrature is analyzed using methods from the theory of analytic functions it is well known that the error term may be expressed in terms of a contour integral against a kernel function k n t we give two methods for computing the coefficients in terms of the moments for the laurent series of k n t and we give explicit expressions for k n t for two particular measures we then use the laurent expansion of k n t to estimate the error for various functions and measures

extremum seeking for static maps with delays in this paper we address the design and analysis of multi variable extremum seeking for static maps subject to arbitrarily long time delays both gradient and newton based methods are considered multi input systems with different time delays in each individual input channel as well as output delays are dealt with the phase compensation of the dither signals and the inclusion of predictor feedback with a perturbation based averaging based estimate of the hessian allow to obtain local exponential convergence results to a small neighborhood of the optimal point even in the presence of delays the stability analysis is carried out using backstepping transformation and averaging in infinite dimensions capturing the infinite dimensional state due the time delay in particular a new backstepping like transformation is introduced to design the predictor for the gradient based extremum seeking scheme with multiple and distinct input delays the proposed newton based extremum seeking approach removes the dependence of the convergence rate on the unknown hessian of the nonlinear map to be optimized being user assignable as in the literature free of delays a source seeking example illustrates the performance of the proposed delay compensated extremum seeking schemes

extremum seeking control with second order sliding mode the extremum seeking control esc approach with a first order sliding mode has been proposed for searching a setpoint by making a sliding mode occur where a performance index tracks a reference signal to reach its extremum the system dynamics is assumed to be much faster than the parameter updating rate determined by the esc rule and thus is omitted in the stability analysis for the esc system with a first order sliding mode as a result only a static optimization can be obtained in this paper an esc approach with a second order sliding mode is proposed with the consideration of the system dynamics such that the performance index can be optimized dynamically even if the system dynamics is not faster than the parameter updating rate for the extremum seeking the asymptotic second order sliding mode relay control algorithm is implemented to guarantee the asymptotic convergence to the second order sliding mode without using the derivative of the performance index simulation results show that the second

nonlinear perturbations of a periodic elliptic problem with critical growth abstract using the well known mountain pass theorem due to ambrosetti and rabinowitz we establish conditions for the existence of a positive solution for a class of nonlinear elliptic problems in r n with critical growth

brief adaptive extremum seeking control of nonlinear dynamic systems with parametric uncertainties we pose and solve an extremum seeking control problem for a class of nonlinear systems with unknown parameters extremum seeking controllers are developed to drive system states to the desired set points that extremize the value of an objective function the proposed adaptive extremum seeking controller is inverse optimal in the sense that it minimizes a meaningful cost function that incorporates penalty on both the performance error and control action simulation studies are provided to verify the effectiveness of the proposed approach

a critical nonlinear fractional elliptic equation with saddle like potential in ℝn in this paper we study the existence of positive solution for the following class of fractional elliptic equation 2s su v z u uq 2u u2s 2uinrn where 0 are positive parameters q 2 2s 2s 2nn 2s n 2s s 0 1 su is the fractional laplacian and v is a saddle like potential the result is proved by using minimizing method constrained to the nehari manifold a special minimax level is obtained by using an argument made by benci and cerami

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

global bifurcation results for a fractional equation in rn abstract in this paper we study the existence of positive solution for the following class of fractional equation p s u g x f u in r n 0 u x 1 x r n lim x u x 0 where n 2 s s 0 1 0 f r r is a continuous function g r n r is a bounded continuous function that changes sign and s u is the fractional laplacian the main tools used are a theorem due to crandall and rabinowitz 14 and the global bifurcation theorem due to rabinowitz 23

on fractional schrodinger equation in rn with critical growth in this paper we study the following nonlinear fractional schrodinger equation with critical exponent h2 u v x u u 2 2u u q 2u x rn where h is a small positive parameter 0 2 0 is a parameter the potential v rn r is a positive continuous function satisfying some natural assumptions by using variational methods we obtain the existence of solutions in the following case if 2 0 such that for all 0 we show that it has one nontrivial solution and there exist at least cat nontrivial solutions if max 2 4 n 2 0

uniqueness of radial solutions for the fractional laplacian we prove general uniqueness results for radial solutions of linear and nonlinear equations involving the fractional laplacian s with s 0 1 for any space dimensions n 1 by extending a monotonicity formula found by cabre and sire we show that the linear equation s u vu 0 in r n has at most one radial and bounded solution vanishing at infinity provided that the potential v is radial and nondecreasing in particular this result implies that all radial eigenvalues of the corresponding fractional schrodinger operator h s v are simple furthermore by combining these findings on linear equations with topological bounds for a related problem on the upper half space r n 1 we show uniqueness and nondegeneracy of ground state solutions for the nonlinear equation s q q q q 0 in r n for arbitrary space dimensions n 1 and all admissible exponents 0 this generalizes the nondegeneracy and uniqueness result for dimension n 1 recently obtained by the first two authors and in particular the uniqueness result for solitary waves of the benjamin ono equation found by amick and toland

fractional schrodinger equation some properties of the fractional schrodinger equation are studied we prove the hermiticity of the fractional hamilton operator and establish the parity conservation law for fractional quantum mechanics as physical applications of the fractional schrodinger equation we find the energy spectra of a hydrogenlike atom fractional bohr atom and of a fractional oscillator in the semiclassical approximation an equation for the fractional probability current density is developed and discussed we also discuss the relationships between the fractional and standard schrodinger equations

unified frameworks for sampled data extremum seeking control global optimisation and multi unit systems two frameworks are proposed for extremum seeking of general nonlinear plants based on a sampled data control law within which a broad class of nonlinear programming methods is accommodated it is established that under some generic assumptions semi global practical convergence to a global extremum can be achieved in the case where the extremum seeking algorithm satisfies a stronger asymptotic stability property the converging sequence is also shown to be stable using a trajectory based proof as opposed to a lyapunov function type approach the former is more straightforward and insightful this allows for more general optimisation algorithms than considered in existing literature such as those which do not admit a state update realisation and or lyapunov functions lying at the heart of the analysis throughout is robustness of the optimisation algorithms to additive perturbations of the objective function multi unit extremum seeking is also investigated with the objective of accelerating the speed of convergence

concentrating standing waves for the fractional nonlinear schrodinger equation abstract we consider the semilinear equation e 2 s s u v x u u p 0 u 0 u h 2 s r n where 0 s 1 1 p n 2 s n 2 s v x is a sufficiently smooth potential with inf r v x 0 and e 0 is a small number letting w be the radial ground state of s w w w p 0 in h 2 s r n we build solutions of the form u e x i 1 k w i x i e e where i v i e and the i e approach suitable critical points of v via a lyapunov schmidt variational reduction we recover various existence results already known for the case s 1 in particular such a solution exists around k nondegenerate critical points of v for s 1 this corresponds to the classical results by floer and weinstein 13 and oh 24 25

error bounds for gaussian quadrature of analytic functions for gaussian quadrature rules over a finite interval applied to analytic or meromorphic functions we develop error bounds from contour integral representations of the remainder term as in previous work on the subject we consider both circular and elliptic contours in contrast with earlier work however we attempt to determine exactly where on the contour the kernel of the error functional attains its maximum modulus we succeed in answering this question for a large class of weight distributions including all jacobi weights when the contour is a circle in the more difficult case of elliptic contours we can settle the question for certain special jacobi weight distributions with parameters pm frac 1 2 and we provide empirical results for more general jacobi weights we further point out that the kernel of the error functional at any complex point outside the interval of integration can be evaluated accurately and efficiently by a recursive procedure the same procedure is useful also t

best constants for sobolev inequalities for higher order fractional derivatives we obtain sharp constants for sobolev inequalities for higher order fractional derivatives as an application we give a new proof of a theorem of w beckner concerning conformally invariant higher order differential operators on the sphere

nodal and multiple solutions of nonlinear problems involving the fractional laplacian abstract this paper is devoted to the existence of nodal and multiple solutions of nonlinear problems involving the fractional laplacian s u f x u in u 0 on where r n n 2 is a bounded smooth domain s 0 1 s stands for the fractional laplacian when f is superlinear and subcritical we prove the existence of a positive solution a negative solution and a nodal solution if f x u is odd in u we obtain an unbounded sequence of nodal solutions in addition the number of nodal domains of the nodal solutions are investigated

on stability and application of extremum seeking control without steady state oscillation to enhance the dynamic and static performance of the extremum seeking control esc scheme a novel fast esc scheme without steady state oscillation is proposed in which the structure of the classic esc scheme is adjusted to make the sinusoidal excitation signal amplitude locally exponentially converge to zero the improved esc scheme can speed up the convergence to shorten the seeking time dramatically and enlarge the search area to avoid falling to local extrema effectively it can also reduce the sinusoidal excitation signal amplitude to eliminate the adverse effects of the steady state oscillation eventually the rigorous stability analysis and proof of the improved esc scheme are provided in detail and the simulation results are presented to illustrate its effectiveness and superiority finally the application of the improved esc scheme to antilock braking systems abs has been discussed through comparing with classical perturbation based esc scheme and sliding mode based esc scheme to illustrate the practicability of the improved scheme

input to state set stability for pulse width modulated control systems with disturbances abstract new results on set stability and input to state stability in pulse width modulated pwm control systems with disturbances are presented the results are based on a recent generalization of two time scale stability theory to differential equations with disturbances in particular averaging theory for systems with disturbances is used to establish the results the nonsmooth nature of pwm systems is accommodated by working with upper semicontinuous set valued maps locally lipschitz inflations of these maps and locally lipschitz parameterizations of locally lipschitz set valued maps

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

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

technical communique on the choice of dither in extremum seeking systems a case study we discuss how the choice of dither excitation signal affects the performance of extremum seeking using a benchmark situation a static scalar map and a simple scalar extremum seeking scheme our comparisons are based on the performance of the system with different dithers in terms of three performance indicators the speed of convergence domain of attraction and accuracy i e the ultimate bound on trajectories our analysis explicitly shows how the dither shape affects each of these performance indicators our study suggests that the practitioners using extremum seeking control should consider the dither shape as an important design parameter computer simulations support our theoretical findings

some observations on gauss legendre quadrature error estimates for analytic functions abstract this paper is concerned with estimates for the error when a gauss legendre quadrature rule is used to numerically integrate an analytic function the error in a standard approximation to the kernel function which appears in a contour integral representation for the quadrature error is investigated for this purpose applications to meromorphic functions with simple poles are discussed