limit cycles of planar quadratic differential equations abstract since hilbert posed the problem of systematically counting and locating the limite cycles of polynomial systems on the plane in 1900 much effort has been expended in its investigation a large body of literature chiefly by chinese and soviet authors has addressed this question in the context of differential equations whose field is specified by quadratic polynomials this paper considers the class of quadratic differential equations which admit a unique equilibrium state and establish sufficient conditions for the existence and uniqueness of limit cycles the work is based upon insights and techniques developed in an earlier analysis of such systems motivated by questions from mathematical control theory

linear systems subject to input saturation and time delay global asymptotic stabilization this note deals with the problem of global asymptotic stabilization of linear systems by bounded static feedbacks subject to time delay under standard necessary conditions we provide two constructions of controllers of nested saturation type which extends to the general case partial results of previous papers for arbitrary small bound on the control and large constant delay to validate the approach a third order integrator and oscillator with multiplicity two example is presented

a family of global stabilizers for quasi optimal control of planar linear saturated systems we propose a family of nonlinear state feedback global stabilizers for all planar linear systems which are globally stabilizable by bounded inputs namely all non exponentially unstable linear systems this family is parametrized by a nonlinear function whose selection can yield quasi time optimal responses where the quasi is required to achieve local exponential stability of the closed loop the arising trajectories are quasi time optimal for arbitrarily large initial conditions so we expect the very simple proposed nonlinear control law to be very useful for embedded control applications with strong computational constraints

l_p stabilization of integrator chains subject to input saturation using lyapunov based homogeneous design consider the n th integrator dot x j nx sigma u e n where x in mathbb r n u in mathbb r j n is the n th jordan block and e n 0 cdots 0 1 t in mathbb r n we provide easily implementable state feedback laws u k x which not only render the closed loop system globally asymptotically stable but also are finite gain l p stabilizing with arbitrarily small gain as in a saberi p hou and a stoorvogel ieee trans automat control 45 2000 pp 1042 1052 these l p stabilizing state feedbacks are built from homogeneous feedbacks appearing in finite time stabilization of linear systems we also provide additional l infty stabilization results for the case of both internal and external disturbances of the n th integrator namely for the perturbed system dot x j nx e n sigma k x d d where d in mathbb r and d in mathbb r n

on the stabilization of persistently excited linear systems we consider control systems of the type dot x ax alpha t bu where u in mathbf r a b is a controllable pair and alpha is an unknown measurable time varying signal with values in 0 1 satisfying a persistent excitation condition of the type int t t t alpha s ds geq mu for every t geq0 with 0 mu leq t independent of t we prove that such a system is stabilizable with a linear feedback depending only on the pair t mu if the eigenvalues of a have a nonpositive real part we also show that stabilizability does not hold for an arbitrary matrix a moreover the question of whether the system can be stabilized or not with an arbitrarily large rate of convergence gives rise to a bifurcation phenomenon depending on the parameter mu t

a general result on the stabilization of linear systems using bounded controls we present two constructions of controllers that globally stabilize linear systems subject to control saturation we allow essentially arbitrary saturation functions the only conditions imposed on the system are the obvious necessary ones namely that no eigenvalues of the uncontrolled system have positive real part and that the standard stabilizability rank condition hold one of the constructions is in terms of a neural network type one hidden layer architecture while the other one is in terms of cascades of linear maps and saturations

some theorems on the existence uniqueness and nonexistence of limit cycles for quadratic systems abstract given a quadratic system qs with a focus or a center at the origin we write it in the form y p 2 x y x dy q 2 x y where p 2 and q 2 are homogeneous polynomials of degree 2 if we define f x y x dy p 2 x y yq 2 x y and g x y xq 2 x y yp 2 x y we give results of existence nonexistence and uniqueness of limit cycles if f x y g x y does not change of sign then by using these results plus the properties on the evolution of the limit cycles of the semicomplete families of rotated vector fields we can study some particular families of qs i e the qs with a unique finite singularity and the bounded qs with either one or two finite singularities

uniqueness of limit cycles for polynomial first order differential equations abstract we study the uniqueness of limit cycles periodic solutions that are isolated in the set of periodic solutions in the scalar ode x k 1 m a k sin i k t cos j k t x n k in terms of i k j k n k our main result characterizes under some additional hypotheses the exponents i k j k n k such that for any choice of a 1 a m r the equation has at most one limit cycle the obtained results have direct application to rigid planar vector fields thus planar systems of the form x y x r x y y x y r x y where r x y k 1 m a k x i k y j k concretely when the set i k j k k 1 m has at least three elements or exactly one and another technical condition is satisfied we characterize the exponents i k j k such that the origin of the rigid system is a center for any choice of a 1 a m r and also when there are no limit cycles surrounding the origin for any choice of a 1 a m r

periodic solutions of polynomial non autonomous differential equations let n denote the set of n 1 tuples of c 1 periodic functions the point p p n p 1 p 0 n is identified with the differential equation we examine the way in which the total number of periodic solutions can vary as p traverses a path in n

on finite gain stabilizability of linear systems subject to input saturation this paper deals with global finite gain input output stabilization of linear systems with saturated controls for neutrally stable systems it is shown that the linear feedback law suggested by the passivity approach indeed provides stability with respect to every l p norm explicit bounds on closed loop gains are obtained and they are related to the norms for the respective systems without saturation these results do not extend to the class of systems for which the state matrix has eigenvalues on the imaginary axis with nonsimple size 1 jordan blocks contradicting what may be expected from the fact that such systems are globally asymptotically stabilizable in the state space sense this is shown in particular for the double integrator

robust stabilization via saturated feedback in this paper we deal with the problem of stabilization of uncertain systems in the presence of input constraint first algebraic conditions are derived for input to state stability of linear system with saturated linear feedback of low dimension then a recursive design procedure is derived for robust stabilization of block upper triangular nonlinear systems with feedforward structure

space time spectral method for a weakly singular parabolic partial integro differential equation on irregular domains abstract the spectral method is proposed for the partial integro differential equations with a weakly singular kernel on irregular domains the space discretization is based on the nodal spectral element method using the lagrange polynomials basis associated with the gauss lobatto legendre quadrature nodes also the model is discretized in time with the legendre spectral galerkin method the discretization leads to conversion of the problem to a sylvester matrix equation which can be solved efficiently by the qz algorithm gardiner et al 1992 the convergence of the method is proven by providing a priori l 2 error estimate numerical results illustrate the efficiency and spectral accuracy of the proposed method

convergence analysis of spectral collocation methods for a class of weakly singular volterra integral equations abstract in this paper we discuss the application of spectral jacobi collocation methods to a certain class of weakly singular volterra integral equations first we use some function transformations and variable changes to transform the equation into a volterra integral equation defined on the standard interval 1 1 then the jacobi gauss quadrature formula is used to approximate the integral operator for the spectral jacobi collocation method a rigorous error analysis in both the l and weighted l 2 norms is given under the assumption that both the kernel function and the source function are sufficiently smooth finally some numerical examples are provided to illustrate the theoretical results

on spectral methods for volterra type integro differential equations this paper considers the spectral methods for a volterra type integro differential equation firstly the volterra type integro differential equation is equivalently restated as two integral equations of the second kind secondly a legendre collocation method is used to solve them then the error analysis is conducted based on the l norm in addition numerical results are presented to confirm our analysis

chebyshev spectral solution of nonlinear volterra hammerstein integral equations abstract in this paper the chebyshev spectral cs method for the approximate solution of nonlinear volterra hammerstein integral equations f o k r g r y r d r 0 t is investigated the method is applied to approximate the solution not to the equation in its original form but rather to an equivalent equation z t g t y t t 1 1 the function z is approximated by the nth degree interpolating polynomial zn with coefficients determined by discretizing g t y t at the chebyshev gauss labatto nodes we then define the approximation to y to be of the form y n t f t 1 1 k t s z n s d s 1 1 and establish that under suitable conditions limn yn t y t uniformly in t finally a numerical experiment for a nonlinear volterra hammerstein integral equation is presented which confirms the convergence demonstrates the applicability and the accuracy of the chebyshev spectral cs method

stable and efficient spectral methods in unbounded domains using laguerre functions stable and efficient spectral methods using laguerre functions are proposed and analyzed for model elliptic equations on regular unbounded domains it is shown that spectral galerkin approximations based on laguerre functions are stable and convergent with spectral accuracy in the usual not weighted sobolev spaces efficient accurate and well conditioned algorithms using laguerre functions are developed and implemented numerical results indicating the spectral convergence rate and effectiveness of these algorithms are presented

post processing discontinuous galerkin solutions to volterra integro differential equations analysis and simulations this paper presents a superconvergence extraction technique for volterra integro differential equations with smooth and non smooth kernels specifically extracting superconvergence is done via a post processed discontinuous galerkin dg method obtained from interpolating the dg solution using lagrange polynomials at the nodal points a global superconvergence error bound in the l norm is established for a non smooth kernel a family of non uniform time meshes is used to compensate for the singular behaviour of the exact solution near t 0 the derived theoretical results are numerically validated in a sample of test problems demonstrating higher than expected convergence rates

laguerre galerkin method for nonlinear partial differential equations on a semi infinite interval summary a laguerre galerkin method is proposed and analyzed for the burgers equation and benjamin bona mahony bbm equation on a semi infinite interval by reformulating these equations with suitable functional transforms it is shown that the laguerre galerkin approximations are convergent on a semi infinite interval with spectral accuracy an efficient and accurate algorithm based on the laguerre galerkin approximations to the transformed equations is developed and implemented numerical results indicating the high accuracy and effectiveness of this algorithm are presented

global stabilization of classes of linear control systems with bounds on the feedback and its successive derivatives in this paper we address the problem of globally stabilizing a linear time invariant lti system by means of a static feedback law whose amplitude and successive time derivatives up to a prescribed order p are bounded by arbitrary prescribed values we solve this problem for two classes of lti systems namely integrator chains and controllable skew symmetric systems with single input for the integrator chains the solution we propose is based on the nested saturations introduced by a r teel we show that this construction fails for skew symmetric systems and propose an alternative feedback law we illustrate these findings by the stabilization of the third order integrator and the harmonic oscillator with prescribed bounds on the feedback and its first two derivatives

universal construction of feedback laws achieving iss and integral iss disturbance attenuation we study nonlinear systems with both control and disturbance inputs the main problem addressed in the paper is design of state feedback control laws that render the closed loop system integral input to state stable iiss with respect to the disturbances we introduce an appropriate concept of control lyapunov function iiss clf whose existence leads to an explicit construction of such a control law the same method applies to the problem of input to state stabilization converse results and techniques for generating iiss clfs are also discussed

limit cycles of a class of polynomial vector fields in the plane on montre que pour 0 suffisamment petit le systeme quadratique dx dt x y x x y dy dt x y xy a une source spirale hyperbolique unique a l origine et un cycle limite unique stable entourant l origine

observability and observers for nonlinear systems this paper deals with observability and observers for general nonlinear systems in the non control affine case we characterize systems that are observable independently of the inputs an exponential observer for these systems is also exhibited

global stabilization of oscillators with bounded delayed input the problem of globally asymptotically stabilizing by bounded feedback of an oscillator with an arbitrary large delay in the input is solved a first solution follows from a general result on the global stabilization of null controllable linear systems with delay in the input by bounded control laws with a distributed term next it is shown through a lyapunov analysis that the stabilization can be achieved as well when neglecting the distributed terms it turns out that this main result is intimately related to the output feedback stabilization problem

linear systems subject to input saturation and time delay finite gain l lowercase p stabilization this paper deals with finite gain stabilization of linear systems by static feedbacks which are bounded and time delayed for neutrally stable continuous time linear systems we provide a saturated linear feedback law ensuring finite gain stability with respect to every l p norm p in 1 infty with an arbitrary small bound on the control and large constant delay moreover we provide upper bounds for the corresponding l p gains which are delay independent

the spectral methods for parabolic volterra integro differential equations in this paper we study the numerical solutions to parabolic volterra integro differential equations in one dimensional bounded and unbounded spatial domains in a bounded domain the given parabolic volterra integro differential equation is converted to two equivalent equations then a legendre collocation method is used to solve them and finally a linear algebraic system is obtained for an unbounded case we use the algebraic mapping to transfer the problem on a bounded domain and then apply the same presented approach for the bounded domain in both cases some numerical examples are presented to illustrate the efficiency and accuracy of the proposed method

limit cycles of polynomial systems with homogeneous non linearities abstract we study the class of vector fields of the form x p1 x y pn x y y q1 x y qn x y where pk and qk are real homogeneous polynomials of degree k n 2 and the origin is a non degenerate critical point of index 1 for this class of vector fields we give four theorems about their limit cycles