a general method for the existence of periodic solutions of differential systems in the plane abstract we propose a general method to prove the existence of periodic solutions for planar systems of ordinary differential equations which can be used in many different circumstances applications are given to some nonresonant cases even for systems with superlinear growth in some direction or with a singularity systems at resonance are also considered provided a landesman lazer type of condition is assumed

positive periodic solutions of singular systems with a parameter abstract the existence and multiplicity of positive periodic solutions for second order non autonomous singular dynamical systems are established with superlinearity or sublinearity assumptions at infinity for an appropriately chosen parameter our results provide a unified treatment for the problem and significantly improve several results in the literature the proof of our results is based on the krasnoselskii fixed point theorem in a cone

weak singularities may help periodic solutions to exist abstract in a periodically forced semilinear differential equation with a singular nonlinearity a weak force condition enables the achievement of new existence criteria through a basic application of schauder s fixed point theorem the originality of the arguments relies in that contrary to the customary situation in the available references a weak singularity facilitates the arguments of the proofs

positive periodic solutions for planar differential systems with repulsive singularities on the axes abstract in this paper we provide an existence result of periodic solutions to planar systems having a repulsive singularity on both the axes we require a nonresonance assumption related to the spectrum of the periodic problem associated to the steen s planar system y c x 3 x x d y 3 y x 0 y 0 which is isochronous see 15 as a particular case we obtain the existence of periodic solutions to a periodically perturbed steen s planar system

positive periodic solutions of second order differential equations with weak singularities abstract we establish the existence of positive periodic solutions of the second order differential equation u a t u f t u c t via schauder s fixed point theorem where a l 1 r t z r c l 1 r t z r f is a caratheodory function and is singular at u 0 our main results generalize some recent results by p j torres

long time versus steady state optimal control this paper analyzes the convergence of optimal control problems for an evolution equation in a finite time horizon 0 t toward the limit steady state ones as t to infty we focus on linear problems we first consider linear time independent finite dimensional systems and show that the optimal controls and states exponentially converge in the transient time as t tends to infinity to the ones of the corresponding steady state model for this to occur suitable observability assumptions need to be imposed we then extend the results to infinite dimensional systems including the linear heat and wave equations with distributed controls

periodic solutions of second order equations via rotation numbers abstract we consider the problem of the existence and multiplicity of periodic solutions associated to a class of scalar equations of the form x f t x 0 the class considered is such that the behaviour of its solutions near zero and infinity can be compared two suitable piecewise linear systems we show how a rotation number approach together with the poincare birkhoff theorem and the phase plane analysis of the spiral properties allows to obtain multiplicity results in terms of the gap between the rotation numbers of the referred piecewise linear systems at zero and at infinity these systems may also be resonant in particular our approach can be used to deal with the problems without both global existence of the cauchy problems associated to the equation and the sign assumption on f the typical example is a partially superlinear second order equation our main result generalizes some classical results of jacobowitz and hartman among others

multiplicity of positive periodic solutions to superlinear repulsive singular equations abstract in this paper we study positive periodic solutions to the repulsive singular perturbations of the hill equations it is proved that such a perturbation problem has at least two positive periodic solutions when the anti maximum principle holds for the hill operator and the perturbation is superlinear at infinity the proof relies on a nonlinear alternative of leray schauder type and on krasnoselskii fixed point theorem on compression and expansion of cones

periodic solutions of the second order differential equations with singularity abstract in this paper we deal with the existence of periodic solutions of the second order differential equations x g x p t with singularity near origin by using the phase plane analysis methods we prove that the given equation has at least one periodic solution when g x exhibits semilinear condition for positive large x and singularity condition at origin and the primitive function g x 1 x g s d s tends to positive infinity as x 0 moreover we construct a counter example to show that our result is no longer valid when g x tends to some finite constant as x 0

existence of one signed periodic solutions of some second order differential equations via a krasnoselskii fixed point theorem this paper is devoted to study the existence of periodic solutions of the second order equation x 00 f t x where f is a caratheodory function by combining some new properties of green s function together with krasnoselskii fixed point theorem on compression and expansion of cones as applications we get new existence results for equations with jumping nonlinearities as well as equations with a repulsive or attractive singularity in this latter case our results cover equations with weak singularities and are compared with some recent results by i rachunkova m tvrdyand i vrkoc r 2002 elsevier science usa all rights reserved

periodic solutions for damped differential equations with a weak repulsive singularity abstract this paper deals with the existence of positive t periodic solutions for the following damped differential equation x p t x q t x f t x c t where p q c l 1 r are t periodic functions and f car r r r is t periodic in the first variable we will prove that a weak repulsive singularity enables the achievement of new existence criteria through a basic application of schauder s fixed point theorem

multiplicity of positive solutions for singular three point boundary value problems at resonance abstract in this paper we study the existence and multiplicity of positive solutions for the three point boundary value problems with singular nonlinear perturbations at resonance the existence results are applicable to both the case of a strong singularity and the case of a weak singularity the proof is based on a nonlinear alternative leray schauder principle and a fixed point theorem in cones for completely continuous operators

multiple positive solutions to superlinear periodic boundary value problems with repulsive singular forces abstract the existence of multiple positive solutions to superlinear periodic boundary value problems with repulsive singular forces is discussed in this paper our nonlinearity may be singular in its dependent variable and our analysis relies on a nonlinear alternative of leray schauder type and on a fixed point theorem in cones

singularly perturbed higher order periodic boundary value problems abstract the existence of positive periodic solutions is proved for the higher order singular differential equation 0 1 x n a n 1 x n 1 a 1 x a 0 t x e t and for some generalizations including a class of differential systems with singular cyclic feedback

certified reduced basis methods for parametrized distributed elliptic optimal control problems with control constraints in this paper we employ the reduced basis method for the efficient and reliable solution of parametrized optimal control problems governed by scalar coercive elliptic partial differential equations we consider the standard linear quadratic problem setting with distributed control and unilateral control constraints for this problem class we propose two different reduced basis approximations and associated error estimation procedures in our first approach we directly consider the resulting optimality system introduce suitable reduced basis approximations for the state adjoint control and lagrange multipliers and use a projection approach to bound the error in the reduced optimal control for our second approach we first reformulate the optimal control problem using a slack variable then develop a reduced basis approximation for the slack problem by suitably restricting the solution space and finally derive error bounds for the slack based optimal control we discuss benefits and drawbacks of b

combined effects of singular and superlinear nonlinearities in some singular boundary value problems abstract this paper deals with a class of singular semilinear elliptic dirichlet boundary value problems where the combined effects of a superlinear and a singular term allow us to establish some existence and multiplicity results

greedy optimal control for elliptic problems and its application to turnpike problems we adapt and apply greedy methods to approximate in an efficient way the optimal controls for parameterized elliptic control problems our results yield an optimal approximation procedure that in particular performs better than simply sampling the parameter space to compute controls for each parameter value the same method can be adapted for parabolic control problems but this leads to greedy selections of the realizations of the parameters that depend on the initial datum under consideration the turnpike property which ensures that parabolic optimal control problems behave nearly in a static manner when the control horizon is long enough allows using the elliptic greedy choice of the parameters in the parabolic setting too we present various numerical experiments and an extensive discussion of the efficiency of our methodology for parabolic control and indicate a number of open problems arising when analyzing the convergence of the proposed algorithms

existence of a pair of periodic solutions of an o d e generalizing a problem in nonlinear elasticity via variational methods on considere l existence des solutions t periodiques de l equation differentielle x g t x ou g r r r est continue et t periodique en t sous certaines conditions sur g on demontre l existence d au moins deux solutions t periodiques on utilise une methode variationnelle

estimating the parameters of exponentially damped sinusoids and pole zero modeling in noise we have presented techniques 1 6 based on linear prediction lp and singular value decomposition svd for accurate estimation of closely spaced frequencies of sinusoidal signals in noise in this note we extend these techniques to estimate the parameters of exponentially damped sinusoidal signals in noise the estimation procedure presented here makes use of backward prediction in addition to svd first the method is applied to data consisting of one and two exponentially damped sinusoids the choice of one and two signal components facilitates the comparison of estimation error in pole damping factors and pole frequencies to the appropriate cramer rao cr bounds and to traditional methods of linear prediction second our method is applied to an example due to steiglitz 8 in which the data consists of noisy values of the impulse response samples composed of many exponentially damped sinusoids of a linear system having both poles and zeros the poles of the system are accurately determined by our method and the zeros are obtained subsequently using shanks method

a relationship between the periodic and the dirichlet bvps of singular differential equations in this paper a relationship between the periodic and the dirichlet boundary value problems for second order ordinary differential equations with singularities is established this relationship may be useful in explaining the difference between the nonresonance of singular and nonsingular differential equations using this relationship we give in this paper an existence result of positive periodic solutions to singular differential equations when the singular forces satisfy some strong force condition at the singularity 0 and some linear growth condition at infinity

greedy controllability of finite dimensional linear systems we analyse the problem of controllability for parameter dependent linear finite dimensional systems the goal is to identify the most distinguished realisations of those parameters so to better describe or approximate the whole range of controls we adapt recent results on greedy and weak greedy algorithms for parameter dependent pdes or more generally abstract equations in banach spaces our results lead to optimal approximation procedures that in particular perform better than simply sampling the parameter space to compute the controls for each of the parameter values we apply these results for the approximate control of finite difference approximations of the heat and the wave equation the numerical experiments confirm the efficiency of the methods and show that the number of weak greedy samplings that are required is particularly low when dealing with heat like equations because of the intrinsic dissipativity that the model introduces for high frequencies

periodic solutions for second order differential equations with a singular nonlinearity abstract this paper deals with the existence of positive t periodic solutions for the following differential equation x a t x f t x c t where a c l 1 0 t and f car 0 t r r the existence results are obtained by using a fixed point theorem in cones

quasi periodic solutions of forced isochronous oscillators at resonance abstract we deal with the existence of quasi periodic solutions of forced isochronous oscillators with a repulsive singularity the nonlinearity is a bounded perturbation using a variant of moser s twist theorem of invariant curves due to ortega r ortega boundedness in a piecewise linear oscillator and a variant of the small twist theorem proc london math soc 79 1999 381 413 we show that there are many quasi periodic solutions and the boundedness of all solutions

a minimax method for a class of hamiltonian systems with singular potentials abstract this paper presents a minimax method which gives existence and multiplicity results for time periodic solutions of a class of hamiltonian systems when a singular potential is present the singularity satisfies the strong force condition of gordon when milder singularities are permitted a notion of generalized t periodic solution is introduced and we get existence and multiplicity results for such solutions

existence and stability of periodic solutions for second order semilinear differential equations with a singular nonlinearity it is proved that a periodically forced second order equation with a singular nonlinearity in the origin with linear growth in infinity possesses a t periodic stable solution for high values of the mean value of the forcing term the method of proof combines a rescaling argument with the analysis of the first twist coefficient of the

reduced basis method and a posteriori error estimation for parametrized linear quadratic optimal control problems we propose the reduced basis method for the solution of parametrized optimal control problems described by parabolic partial differential equations in the unconstrained case the method which is based on an off line on line decomposition procedure allows at the on line step large computational cost savings with respect to the truth approximation used for defining the reduced basis an a posteriori error estimate is provided by means of the goal oriented analysis thus associating an error bound to each optimal solution of the parametrized optimal control problem and answering to the demand for a reliable method an adaptive procedure led by the a posteriori error estimate is considered for the generation of the reduced basis space which is set according to the optimal primal and dual solutions of the optimal control problem at hand

existence of overtaking solutions to infinite dimensional control problems on unbounded time intervals the optimal control of an infinite dimensional linear control system with unbounded time interval is considered the results obtained are an extension to a hilbert space setting of the results reported in brock and haurie math oper res 1 1976 pp 337 346 in particular it is shown that under appropriate conditions overtaking optimal trajectories asymptotically approach a unique optimal steady state this is the so called turnpike property found in the economics literature when this condition is combined with an associated optimal control problem results concerning the existence of overtaking optimal solutions are obtained both distributed and boundary controls are considered finally we demonstrate the applicability of our results by considering several examples arising in the context of economics and resources management

dynamical systems with effective like potentials on etudie l existence des solutions t periodiques pour des systemes hamiltoniens d ordre 2 de la forme x v x ou v r n 0 r est de classe c 2 et v x quand x 0

radially symmetric systems with a singularity and asymptotically linear growth abstract we prove the existence of infinitely many periodic solutions for radially symmetric systems with a singularity of repulsive type the nonlinearity is assumed to have a linear growth at infinity being controlled by two constants which have a precise interpretation in terms of the dancer fucik spectrum our result generalizes an existence theorem by del pino et al 1992 4 obtained in the case of a scalar second order differential equation

reduced basis method for parametrized elliptic optimal control problems we propose a suitable model reduction paradigm the certied reduced basis method rb for the rapid and reliable solution of parametrized optimal control problems governed by partial dierential equations pdes in particular we develop the methodology for parametrized quadratic optimization problems with elliptic equations as constraint and infinite dimensional control variable firstly we recast the optimal control problem in the framework of saddle point problems in order to take advantage of the already developed rb theory for stokes type problems then the usual ingredients of the rb methodology are called into play a galerkin projection onto a low dimensional space of basis functions properly selected by an adaptive procedure an affine parametric dependence enabling to perform competitive offline online splitting in the computational procedure an efficient and rigorous a posteriori error estimate on the state control and adjoint variables as well as on the cost functional finally we address some numerical tests that confirm our theoretical results and show the efficiency of the proposed technique

the turnpike property in finite dimensional nonlinear optimal control turnpike properties have been established long time ago in finite dimensional optimal control problems arising in econometry they refer to the fact that under quite general assumptions the optimal solutions of a given optimal control problem settled in large time consist approximately of three pieces the first and the last of which being transient short time arcs and the middle piece being a long time arc staying exponentially close to the optimal steady state solution of an associated static optimal control problem we provide in this paper a general version of a turnpike theorem valuable for nonlinear dynamics without any specific assumption and for very general terminal conditions not only the optimal trajectory is shown to remain exponentially close to a steady state but also the corresponding adjoint vector of the pontryagin maximum principle the exponential closedness is quantified with the use of appropriate normal forms of riccati equations we show then how the property on the adjoint vector can be adequately used in order to initialize successfully a numerical direct method or a shooting method in particular we provide an appropriate variant of the usual shooting method in which we initialize the adjoint vector not at the initial time but at the middle of the trajectory

periodic solutions of duffing equations with semi quadratic potential and singularity abstract in this paper we deal with the existence of periodic solutions of the second order differential equations x g x p t with singularity we prove that the given equation has at least one periodic solution when g x has singularity at origin g x 1 x g s d s satisfies n 2 4 lim inf x 2 g x x 2 lim sup x 2 g x x 2 n 1 2 4 for large positive x where n 0 is an integer and g x tends to positive infinity as x 0

a multiplicity result for periodic solutions of second order differential equations with a singularity abstract by the use of the poincare birkhoff fixed point theorem we prove a multiplicity result for periodic solutions of a second order differential equation where the nonlinearity exhibits a singularity of repulsive type at the origin and has linear growth at infinity our main theorem is related to previous results by rebelo 1996 1997 4 5 and rebelo and zanolin 1996 6 7 in connection with a problem raised by del pino et al 1992 1

on the 2 dimensional dual minkowski problem abstract we study the 2 dimensional dual minkowski problem which is the following nonlinear problem on unit circle u u g u 1 u 2 u 2 2 k 2 s for any given positive continuous function g with 2 m periodic we prove that it is solvable for all k 1 and m 3 4 5

periodic orbits of radially symmetric keplerian like systems a topological degree approach abstract we are concerned with non autonomous radially symmetric systems with a singularity which are t periodic in time by the use of topological degree theory we prove the existence of large amplitude periodic solutions whose minimal period is an integer multiple of t precise estimates are then given in the case of keplerian like systems showing some resemblance between the orbits of those solutions and the circular orbits of the corresponding classical autonomous system

twist periodic solutions of second order singular differential equations abstract we study the twist periodic solutions of second order singular differential equations such twist periodic solutions are stable in the sense of lyapunov and present much interesting dynamical features around them the proof is based on the third order approximation method the estimates of periodic solutions of ermakov pinney equations and the estimates on rotation numbers of hill equations play an important role in the analysis

estimating the steady state frequency of a sine wave burst with extremely short record length we present a technique based on modified linear prediction for accurate estimation of the steady state frequency of a sine wave burst with extremely short record length the estimation procedure presented here makes use of automatic windowing of the burst location within the observation interval such a windowing technique eliminates the transition effects which accompany the estimators derived using the traditional lpc technique to implement this windowing a set of artificial state variables given by the product of output samples and their delayed versions is used the advantages of this method are demonstrated using a computer simulation

positive solutions to a singular neumann problem abstract we show the existence of positive solution for the following class of singular neumann problem u a x u h x u p in b r with u 0 on b r where r 0 0 is a positive parameter 0 p 0 1 b r b r 0 r n a b r r and h b r r are radially symmetric nonnegative c 1 functions using variational methods and sub and supersolutions we obtain a solution for an approximated problem involving mixed boundary conditions the limit of the approximated solutions is a positive solution

a neumann problem with logarithmic nonlinearity in a ball abstract the nonlinear elliptic problem u log u h x u p in a ball b r r n with neumann boundary conditions u 0 on b r were studied by means of a double perturbation argument we obtain a positive radial solution u c 2 b r 0 c b r

positive bound states of systems of nonlinear schrodinger equations abstract we study the existence of positive bound states of non autonomous systems of nonlinear schrodinger equations both the singular case and the regular case are discussed the proof is based on a nonlinear alternative principle of leray schauder

periodic solutions of superlinear impulsive differential equations a geometric approach abstract a geometric method is introduced to study superlinear second order differential equations with impulsive effects basing on a reference continuous polar lifting of a planar orientation preserving homeomorphism we prove via the poincare birkhoff twist theorem the existence of infinitely many periodic solutions of conservative superlinear second order equations with the finite twist impulsive terms we also prove by developing a new twist fixed point theorem the existence of periodic solutions for non conservative superlinear second order equations with degenerate impulsive terms

positive solutions to superlinear semipositone periodic boundary value problems with repulsive weak singular forces this paper is devoted to study the existence of positive solutions to the second order semipositone periodic boundary value problem x a t x f t x x 0 x 1 x 0 x 1 here f t x may be singular at x 0 and may be superlinear at x our analysis relies on a fixed point theorem in cones

existence of nonnegative and nonpositive solutions for second order periodic boundary value problems we are interested in nonnegative and nonpositive solutions of the boundary value problem u f t u u 0 u 1 u 0 u 1 where f fulfils the caratheodory conditions on 0 1 r we generalize the results reached by m n nkashama j santanilla and l sanchez and present estimates for solutions in addition we apply our existence theorems to periodic boundary value problems for nonlinear duffing equations whose right hand sides have a repulsive or attractive singularity at the origin we extend or generalize existence results by a c lazer and s solimini and other authors moreover we get some multiplicity results and in the case of a repulsive singularity we also admit a weak singularity in constrast to the previous papers on this subject our proofs are based on the method of lower and upper functions and topological degree arguments and the results are tested on examples

periodic motion of a system of two or three charged particles we provide necessary and sufficient conditions for the existence of t periodic solutions of a system of second order ordinary differential equations that models the motion of two or three collinear charged particles of the same sign

periodic solutions for second order singular damped differential equations abstract we study the existence of positive periodic solutions for second order singular damped differential equations by combining the analysis of the sign of green s functions for the linear damped equation together with a nonlinear alternative principle of leray schauder recent results in the literature are generalized and significantly improved

an elliptic singular system with nonlocal boundary conditions abstract we study the existence of solutions for the nonlinear second order elliptic system u g u f x where g c r n s r n with s r n bounded using topological degree methods we prove an existence result under a geometric condition on g moreover we analyze the particular case of an isolated repulsive singularity under a nirenberg type condition we prove the existence of a sequence of solutions of appropriate approximated problems that converges to a generalized solution

nontrivial solutions of operator equations and morse indices of critical points of min max type on montre que si les hypotheses du theoreme de point selle sont satisfaites si dim y 2 et si f 1 c contient seulement des points critiques non degeneres alors f 1 c contient au moins un point critique u 0 tel que 2 indice de morse u 0 dim y

a parameter estimation approach to estimation of frequencies of sinusoids it is shown that with uniform sampling a sample of the sum of m sinusoids at time nt can be uniquely expressed as a linear combination of the 2m samples at times n 1 t n 2m t the 2m coefficients of linear combination are also the coefficients of a 2m order polynomial whose roots are e pmj omega i omega i being the frequencies of the sinusoids given the samples of sinusoids plus noise a consistent estimate of the 2m coefficients are obtained by the instrumental variable method of parameter estimation it is also possible to track time varying frequencies by a recursive algorithm that exponentially weighs out the past data simulation examples of one and two sinusoids are given

positive periodic solutions of hill s equations with singular nonlinear perturbations abstract we study the existence and multiplicity of positive periodic solutions of hill s equations with singular nonlinear perturbations the new results are applicable to the case of a strong singularity as well as the case of a weak singularity the proof relies on a nonlinear alternative principle of leray schauder and a fixed point theorem in cones some recent results in the literature are generalized and improved

non collision periodic solutions of second order singular dynamical systems we study the existence of non collision periodic solutions for second order singular dynamical systems the repulsive case and the attractive case are dealt with using a unified topological approach the proof is based on a well known fixed point theorem for completely continuous operators involving a new type of cone we do not need to consider so called strong force conditions moreover for the repulsive case the critical case can be covered recent results in the literature even in the scalar case are complemented generalized and improved

periodic solutions of second order non autonomous singular dynamical systems abstract in this paper we establish two different existence results of positive periodic solutions for second order non autonomous singular dynamical systems the first one is based on a nonlinear alternative principle of leray schauder and the result is applicable to the case of a strong singularity as well as the case of a weak singularity the second one is based on schauder s fixed point theorem and the result sheds some new light on problems with weak singularities and proves that in some situations weak singularities may help create periodic solutions recent results in the literature are generalized and significantly improved