Scanning control of a vibrating string

We construct scanning feedback controls {γ _i (t)} for the vibrating string equation $$\begin{gathered} y_{tt} (x,t) = y_{xx} (x,t) + Ry(x,t) + \sum\limits_{i = 1}^N {\phi (x - \gamma _i } (t))y(x,t), \hfill \\ 0< x< 1,y = 0 at x = 0,1. \hfill \\ \end{gathered} $$ so that (y, y _t ) → (0,0) ast → ∞ in the weak topology ofH ₀ ¹ (0,1) ×L ₂ (0,1). In particular we show that ifφ is an even polynomial of degreeN with nonpositive coefficients that forR <π ² we can find such stabilizingγ _i (t), i=1,?,N.     

Analysis of a nonlinear model of the vibrating string

A nonlinear model of the vibrating string is studied under the assumption that the motion is transversal and existence and uniqueness theorems are given for the Cauchy-Dirichlet problems. Some numerical experiments are also described, illustrating the behaviour of this model with respect to the nonlinear Kirchhoff model and the classical linear model of D'Alembert.The research has been supported by MURST 40% and 60% Research Contracts     

Flatness-based control without prediction: example of a vibrating string

Stable trajectory tracking by boundary control is discussed for a string with a mass at its free end. Based on the known fact that the ring of operators used to describe the system is a Bézout ring it is shown that predictions are not required for stabilization if distributed delays are admitted. The method is rather general for systems of boundary coupled wave equations with boundary control that can be modeled as delay systems with commensurate delays. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Motion of a string vibrating against a rigid fixed obstacle

We consider a string, fixed at both ends and moving in a plane in presence of a straight fixed obstacle placed on the equilibrium position of the string; the rebound of the string on the obstacle obeys the law of perfect reflection. The string being initially at rest in an arbitrary shape, we prove that the motion is periodic with the same period that the free oscillations.     

On the characterizations of the breakdown points of nonlinear vibrating string

We consider the mixed initial and boundary value problem of a hyperbolic 2-conservation law which describes the motion of a model of nonlinear vibrating string. It is known that solutions of such problems eventually break down in the sense that some of their first-order derivatives become unbounded at finite time. We call a point at which the breakdown first occurs a breakdown point. We prove that there are at most finitely many breakdown points. We also characterize such points in regard to existence or nonexistence of shock curves.     

Formation of singularities for wave equations including the nonlinear vibrating string

Under very general assumptions, the authors prove that smooth solutions of quasilinear wave equations with small-amplitude periodic initial data always develop singularities in the second derivatives in finite time. One consequence of these results is the fact that all solutions of the classical nonlinear vibrating string equation satisfying either Dirichlet or Neumann boundary conditions and with sufficiently small nontriviai initial data necessarily develop singularities. In particular, there are no nontrivial smooth small-amplitude time-periodic solutions.     

Study of motions of a vibrating string subject to solid friction

In this paper we consider a string moving in its plane and subject to solid friction (Coulomb's law). It is known that when the time increases indefinitely the string reaches an equilibrium position and by analogy with the case of a mass point we ask if the equilibrium is reached after a finite time. We prove that this is the case when the string is initially at rest and 1. when the initial shape possesses a second derivative bounded by certain limits, or 2. when the initial shape is formed by two straight line segments. In the last section we obtain some partial results when the string is initially at rest in the shape of a polygonal line. The case of an arbitrary initial position is still an open problem.     

The maximal Lyapunov exponent for a three-dimensional system driven by white noise

In this paper, based on a perturbation method, the asymptotic expansions of the invariant measure and the maximal Lyapunov exponent for a three-dimensional system excited by a white noise are evaluated. All possible singular boundaries of the first or the second kind that exist in the one-dimensional phase diffusion process are considered and the results of the maximal Lyapunov exponent are obtained. In addition, the P-bifurcation behaviors are investigated.     

Boundary control of forced string vibrations by first directional derivatives on a short time interval

We solve a problem on the boundary control of forced vibrations of a homogeneous string by two first directional derivatives with time-dependent coefficients and noncharacteristic directions in the boundary conditions on a short time interval in the set of classical solutions. We obtain smoothness, coordination, and controllability conditions on the right-hand side of the equation and the initial and terminal data necessary and sufficient for the unique boundary control of the wave process.     

A delta white noise functional

The additive renormalization% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiaabs7adaWgaaWcbaGaaeySdiaab6cacaqG0bqefeKCPfgBaGqb% diaa-bcaaeqaaOGaeyypa0Jaa8hiaiaacIcacaaIYaGaeqiWdaNaai% ykamaaCaaaleqabaGaeyOeI0IaaGymaiaac+cacaaIYaaaaGqadOGa% a4hiaiGacwgacaGG4bGaaiiCaiaacIcacqGHsislcaqGXoWaaWbaaS% qabeaacaqGYaaaaOGaai4laiaaikdacaGGPaGaa4hiaiaacQdaciGG% LbGaaiiEaiaacchacqGHXcqSdaWadiqaaiabgkHiTiaadkeacaGGNa% GaaiikaiaadshacaGGPaWaaWbaaSqabeaacaaIYaaaaOGaai4laiaa% ikdacaGFGaGaey4kaSIaa4hiaiaabg7acaWGcbGaai4jaiaacIcaca% WG0bGaaiykaaGaay5waiaaw2faaiaacQdaaaa!6C5C!\[{\rm{\delta }}_{{\rm{\alpha }}{\rm{.t}} } = (2\pi )^{ - 1/2} \exp ( - {\rm{\alpha }}^{\rm{2}} /2) :\exp \pm \left[ { - B'(t)^2 /2 + {\rm{\alpha }}B'(t)} \right]:\]is shown to be a generalized Brownian functional. Some of its properties are derived. is shown to be a generalized Brownian functional. Some of its properties are derived.On leave from Universidade do Minho, Area de Matematica, Largo Carlos Amarante, P-4700 Braga, Portugal.     

Periodic motions of a string vibrating against a fixed point-mass obstacle: II

A string fixed at both ends A and B, can oscillate in a plane which there is a fixed point obstacle, placed in the middle of the line AB. The string is initially at rest with a prescribed shape, symmetric with respect to the normal mid-plane of the segment AB. Using results established before [9] we find new periodic motions.     

A hyperbolic problem of second order with unilateral constraints: The vibrating string with a concave obstacle

The motion of a vibrating string constrained to remain above a material concave obstacle is studied. It is assumed that the string does not lose energy when it hits the obstacle. A set of natural inequations describes this model; an energy condition in an ad hoc form must be added to ensure uniqueness. Existence and uniqueness are proved for the Cauchy problem; the case of an infinite string and the case of a finite string with fixed ends are considered.