Chaotic dynamical systems associated with tilings of RN\R^N

In this chapter, we consider a class of discrete dynamical systems defined on the homogeneous space associated with a regular tiling of $\R^N$, whose most familiar example is provided by the $N-$dimensional torus \T ^N. It is proved that any dynamical system in this class is chaotic in the sense of Devaney, and that it admits at least one positive Lyapunov exponent. Next, a chaos-synchronization mechanism is introduced and used for masking information in a communication setup

On the Benjamin-Bona-Mahony equation with a localized damping

We introduce several mechanisms to dissipate the energy in the Benjamin-Bona-Mahony (BBM) equation. We consider either a distributed (localized) feedback law, or a boundary feedback law. In each case, we prove the global wellposedness of the system and the convergence towards a solution of the BBM equation which is null on a band. If the Unique Continuation Property holds for the BBM equation, this implies that the origin is asymp-totically stable for the damped BBM equation

Exploring mechanisms responsible for tidal modulation in flow of the Filchner–Ronne Ice Shelf

An extensive network of GPS sites on the Filchner–Ronne Ice Shelf and adjoining ice streams shows strong tidal modulation of horizontal ice flow at a range of frequencies. A particularly strong (horizontal) response is found at the fortnightly (Msf) frequency. Since this tidal constituent is absent in the (vertical) tidal forcing, this observation implies the action of some non-linear mechanism. Another striking aspect is the strong amplitude of the flow perturbation, causing a periodic reversal in the direction of ice shelf flow in some areas and a 10 %–20 % change in speed at grounding lines. No model has yet been able to reproduce the quantitative aspects of the observed tidal modulation across the entire Filchner–Ronne Ice Shelf. The cause of the tidal ice flow response has, therefore, remained an enigma, indicating a serious limitation in our current understanding of the mechanics of large-scale ice flow. A further limitation of previous studies is that they have all focused on isolated regions and interactions between different areas have, therefore, not been fully accounted for. Here, we conduct the first large-scale ice flow modelling study to explore these processes using a viscoelastic rheology and realistic geometry of the entire Filchner–Ronne Ice Shelf, where the best observations of tidal response are available. We evaluate all relevant mechanisms that have hitherto been put forward to explain how tides might affect ice shelf flow and compare our results with observational data. We conclude that, while some are able to generate the correct general qualitative aspects of the tidally induced perturbations in ice flow, most of these mechanisms must be ruled out as being the primary cause of the observed long-period response. We find that only tidally induced lateral migration of grounding lines can generate a sufficiently strong long-period Msf response on the ice shelf to match observations. Furthermore, we show that the observed horizontal short-period semidiurnal tidal motion, causing twice-daily flow reversals at the ice front, can be generated through a purely elastic response to basin-wide tidal perturbations in the ice shelf slope. This model also allows us to quantify the effect of tides on mean ice flow and we find that the Filchner–Ronne Ice Shelf flows, on average, ∼ 21 % faster than it would in the absence of large ocean tides

Unique continuation property and control for the Benjamin-Bona-Mahony equation on the torus

We consider the Benjamin-Bona-Mahony (BBM) equation on the one dimensional torus T = R/(2{\pi}Z). We prove a Unique Continuation Property (UCP) for small data in H^1(T) with nonnegative zero means. Next we extend the UCP to certain BBM-like equations, including the equal width wave equation and the KdV-BBM equation. Applications to the stabilization of the above equations are given. In particular, we show that when an internal control acting on a moving interval is applied in BBM equation, then a semiglobal exponential stabilization can be derived in H^s(T) for any s \geq 1. Furthermore, we prove that the BBM equation with a moving control is also locally exactly controllable in H^s(T) for any s \geq 0 and globally exactly controllable in H s (T) for any s \geq 1

Control and Stabilization of the Nonlinear Schroedinger Equation on Rectangles

This paper studies the local exact controllability and the local stabilization of the semilinear Schr\"odinger equation posed on a product of $n$ intervals ($n\ge 1$). Both internal and boundary controls are considered, and the results are given with periodic (resp. Dirichlet or Neumann) boundary conditions. In the case of internal control, we obtain local controllability results which are sharp as far as the localization of the control region and the smoothness of the state space are concerned. It is also proved that for the linear Schr\"odinger equation with Dirichlet control, the exact controllability holds in $H^{-1}(\Omega)$ whenever the control region contains a neighborhood of a vertex

On the reachable states for the boundary control of the heat equation

We are interested in the determination of the reachable states for the boundary control of the one-dimensional heat equation. We consider either one or two boundary controls. We show that reachable states associated with square integrable controls can be extended to analytic functions onsome square of C, and conversely, that analytic functions defined on a certain disk can be reached by using boundary controlsthat are Gevrey functions of order 2. The method of proof combines the flatness approach with some new Borel interpolation theorem in some Gevrey class witha specified value of the loss in the uniform estimates of the successive derivatives of the interpolating function

Portable device for use in starting air-start-units for aircraft and having cable lead testing capability

A portable device for starting aircraft engines and the like is disclosed. The device includes a lead testing and motor starting circuit characterized by: (1) a direct current voltage source, (2) a pair of terminal plugs connected with the circuit (each being characterized by a first, second, and third terminal) (3) a pair of manually operable switches for connecting the first terminal of each plug of the pair to the positive side of the voltage source, (4) a circuit lead connecting to the second terminal of each plug the negative side of said source, (5) a pair of electrical cables adapted to connect said first and second terminals of each plug to an air-start unit, and means for connecting each cable of the pair of cables between the first terminal of one plug and the third terminal of the other plug of the pair, and (6) a second pair of manually operable switches for selectivity connecting the third terminal of each plug of the pair to the negative side of the voltage source

Null controllability of one-dimensional parabolic equations by the flatness approach

We consider linear one-dimensional parabolic equations with space dependent coefficients that are only measurable and that may be degenerate or singular.Considering generalized Robin-Neumann boundary conditions at both extremities, we prove the null controllability with one boundary control by following the flatness approach, which providesexplicitly the control and the associated trajectory as series. Both the control and the trajectory have a Gevrey regularity in time related to the $L^p$ class of the coefficient in front of $u\_t$.The approach applies in particular to the (possibly degenerate or singular) heat equation $(a(x)u\_x)\_x-u\_t=0$ with a(x)\textgreater{}0 for a.e. $x\in (0,1)$ and $a+1/a \in L^1(0,1)$, or to the heat equation with inverse square potential $u\_{xx}+(\mu / |x|^2)u-u\_t=0$with $\mu\ge 1/4$

Null controllability of the 1D heat equation using flatness

We derive in a straightforward way the null controllability of a 1-D heat equation with boundary control. We use the so-called {\em flatness approach}, which consists in parameterizing the solution and the control by the derivatives of a "flat output". This provides an explicit control law achieving the exact steering to zero. We also give accurate error estimates when the various series involved are replaced by their partial sums, which is paramount for an actual numerical scheme. Numerical experiments demonstrate the relevance of the approach