13,175 research outputs found
Boundary controllability of Maxwell's equations with nonzero conductivity inside a cube, I: Spectral controllability
AbstractThis is a first paper in a series of two. In both papers, we consider the question of control of Maxwell's equations in a homogeneous medium with positive conductivity by means of boundary surface currents. The domain under consideration is a cube, where the conductivity is allowed to take on any nonnegative value. An additional restriction imposed in order to make this problem more suitable for practical implementations is that the controls are applied over only one face of the cube. In this paper, the method of moments is employed to establish spectral controllability for the above case (meaning that any finite combination of eigenfunctions is controllable). In the companion paper [S.S. Krigman, C.E. Wayne, Boundary controllability of Maxwell's equations with nonzero conductivity inside a cube, II: Lack of exact controllability and controllability for very smooth solutions, J. Math. Anal. Appl. (2006), doi:10.1016/j.jmaa2006.02.102] it will be established, by modifying the calculations in [H.O. Fattorini, Estimates for sequences biorthogonal to certain complex exponentials and boundary control of the wave equation, in: New Trends in Systems Analysis, Proceedings of the International Symposium, Versailles, 1976, in: Lecture Notes in Control and Inform. Sci., vol. 2, Springer, Berlin, 1977, pp. 111–124], that exact controllability fails for this geometry regardless of the size of the conductivity term. However, we will also establish in [S.S. Krigman, C.E. Wayne, Boundary controllability of Maxwell's equations with nonzero conductivity inside a cube, II: Lack of exact controllability and controllability for very smooth solutions, J. Math. Anal. Appl. (2006), doi:10.1016/j.jmaa2006.02.102] controllability of solutions that are smooth enough that the Fourier coefficients of their initial data decay at a suitable exponential rate
Remarks on global controllability for the shallow-water system with two control forces
In this paper we deal with the compressible Navier-Stokes equations with a
friction term in one dimension on an interval. We study the exact
controllability properties of this equation with general initial condition when
the boundary control is acting at both endpoints of the interval. Inspired by
the work of Guerrero and Imanuvilov in \cite{GI} on the viscous Burger
equation, we prove by choosing irrotational data and using the notion of
effective velocity developed in \cite{cpde,cras} that the exact global
controllability result does not hold for any time
Fast global null controllability for a viscous Burgers' equation despite the presence of a boundary layer
In this work, we are interested in the small time global null controllability
for the viscous Burgers' equation y_t - y_xx + y y_x = u(t) on the line segment
[0,1]. The second-hand side is a scalar control playing a role similar to that
of a pressure. We set y(t,1) = 0 and restrict ourselves to using only two
controls (namely the interior one u(t) and the boundary one y(t,0)). In this
setting, we show that small time global null controllability still holds by
taking advantage of both hyperbolic and parabolic behaviors of our system. We
use the Cole-Hopf transform and Fourier series to derive precise estimates for
the creation and the dissipation of a boundary layer
Controllability cost of conservative systems: resolvent condition and transmutation
This article concerns the exact controllability of unitary groups on Hilbert
spaces with unbounded control operator. It provides a necessary and sufficient
condition not involving time which blends a resolvent estimate and an
observability inequality. By the transmutation of controls in some time L for
the corresponding second order conservative system, it is proved that the cost
of controls in time T for the unitary group grows at most like \exp(\alpha
L^{2}/T) as T tends to 0. In the application to the cost of fast controls for
the Schr{\"o}dinger equation, L is the length of the longest ray of geometric
optics which does not intersect the control region. This article also provides
observability resolvent estimates implying fast smoothing effect
controllability at low cost, and underscores that the controllability cost of a
system is not changed by taking its tensor product with a conservative system.Comment: 20 pages, a4paper, typos corrected in lem.5.2, lem.5.3, th.10.
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