In this paper we study linear conservative systems of finite dimension coupled with an infinite dimensional system of diffusive type. Computing the time-derivative of an appropriate energy functional along the solutions helps us to prove the well-posedness of the system and a stability property. But in order to prove asymptotic stability we need to apply a sufficient spectral condition. We also illustrate the sharpness of this condition by exhibiting some systems for which we do not have the asymptotic...
In this paper we study linear conservative systems of finite
dimension
coupled with an infinite dimensional system of diffusive type.
Computing the time-derivative of an
appropriate energy functional along the solutions helps us to
prove the well-posedness of the system
and a stability property.
But in order to prove asymptotic stability we need to apply
a sufficient spectral condition. We also illustrate the sharpness of this
condition by exhibiting some systems for which we do not have the asymptotic
property.
...
We consider a controllability problem for a beam, clamped at one boundary and
free at the other boundary, with an attached piezoelectric actuator. By
Hilbert Uniqueness Method (HUM)
and new results on diophantine approximations, we
prove that the space of exactly initial controllable data depends on the
location of the actuator. We also illustrate these results with numerical
simulations.
We consider a tank containing a fluid. The tank is subjected to a one-dimensional horizontal move and the motion of the fluid is described by the shallow water equations. By means of a Lyapunov approach, we deduce control laws to stabilize the fluid's state and the tank's position. Although global asymptotic stability is yet to be proved, we numerically simulate the system and observe the stabilization for different control situations.
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