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CPAA

We study the exponential stability of the Timoshenko beam system by interior
time-dependent delay term feedbacks. The beam is
clamped at the two hand points subject to two internal feedbacks: one
with a time-varying delay and the other without delay.
Using the variable norm technique of Kato, it is proved that the system is well-posed
whenever an hypothesis between the weight of the delay term in the feedback,
the weight of the term without delay and the wave speeds.
By introducing an appropriate Lyapunov functional the exponential stability of the system is proved.
Under the imposed constrain on the weights of the
feedbacks and the wave speeds, the exponential decay of the energy is
established via a suitable Lyapunov functional.

EECT

In this paper, we investigate a backward problem for a fractional abstract evolution equation for which we wants to extract the initial distribution from the observation data provided along the final time $t = T.$ This problem is well-known to be ill-posed due to the rapid decay of the forward process. We consider a final value problem for fractional evolution process with respect to time. For this ill-posed problem, we construct two regularized solutions using quasi-reversibility method and quasi-boundary value method. The well-posedness of the regularized solutions as well as the convergence property is analyzed. The advantage of the proposed methods is that the regularized solution is given analytically and therefore is easy to be implemented. A numerical example is presented to show the validity of the proposed methods.

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