Exponential stabilization of solutions for the 1-D transmission wave equation with boundary feedback

The purpose of this work is to study the exponential decay of the energy for the one-dimensional transmission wave equation with a boundary velocity feedback. Thanks to the perturbed energy method developed by some authors in several contexts, and under certain conditions, we prove that the feedback controller exponentially stabilizes the equilibrium to zero of the system below, i.e. the feedback leads to faster energy decay.

The two constants a and b called the wave speeds in (0, L/2), (L/2, L) respectively, λ is the control gain, and the function φ = −λv t (L, t) represents the feedback control.
Let us point out that in physics, feedback means the return of a portion of the output of a circuit or device to its input, and a system in which the value of some output quantity is controlled by feeding back the value of the controlled quantity and using it to manipulate an input quantity so as to bring the value of the controlled quantity closer to a desired value.Also known as closed-loop control system (see [15]).
In recent years, questions of stabilization and decay of energy of solutions for hyperbolic equations, in particular, wave models, have been studied by many mathematicians, by using methods different.
There exists several degrees of stability that one can study.The first degree consists at analyze merely the decreasing of the energy of the solutions towards zero, i.e. : For the second, one studies intermediate situations in which the solutions decreases of the polynomial type for example: , for t > 0, Where C And α Are positive constants with C depends on the initial data.In this case, one must take initial data more regular in the operator's domain.As for the third, one is been interested in the decreasing of the fastest energy, namely when this one tends to 0 in an exponential manner i.e. : where C and δ are positive constants with C depends on the initial data.We wish to stabilze the system ((1.1)-(1.5)), we seek a suitable feedback such that for any initial data (of finite energy E(0) < ∞), the energy of the solution of the problem ((1.1) -(1.5)) tends to zero exponentially as t → 0 (see [8]).
In this research we show how the feedback controller exponentially stabilizes the system ((1.1)-(1.5)), under suitable conditions.The well-posedness of problem ((1.1) -(1.5)) is by now well known in the case where a = b (see [2], [10]), and can be similarly treated without any difficulty in the case where a = b.We define the energy functional E(t) of the system ((1.1)-(1.5)):(see [16]) and construct the following perturbed energy functional E (see [7]) where is a positive constant, choosing sufficiently small.

Preliminaries
Before proving the below main result theorem, we first establish the following lemmas.
Proof.We examine the derivative of the energy x (x, t) dx, using the identities we get and then the energy is decreasing with time, i.e., E(t) ≤ E(0) for all t ≥ 0.
Proof.We have by applying Young's inequality 2.1, we derive that finally, we get

Main results
We now in position to announce our result.
Proof.Differentiating (1.6) with respect to t, we obtain Moreover, by (1.1) yields x dx, by integrating by parts, we obtain Similarly, we have By