Browsing by Subject "Wave equation"

Now showing 1 - 4 of 4

Open Access
Decay rate estimates for the wave equation with subcritical semilinearities and locally distributed nonlinear dissipation
(Springer, 2023-11-07) Cavalcanti, M. M.; Domingos Cavalcanti, V. N.; Gonzalez Martinez, V. H.; Özsarı, Türker
We study the stabilization and the wellposedness of solutions of the wave equation with subcritical semilinearities and locally distributed nonlinear dissipation. The novelty of this paper is that we deal with the difficulty that the main equation does not have good nonlinear structure amenable to a direct proof of a priori bounds and a desirable observability inequality. It is well known that observability inequalities play a critical role in characterizing the long time behaviour of solutions of evolution equations, which is the main goal of this study. In order to address this, we truncate the nonlinearities, and thereby construct approximate solutions for which it is possible to obtain a priori bounds and prove the essential observability inequality. The treatment of these approximate solutions is still a challenging task and requires the use of Strichartz estimates and some microlocal analysis tools such as microlocal defect measures. We include an appendix on the latter topic here to make the article self contained and supplement details to proofs of some of the theorems which can be already be found in the lecture notes of Burq and Gérard (http://www.math.u-psud.fr/~burq/articles/coursX.pdf, 2001). Once we establish essential observability properties for the approximate solutions, it is not difficult to prove that the solution of the original problem also possesses a similar feature via a delicate passage to limit. In the last part of the paper, we establish various decay rate estimates for different growth conditions on the nonlinear dissipative effect. We in particular generalize the known results on the subject to a considerably larger class of dissipative effects.
Open Access
Robust stabilization of the wave equation against small delays
(IEEE, 1994) Morgül, Ömer
In this paper we consider a system which can be modeled by (undamped) wave equation in a bounded domain. We assume that the system is fixed at one end and is controlled by a boundary controller at the other end. We also considered two damped versions of this system, both parameterized by a nonnegative damping constant. We study two problems for these models, namely the stabilization by means of a boundary controller, and the stability robustness of the closed-loop system against small time delays in the feedback loop. We propose a class of finite dimensional dynamic boundary controllers to solve these problems. One basic feature of these controllers is that the corresponding controller transfer functions are required to be strictly positive real functions. We show that these controllers stabilize both damped and undamped models and solve the stability robustness problem for the damped models. It is also shown that while strict positive realness of the controller transfer functions is important for closed-loop stability, the strict properness is important for the stability robustness against small time delays in the feedback loop.
Open Access
Stabilization and disturbance rejection for the wave equation
(IEEE, 1994) Morgül, Ömer
We consider a system described by the one dimensional linear wave equation in a bounded domain with appropriate boundary conditions. To stabilize the system, we propose a dynamic boundary controller applied at the free end of the system. We also consider the case where the output of the controller is corrupted by a disturbance and show that it may be possible to attenuate the effect of the disturbance at the output if we choose the controller transfer function appropriately.
Open Access
Ultimate intrinsic SNR in magnetic resonance imaging by optimizing the EM field generated by internal coils
(2000) Abdel-Hafez, Imad Amin
A method to find the ultimate intrinsic signal-to-noise ratio (ISNR) in a magnetic resonance imaging experiment is applied to a human body model. The method uses cylindrical wave expansion to represent an arbitrary electromagnetic field inside the body. This field is optimized to give the maximum possible ISNR for some point of interest from which the signal is received, and repeated for all points inside the body. Optimization is conducted by finding the set of coefficients associated with expansion modes that give the maximum ISNR. Application of this method enables the determination of the ultimate ISNR and the associated optimum electromagnetic field without the necessity of finding the receiving coil configuration needed to obtain the ultimate value of ISNR. Results of this work can be used to examine the efficiency of already available commercial coils and how far they can be improved. Moreover, the solution can be used to determine the performance difference between internal and external Magnetic Resonance Imaging (MRI) coils. Finally, knowledge of the optimum electromagnetic field inside the human body can be used to find the coil configuration that can radiate this field by solving an inverse problem.