关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

https://repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/232625/1/j.automatica.2017.04.017.pdf

Boundary cooperative control by flexible Timoshenko arms

Motivated by potential applications to partial differential equations, we develop a theory of fine scales of decay rates for operator semigroups. The theory contains, unifies, and extends several notable results in the literature on decay of operator semigroups and yields a number of new ones. Its core is a new operator-theoretical method of deriving rates of decay combining ingredients from functional calculus, and complex, real and harmonic analysis. It also leads to several results of independent interest.

https://www.ems-ph.org/journals/show_pdf.php?issn=1435-9855&vol=18&iss=4&rank=6

Fine scales of decay of operator semigroups

In this paper, we study the well-posedness and asymptotic stability of a thermoelastic laminated beam with past history. For the system with structural damping, without any restriction on the speeds of wave propagations, we prove the exponential and polynomial stabilities which depend on the behavior of the kernel function of the history term, by using the perturbed energy method. For the system without structural damping, we prove the exponential and polynomial stabilities in case of equal speeds and lack of exponential stability in case of non-equal speeds by using the perturbed energy method and Gearhart–Herbst–Pruss–Huang theorem, respectively. Furthermore, the well-posedness of the system is also obtained by using Lumer–Philips theorem.

Stabilization of a Thermoelastic Laminated Beam with Past History

On the stability of Timoshenko systems with Gurtin–Pipkin thermal law

The stability number of the Timoshenko system with second sound

In this paper, we consider the Timoshenko systems with frictional dissipation working only on the vertical displacement. We prove that the system is exponentially stable if and only if the wave speeds are the same. On the contrary, we show that the Timoshenko systems is polynomially stable giving the optimal decay rate. Copyright © 2013 John Wiley & Sons, Ltd.

Stability to weakly dissipative Timoshenko systems

In this paper we analyze the porous elastic system. We show that viscoelasticity is not strong enough to make the solutions decay in an exponential way, independently of any relationship between the coefficients of wave propagation speed. However, it decays polynomially with optimal rate. When the porous damping is coupled with microtemperatures, we give an explicit characterization on the decay rate that can be exponential or polynomial type, depending on the relation between the coefficients of wave propagation speed. Numerical experiments using finite differences are given to confirm our analytical results. It is worth noting that the result obtained here is different from all existing in the literature for porous elastic materials, where the sum of the two slow decay processes determine a process that decay exponentially.

On the Decay Rates of Porous Elastic Systems

In this article, we are considering the one-dimensional equations of an homogeneous and isotropic porous elastic solid, where the localized damping involves the sum of displacement velocity of a solid elastic material and the volume fraction velocity. First we show, using a result due to Benchimol (SIAM J Control Optim 16:373–379, 1978), that the semigroup associated with the system is strongly stable if and only if the boundary of the support of feedback control intersects that of the interval under consideration. Then we use the frequency domain method combined with careful inequalities obtained using multiplicative techniques to prove that the semigroup under consideration is exponentially stable.

On porous-elastic system with localized damping

In the present work, we prove that there exists a relation between a physical inconsistence known as second spectrum of frequency or non-physical spectrum and the exponential decay of a dissipative Timoshenko system where the damping mechanism acts on angle rotation. The so-called second spectrum is addressed into stabilization scenario and, in particular, we show that the second spectrum of the classical Timoshenko model can be truncated by taking a damping mechanism. Also, we show that dissipative Timoshenko type systems which are free of the second spectrum [based on important physical and historical observations made by Elishakoff (Advances mathematical modeling and experimental methods for materials and structures, solid mechanics and its applications, Springer, Berlin, pp 249–254, 2010), Elishakoff et al. (ASME Am Soc Mech Eng Appl Mech Rev 67(6):1–11 2015) and Elishakoff et al. (Int J Solids Struct 109:143–151, 2017)] are exponential stable for any values of the coefficients of system. In this direction, we provide physical explanations why weakly dissipative Timoshenko systems decay exponentially according to equality between velocity of wave propagation as proved in pioneering works by Soufyane (C R Acad Sci 328(8):731–734, 1999) and also by Munoz Rivera and Racke (Discrete Contin Dyn Syst B 9:1625–1639, 2003). Therefore, the second spectrum of the classical Timoshenko beam model plays an important role in explaining some results on exponential decay and our investigations suggest to pay attention to the eventual consequences of this spectrum on stabilization setting for dissipative Timoshenko type systems.

D. S. Almeida Júnior

Papers

The stability number of the Timoshenko system with second sound

Stability to weakly dissipative Timoshenko systems

On the Decay Rates of Porous Elastic Systems

On porous-elastic system with localized damping

On the nature of dissipative Timoshenko systems at light of the second spectrum of frequency