Of concern is a viscoelastic beam modelled using the Timoshenko theory. It is well-kimwn that the system is exponentially stable if the kernel in the memory term is sub- exponential. That is, if the product of the ker...Of concern is a viscoelastic beam modelled using the Timoshenko theory. It is well-kimwn that the system is exponentially stable if the kernel in the memory term is sub- exponential. That is, if the product of the kernel with an exponential function is a summable function. In this article we address the questions： What if the kernel is tested against a different function （say Gamma） other than the exponential function？ Would there still be stability？ In the affirmative, what kind of decay rate we get？ It is proved that for a non- decreasing function ＂Gamma＂ whose ＂logarithmic derivative＂ is decreasing to zero we have a decay of order Gamma to some power and in the case it decreases to a different value than zero then the decay is exponential.展开更多
The stabilization problem for the Schr?dinger equation with an input time delay is considered from the view of system equivalence.First,a linear transform from the original system into an exponentially stable system w...The stabilization problem for the Schr?dinger equation with an input time delay is considered from the view of system equivalence.First,a linear transform from the original system into an exponentially stable system with arbitrary decay rate,also called"target system",is introduced.The linear transform is constructed via a kind of Volterra-type integration with singular kernels functions.As a result,a feedback control law for the original system is obtained.Secondly,a linear transform from the target system into the original closed-loop system is derived.Finally,the exponential stability with arbitrary decay rate of the closed-loop system is obtained through the established equivalence between the original closed-loop system and the target one.The authors conclude this work with some numerical simulations giving support to the results obtained in this paper.展开更多
基金the financial support and the facilities provided by King Fahd University of Petroleum and Minerals through project No. IN111034
文摘Of concern is a viscoelastic beam modelled using the Timoshenko theory. It is well-kimwn that the system is exponentially stable if the kernel in the memory term is sub- exponential. That is, if the product of the kernel with an exponential function is a summable function. In this article we address the questions： What if the kernel is tested against a different function （say Gamma） other than the exponential function？ Would there still be stability？ In the affirmative, what kind of decay rate we get？ It is proved that for a non- decreasing function ＂Gamma＂ whose ＂logarithmic derivative＂ is decreasing to zero we have a decay of order Gamma to some power and in the case it decreases to a different value than zero then the decay is exponential.
基金supported by the Doctoral Scientific Research Foundation of Henan Normal University under Grant No.qd18088the Natural Science Foundation of China under Grant No.61773277the Central University Basic Scientific Research Project of Civil Aviation University of China under Grant No.3122019140。
文摘The stabilization problem for the Schr?dinger equation with an input time delay is considered from the view of system equivalence.First,a linear transform from the original system into an exponentially stable system with arbitrary decay rate,also called"target system",is introduced.The linear transform is constructed via a kind of Volterra-type integration with singular kernels functions.As a result,a feedback control law for the original system is obtained.Secondly,a linear transform from the target system into the original closed-loop system is derived.Finally,the exponential stability with arbitrary decay rate of the closed-loop system is obtained through the established equivalence between the original closed-loop system and the target one.The authors conclude this work with some numerical simulations giving support to the results obtained in this paper.
