摘要
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.
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
the financial support and the facilities provided by King Fahd University of Petroleum and Minerals through project No. IN111034
