Abstract
AbstractUsing the frequency domain approach, we prove the rational stability for a wave equation with distributed delay on the dynamical control, after establishing the strong stability and the lack of uniform stability.
Highlights
There are full practical processes that might be modelled by distributed delay systems, which present a wide range of applications in various elds such as micro-organism growth [25], hematopoiesis
In the case of the wave equations, Nicaise and Pignotti [16] investigated exponential stability results with delay concentrated at τ for the system utt(x, t) − ∆u(x, t) = in Ω × (, +∞)
In this paper, staying on the one dimensional space, we purpose a dynamical boundary moment control η with a distributed delay term, and we look for the possible ways to stabilize the system (1)
Summary
There are full practical processes that might be modelled by distributed delay systems, which present a wide range of applications in various elds such as micro-organism growth [25], hematopoiesis [1, 2], logistics [4] and tra c ow [21]. In the recent past (last four decades), many researchers have fruitfully investigated on that subject, and successfully applied them in more widespread other areas They have developed mathematical tools in order to establish polynomial or exponential decays of these systems. We consider the following wave equation with a distributed delay term on the dynamical control : utt(x, t) − uxx(x, t) =. In the case of the wave equations, Nicaise and Pignotti [16] investigated exponential stability results with delay concentrated at τ for the system utt(x, t) − ∆u(x, t) = in Ω × ( , +∞). In this paper, staying on the one dimensional space, we purpose a dynamical boundary moment control η with a distributed delay term, and we look for the possible ways to stabilize the system (1). The paper is organized as follows: section 2 is devoted to the well posedness of problem (1), while the section 3 deals with the strong stability of problem (1) ; section 4 establishes the lack of uniform stability, and nally in section 5 stands on the polynomial stability of problem (1)
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