Global Asymptotic Stability of Solutions to Nonlinear Marine Riser Equation
© Şevket Gür. 2010
Received: 28 May 2010
Accepted: 14 September 2010
Published: 15 September 2010
This paper studies initial boundary value problem of fourth-order nonlinear marine riser equation. By using multiplier method, it is proven that the zero solution of the problem is globally asymptotically stable.
where is the flexural rigidity of the riser, is the "effective tension", is the coefficient of the Coriolis force, is the coefficient of the nonlinear drag force, and is the mass line density. represents the riser deflection.
By using the Lyapunov function technique, Köhl has shown that the zero solution of the problem is stable.
under boundary conditions (1.2). Here , and are given positive numbers, is given real number, is a function, and . It is shown that the zero solution of the problem (1.3)-(1.2) is globally asymptotically stable, that is, the zero solution is stable and all solutions of this problem are tending to zero when . Moreover the polynomial decay rate for solutions is established.
There are many articles devoted to the investigation of the asymptotic behavior of solutions of nonlinear wave equations with nonlinear dissipative terms (see, e.g. [3, 4]), where theorems on asymptotic stability of the zero solution and estimates of the zero solution and the estimates of the rate of decay of solutions to second order wave equations are obtained.
Similar results for the higher-order nonlinear wave equations are obtained in .
2. Decay Estimate
From this inequality it follows that the zero solution (1.4)–(1.6) is globally asymptotically stable.
Special thanks to Prof. Dr. Varga Kalantarov.
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