Spectral analysis and stabilization of a chain of

Spectral analysis and stabilization of a chain of serially connected Euler-Bernoulli beams and strings Kaıs Ammari ?, Denis Mercier †, Virginie Regnier † and Julie Valein ‡ Abstract. We consider N Euler-Bernoulli beams and N strings alternatively connected to one another and forming a particular network which is a chain begin- ning with a string. We study two stabilization problems on the same network and the spectrum of the corresponding conservative system: the characteristic equation as well as its asymptotic behavior are given. We prove that the energy of the so- lution of the first dissipative system tends to zero when the time tends to infinity under some irrationality assumptions on the length of the strings and beams. On another hand we prove a polynomial decay result of the energy of the second sys- tem, independently of the length of the strings and beams, for all regular initial data. Our technique is based on a frequency domain method and combines a con- tradiction argument with the multiplier technique to carry out a special analysis for the resolvent. 2010 Mathematics Subject Classification. 35L05, 35M10, 35R02, 47A10, 93D15, 93D20. Key words and phrases. Network, wave equation, Euler-Bernoulli beam equation, spec- trum, resolvent method, feedback stabilization.

• feedback law


• kaıs ammari


• explicit decay


• decay rate


• ing conservative


• euler-bernoulli beams


• interesting stability results


• system