| The nonlinear rotating disk beam system is a kind of rigid-flexible coupled dynamic model describing the aircraft,which is widely used in aerospace and practical engineering.Based on the disk-beam system proposed by Baillieul and Levi,we consider the stability problem of disk-beam system with localized thermal effect and infinite memory term(a moment control case),respectively.Firstly,we discuss the stability problem of the disk-beam system with localized thermoelastic damping.Assume the thermoelastic damping applied to localized domain of the beam and a torque feedback control applied to the disk,we study the stability of the corresponding closed-loop system.We first choose the appropriate state space and define the system operator so that to rewrite the system into the form of abstract evolution equation,the well-posedness of the system is proved by semigroup theory and related nonlinear knowledge.The proof of the exponential stability is divided into linear beam part and nonlinear disk part.Stability of the linear part is mainly proved by the frequency domain method.We obtain the exponential stability of linear part by judging that there are no spectrums of system operator on the imaginary axis and the norm of the resolvent operator along the imaginary axis is bounded uniformly.The stability of the nonlinear part need to prove step by step,by the differential equation of the disk,we find the specific expression of the difference between the angular velocity and the desired value,establishing inequality relationship between the difference and the linear partial solution.Then,exploiting indirectly the exponential stability of the linear solution to obtain the stability of the nonlinear part.Finally,we give the stability result of the model: Using only torque control,the system can be stabilized exponentially under certain condition on angular velocity,no matter how small the part with thermal effect of the beam is.Secondly,we study the stability problem of the disk-beam system under the infinite memory.The feedback control consist of a linear torque control acting on the disk and a moment control with infinite memory term.Firstly,the infinite memory term is transformed by the minimal state space method.Then,similar to the above proof,we obtain the the well-posedness of this system.We prove the stability of the linear part by the frequency domain method mentioned above,that is,to verify there are no spectrums of system operator on the imaginary axis and the norm of the resolvent operator along the imaginary axis is bounded uniformly.At the same time,we give the asymptotic distribution of spectrum to the linear subsystem and show that the distribution of the high-frequency spectrum is not affected by the memory term.The proof of the nonlinear part is the same as above.Similarly,we give the stability result of the system: the system is exponentially stable provided that the angular velocity and the memory kernel function satisfy certain conditions.Finally,we present some numerical simulations on the dynamic behavior of the system to support the theoretical results by Matlab software. |
