The Long Time Asymptotic Behaviour Of A Class Of Second Order Dissipative System

Dissipative system is an important class of dynamical system,which has beenapplied in mechanics,control and optimization,phisics and many other fields.Inrecent years, many researchers have a lot of fruitful results in the dynamicalbehaviour of the dissipative system. However, most dissipative system theystudied contains the dissipative term all the time. If the dissipative term graduallydisappears over time, the long time behaviour of the system is not very clear. In2008, Cabot gave a conjecture that whether the solution of the dissipative systemwith the disappearing dissipative term was convergent. To address this conjecture,in this paper,we mainly discuss the long time behaviour of a class of dissipativesystem with gradually disappearing dissipative term.Firstly, the research background and development process of the dissipativesystem are introduced, and a class of second order dissipative system which isstudied in this paper is given. Then, the global existence and uniqueness of thesolution of the system is proved by using the theory of functional analysis anddifferential equation, and an energy function is constructed to prove someproperties of the system. Further more, if the dissipative term disappears slowlyenough, the convergence of the solution trajectory is proved under the conditionthat the objective function is real analytic by the Lojasiewicz inequality. Finally,we discuss the asymptotic behaviour of the system under the condition that theobjective function is convex and the objective function has finite critical points,respectively.