The Exact Controllability Of Partial Differential Equations With Memory Terms

In nature,many phenomena can be studied by partial differential equations.In addition,many dynamic phenomena are affected by the past history of one or more variables,which can be described by partial differential equations with memory terms.Therefore,it is of great scientific significance and practical value to study the control problem of partial differential equations with memory terms.This paper studies the exact controllability of systems with memory terms.First of all,we concerns exact controllability of coupled wave equations with memory and defines the energy of corresponding dual system.Using multiplier method,the compactness and uniqueness,to obtain the dual system of some important estimate type and regularity,especially to see inequalities of dual system,then we can prove the exact controllability of coupled wave equations with memory,based on the thought of HUM.Secondly,we shall consider exact exact controllability of thermoelastic plate equation with memory.Using the thought of the multiplier method to construct the observability inequality of the dual system,then we prove that thermoelastic plate equation with memory is exactly controllable by the HUM.Last but not,we study the null controllability of weakly degenerate wave equation with memory terms.After simplifying the weakly degenerate wave equation with memory terms by taking a special memory function and using the multiplier method,we can get the observability inequality of the dual system and prove that the weakly degenerate wave equation with memory term is exactly null controllable.