Finite Codimensional Controllability Of Linear Systems With Unbounded Control Operators And Application

Controllability and optimal control problem are two fundamental issues in control theory.Controllability mainly focuses on the feasibility problem,that is,whether a control system can achieve or approach an expected goal under the action of control.Optimal control problem is a special kind of optimization problems,which is mainly concerned with how to achieve the goal in the best way.Although they are two different types of control problems,they are closely related.In the existing work,the concept of finite codimensional controllability is introduced in the case of bounded control operators.Also,the controllability method is used to study optimal control problems with state constraint.However,this work cannot be directly applied to control systems with unbounded control operators,such as boundary control or point control.In order to solve this problem,this dissertation is devoted to studying the finite codimensional exact controllability of abstract linear systems in the case of unbounded control operators.An equivalent characterization of finite codimen-sional exact controllability is given,which is transformed into suitable a priori estimates for homogeneous equations/systems.As an application of this result,this dissertation takes a quadratic optimal control problem of linear systems with state constraint as an example,and presents the first order necessary conditions for optimal controls.This method is also applicable to nonlinear control sys-tems and general cost functionals.Moreover,it provides a more easily verifiable method for the finite codimensionality condition in the necessary condition.In order to illustrate the verifiability of this method,three examples on un-bounded control operators are given in this dissertation.Optimal control prob-lems for wave equations with boundary control,wave equations with point con-trol,and coupled wave systems with one single boundary control are studied,respectively.By establishing proper a priori estimates for the corresponding equa-tion/system,respectively,the first order necessary conditions for optimal controls are given.