On Optimal Control Problems In Infinite Time Horizon

The optimal control problem over infinite time horizon is an important problem in optimal control theory,which has a wide range of applications in economics,physics,engineering and other fields.In this dissertation,we mainly consider the optimal control problems of linear system with quadratic cost functional over an infinite time horizon without assuming the controllability/stabilizability condition and the global integrability condition for the nonhomogeneous term of the state equation and the weight functions in the linear terms in the running cost rate function.At this time,the index functional may not exist in the whole infinite time horizon,so the classical approaches of the linear quadratic optimal control problem do not apply for such kind of problems.Therefore,the overtaking optimal control theory will be used to study the optimal control problem at this time.Then the existence and non-existence of the overtaking optimal control will be characterized.The main contents are as follows:Firstly,the research background,research situation and latest progress of the optimal control problem in infinite time horizon are systematically introduced.And the main research work to be carried out in this thesis is expounded,and the required preliminary knowledge is also given.Then,the relevant results of the quadratic optimal control problems of linear ordinary differential equations in finite time horizon and infinite time horizon under classical conditions are introduced.The quadratic optimal control problems of linear Caputo type fractional order equations in finite time horizon are studied.According to the properties of the fractional integral operator,we obtain the existence of the optimal control.Under some mild conditions,the quadratic optimal control problem of the linear singular Volterra integral equations in infinite time horizon is studied.These research results will lay the groundwork for further discussions.Furthermore,the quadratic overtaking optimal control problem of linear ordinary differential equations is studied.Under the condition that the controllability condition is generally not assumed,the original optimal control problem can be transformed into a controllable system by decomposing the state space,and its objective functional has a locally integrable linear weight function.For such a problem,under special assumptions,it can be obtained that there is a unique classical optimal solution for this problem,and this optimal solution is a unique overtaking optimal solution of the original problem.For the general case,through the characterization of the linear weight function,and by means of the definition method,inequality and Fredholm integral equation theory,we have obtained a series of results about the existence and non-existence of overtaking optimal control on convex sets.In addition,examples are given to illustrate the reasonability of the conclusion.On the other hand,the quadratic index overtaking optimal control problem of linear Caputo fractional differential equations is studied.Mainly based on the definition method and inequality estimation,we have obtained the existence and non-existence of the quadratic overtaking optimal control of the linear Caputo fractional differential equations.Finally,the research work of this thesis is summarized,and the prospect of the work is proposed.