Quadratic Optimal Control Problems For Fractional Linear Systems

In recent years,the theory of fractional differential equations has attracted the in-depth and systematic research of a large number of mathematicians.The optimal control problems of fractional differential equations have been developed into an important research direction of the fractional differential equation theory.Since fractional differential systems can reflect the dynamic changing processes more reasonably and accurately than integer order ones,such systems will have a wider application prospect.However,there have been a few papers making the preliminary research on the quadratic optimal control problems for fractional systems.The techniques are approximating the fractional systems by the integer order ones.In this thesis,the method of integer order approximation will not be used.We try to directly study the quadratic optimal control problems of fractional order by the newly introduced operators.In the introduction,we summarize the development of fractional calculus.The preliminary section introduces the definition of Riemann-Liouville calculus and its properties,the definition of Caputo differential,the definition of MittagLeffler functions and their properties.In the third part,we show the necessary conditions of the finiteness of the quadratic optimal control problems for Caputo differential systems,and the sufficient and necessary conditions under which there exists a unique optimal control.Similarly as in the third part,we give a brief comment in the fourth part for the quadratic optimal control problems of RiemannLiouville differential systems.Finally,we summarize the whole thesis and point out the forecast of this paper.