Research On The Optimal Control Problems To Nonlinear Fractional Evolution Equations

The objective world is made up of various systems.With the development of science and technology and the need of actual engineering control system design,various complex systems are constantly emerging in front of people,and expect people to know and study.As one of the objective systems,fractional evolution equation is a mathematical discipline that studies the properties and applications of differential equations containing non-integer order derivatives and is a generalization of classical integer evolution equation.In recent years,researchers have found that fractional models can more accurately simulate and describe problems with memory and genetic characteristics in real life,such as dispersion in porous media,turbulence in fluid motion and viscoelastic systemsOn the other hand,as an important branch of modern control theory,the optimal control problem is a discipline that studies and solves the problem of finding optimal control from all possible control schemes.From a mathematical point of view,the optimal control problem can be expressed as:under the constraints of the motion equation and the allowable control range,the extreme value(maximum value or minimum value)of the cost function with the control function and the motion state as variables is obtained.At present,The principle and method of optimal control theory have been widely applied in many fields in real life.For example,determining an optimal control mode to minimize fuel consumption in the process of space vehicle switching from one orbit to another;selecting a regulation law of temperature and the corresponding ratio of raw materials to maximize the output of chemical reaction process,etc These are some typical optimal control problems.It can be said that the optimal control theory is a very vital discipline with wide development prospects.This thesis will study the optimal control problem of fractional evolution equation,mainly including the following contents:(1)This thesis study the impulsive control systems of fractional order and their optimal controls in Banach spaces.Firstly,the thesis show the existence and uniqueness of mild solu-tions for a broad class of impulsive fractional infinite dimensional control systems under suit-able assumptions.Then by using generally mild conditions of cost functionals,the thesis extend the existence result of optimal controls to the impulsive fractional control systems.Finally,an application is given to illustrate the effectiveness of our main results(2)This thesis deal with the minimization problem of an integral functional with an inte-grand that is not convex in the control,on solutions of a control system described by a nonau-tonomous fractional evolution equation with mixed nonconvex constraints on the control.A relaxation problem is treated along with the original problem.It is proved that the relaxation problem has an optimal solution and that for each optimal solution there is a minimizing se-quence of the original problem that converges to the optimal solution with respect to the trajec-tory,the control and the functional in appropriate topologies simultaneously(more details,one can see Theorem 4.10)(3)In this thesis,a sensitivity analysis of optimal control problem for a class of systems described by nonlinear fractional evolution inclusions on Banach spaces is investigated.Firstly,the nonemptiness as well as the compactness of the mild solutions set S(?)(? being the initial condition)for the nonlinear fractional evolution inclusions are obtained,and we also present an extension Filippov's theorem and whose proof differs from previous work only in some tech-nical details.Finally,the optimal control problems described by nonlinear fractional evolution inclusions depending on the initial condition ? and the parameter ? are considered and the sen-sitivity properties of the optimal control problem are also established.