Time Optimal Control Problems Of Several Kinds Of Differential Coupled Systems

The world is connected and the connections are mutual,which determines that many phenomena in actual production and real life can be described by mathematical models of differential coupling systems,such as microwave heating and chemical reactions.How to select the control function so that the differential coupling controlled system can reach the desired goal in the shortest time is the time optimal control problem of the coupling system.This kind of problems are usually concerned by researchers for control theory and engineering control problems.In this paper,three kinds of time optimal control problems of differential coupled systems are studied: linear ordinary differential coupled systems,microwave heating system,and the Pestrowsky system,where the systems are from finite dimension’s ODE to infinite dimension’s PDE.For all of them,the existence and the bang-bang property of the time optimal control are mainly studied.the content included the following aspects:1.For the time optimal control problems of linear ordinary differential coupled systems,the weakly coupled systems,the two equations are regarded as a series system,and the first state variable is considered as the control for the second system.Thus the time optimal control problem of weakly coupled systems are transformed into two time optimal control problems of single equation system.To the strongly coupled system,the two state variables and controls can be integrated into higher-dimensional variables,then the time optimal control problems of strongly coupled systems can be transformed into the time optimal control problems of single linear ordinary differential equation through higher order vector and matrix.The existence and the bang-bang properties of time optimal control problems for all the linear ordinary coupling systems are studied.2.For the time optimal control of microwave heating system.Firstly,according to the microwave heating theory,the mathematical model of time optimal control of microwave heating system is established by the electromagnetic theory.Then,according to the microwave heating intensity,the time optimal control problems of two types,the weakly coupled systems and the strongly coupled system are explored.To the time optimal control problem of weakly coupled system on microwave heating,at first,the Hahn-Banach theorem and the Riesz representation theorem are used to show the nullcontrollability of the state of the controlled system,and then the existence of the time optimal control for the microwave heating weakly coupled system,is proved through the null-controllability of the controlled state,the minimization sequence,the correlation convergence and the embedding theorem.At last,the bang-bang property of the time optimal control of the weakly coupled microwave heating system is proved by contract through the null controllability.To the time optimal control problem of strongly coupled system about microwave heating,due to the nonlinearity of the system,the null controllability of the state is shown by Kakutani fixed point theorem,then by the means of the minimization sequence,convergence and the tight embedding theory,it is proved that there exists the time optimal control for the strongly coupled system.Finally,with the Carlman inequality,the relation between the action time and the norm of the control are established,through which the bang-bang property for the time optimal control of the strong coupled microwave heating system is shown by contract.3.For the time optimal control problem of the Pestowsky system,according to the equivalence between the necessary and sufficient condition for null controllability and the necessary condition for the extreme value of a linear functional,we first give the sufficient and necessary condition to null controllability of the controlled Petrowsky system.Nevertheless,the necessary and sufficient condition for the null controllability of the Pestrowsky system can be regarded as a necessary condition for the minimum of a functional.Then the controllability of the controlled Petrowsky system is obtained through proving the existence of minimum value for the functional.Furthermore,using the Lebesgue Convergence lemma,get the existence of the time optimal control of the Petrowsky system.At last,by contract,the bang-bang property of the time optimal control problem governed by the Petrowsky equation through the formula of weak solution of the Petrowsky equation by contrast.The innovations of this paper lies in the following: To the time optimal control of ordinary differential coupling system,its research will be applied to study on the optimal control of complex system or the optimal control problems with multi-control and multi-object.To the time optimal control of microwave heating system,its time optimal control problems will be applied well.The method for discussing the time optimal control,especially,the proof of the bang-bang property is novel.To the time optimal control of the Petrowsky system,the necessary and sufficient condition of controllability is considered as the necessary condition of extreme value of a functional,which changes the angle of view and the method of solving the problem.To all the three types of time optimal control,the bang-bang property for the time optimal controls are all discussed.At last,we list the research work which will be done in the future.