The Optimal Control, Approximate Optimal Solution And Stability Of The Degenerate System With Delay

Because of the extensive and important uses of the optimal control theory and stability theory in many different systems such as ecosystems, social systems,economic systems, management systems and so on, many scholars have studied them devotedly and obtained many results about them, such as literatures[1-17].However, when they dicussed the theories about the optimal, control and stability , they usually attached more importance to these nonlinear and linear differential systems without taking the effects of the delay and being degenerate into consideration.And until now, the optimal control theory and stability theory of systems mentioned above have been considerably developed all-rounded and complete , such as literatures[1-6],[13-14].Of course, the optimal control theory and stability theory about only either systems with delays or degenerate ones have been discussed in many literatures[7,8,9,11,12,16,17,23]. There are also some discussions about the degenerate systems with delay such as literatures [10,16,24]. In the paper, many further discussions about the optimal control problem,the approximate optimal solution to generalized time optimal control problem and stability of the degenerate systems with delay are considered and some important and meaningful results are obtained in it.And then, a brief account of the paper is offered followingly.In the first chapter, concepts and their relevant knowledge about the optimal control, approximate optimal solution and stability are introduced in the chapter.In the second chapter, owing to the complexity of the degenerate system with delay itself, it is somewhat difficult to discuss the optimal control aboutthe general form of the system, thus, the optimal control of some typical degenerate systems with delay is considered and some results are obtained in the chapter.In the third chapter, because there are theories about the systems without being generate and delay when they are of restricted state right endpoint, and thus based on that, the optimal control problems of the generate systems with delay with restricted state right endpoint is considered in the chapter.In the fourth chapter, the approximate optimal solution to the generalized time optimal control problem is considered in it. Because it is possible and allowable that there is no solution to the time optimal control,from an pratical point, an approximate optimal solution being satisfied with some precision is obtained in the chapter.In the fifth chapter, the stability of the degenerate system with delay were discussed, many scholars have studied on the linear degenerate system with delay,however, in the paper, the stability of the degenerate nonlinear system with delay is discussed and a series of important results are obtaind in it.