Optimal Control Theory And Application Of Hybrid Dynamic Systems

In this dissertation, optimal control theory and applications of hybrid dynamic time-varying systems in finite horizon are researched. At first, two optimal control problems for hybrid dynamic time-varying systems with free-terminal and terminal constraint are considered, value functions of these two optimal problems are respectively proved to be the unique viscosity solution and the unique lower semicontinuous solution to a type of Quasi-Variational Inequality(hereinafter as QVI) and the relationship between them is addressed. Consequently, a new unified hybrid dynamic model for inventory systems is formulated, and a concrete example of switching optimal problem arising from actual inventory systems is solved by the use of previous theoretical results. Finally, the controllability problem for three dimensional parabolic control systems with nonlinear boundary is discussed. There are seven chapters in this dissertation.In Chapter 1, Hybrid Dynamic Systems and their research background are introduced, and mathematical models for Hybrid Dynamic Systems are simply described. From the view of Maximum Principle and Dynamic Programming, some advances in the study of optimal control problem of hybrid dynamic systems are recalled. Some open problems in this field are pointed out as well. At the end, the main work of this dissertation is listed.In Chapter 2, an optimal switching problem for a class of deterministic switched control time-varying systems with free terminals in finite horizon is concerned. By employing the properties of value functions for the optimal control problem, the optimal conditions are obtained and an optimal switching policy is constructed. A new rigorous and exhaustive proof about the statement that value functions of the optimal control problem is a unique viscosity solution to a Bensoussan-Lions type QVI is given. This conclusion about switching systems with free terminal has established theoretical foundation for discussing optimal problem of switching systems with terminal constraints in the next two chapters, and the comparison of method applied in this chapter with that of the next two chapters shows this more clearly.Chapter 3 is devoted to the optimal control problem for a class of switched control time-varying systems with terminal constraints. The presence of an end-point constraint makes the value functions neither be continuously differentiable, nor be finite valued on the interior of its effective domain. Since the extended value functions are merely semicontinuous, an extension of viscosity solutions to semicontinuous functions is needed. Viability Theory is applied to the optimal control problem for switching time varying systems in this dissertation for the first time. The statement about a characterization of the extended value functions as the unique lower semicontinuous solution to a Bensoussan-Lions type QVI is proved. It is also worth noting that the commonly used hypothesis in [57, 58] about a switching cost function is removed. And only the hypothesis, that switching cost is more than zero, is preserved. Finally, we conclude that the value function of optimal switching control problem is a viscosity solution to this type QVI under some additional assumptions.In Chapter 4, a class of hybrid dynamic time-varying systems with terminal constraint, which including two kinds of discrete events "system switching" and "state jumping", are considered. The optimal control problem of these systems is more complex than the optimal switching problem in Chapter 3. Viability Theory is applied to the optimal control problem for this class of hybrid dynamic time varying systems in this dissertation for the first time. And the statement about a characterization of the extended value functions as the unique lower semicontinuous solution to a Bensoussan-Lions type QVI is proved. Similar to Chapter 3, the commonly used hypotheses in [3] [57, 58] about a switching cost function and a jumping cost function are removed. Compared with previous references [68] and Chapter 3, the relationships between viscosity solutions and lower semicontinuous solutions to QVI are further discussed in this chapter.In chapter 5, a new and unified hybrid dynamic model for real-world inventory systems is proposed on the basis of Hybrid Dynamic Systems theory. This hybrid dynamic model characterizes inventory systems as interacting collections of dynamic systems, evolving on continuous-variable state spaces and subjecting to continuous controls and discrete transitions. This new model is called unified model because it could capture all kinds of discrete phenomena in inventory systems, and subsume all previous models for inventory systems.Chapter 6 deals with a switching optimal problem of a concrete example arising from actual inventory systems. As shown in Chapter 3, the description of lower semicontinuous functions as the unique generalized solutions to QVI has been advanced. However, there are few examples available in the previous literatures to illustrating the use of this analytic machinery. The purpose of this chapter is to provide such an example. Our analysis does not directly solve the H-J-B equations, but provides a candidate for the value function. After some steps involved in calculating subdifferentials and verifying satisfaction of QVI, the fact that the candidate is indeed the value function is confirmed. The optimal switching policy is thereupon obtained. This method of solving switching problem is different from previous literatures, and the switching sequence in this chapter is also unknown.In Chapter 7, a controllability problem for a three dimensional parabolic control system with nonlinear boundary is discussed. The nonlinear items only satisfy the local Lipschitz condition, but not the uniformly Lipschitz condition. The step function is applied to the approximate controllability problem of the corresponding linear system. The explicit expression of the control is provided. Based on the result of linear systems, the approximate controllability problem of nonlinear systems is solved.