Optimal Control Problems Governed By Dynamic Differential Systems On Arbitrary Time Scales And Applications

Dynamic equations on the so-called time scales unifies theory of differential equations and dif-ference equations and it even allows dynamic systems to be of partly continuous-time and partly discrete-time. With this, it is possible to investigate optimal control problems on arbitrary time-scales. This will bring advantage to establish theory for continuous-time and discrete-time prob-lems in a unified way. Also it provides more accurate and flexible information for systems with the domain that could be, for example, a union of disjoint connected time intervals with some discrete instances.In this thesis, we investigate optimal control problems governed by dynamic differential sys-tems on arbitrary time scales. The main results of this thesis are as follows:First, one of our current works focuses on the existence for optimal control problem governed by a class of semi-linear dynamical system with terminal constraint on time scales. A definition of solution of semi-linear control system involving Sobolev space WT1,2is introduced and new existence and uniqueness results of this kind of dynamic systems on time scales are presented. Based on LT2strong-weak lower semi-continuity of integral functional, we establish the existence of optimal controls. In particular, the existence for calculus of variations on time scales are also derived.Next, based on the definitions and propositions of Lebesgue Δ-integral functions and abso-lutely continuous functions on time scales, a definition of absolutely continuous solution is intro-duced and existence and uniqueness results of nonlinear dynamic systems on time scales are es-tablished by Contraction mapping theorem. Meanwhile, using Dunford-Pettis on time scales and Cesari property for a multi function, existence result of optimal control problem governed by this kind of system are established.Further, dynamic programming method is a mainstream tool to deal with continuous or dis-crete optimal control problems separately. That motivates us to analyze the corresponding theory on times scales. we derive Hamilton-Jacobi-Bellman Equations on a time scale for a class of optimal control problems with fixed terminal state and establish a sufficient condition for an admissible pair to be optimal. With this, we disclose the intrinsical characteristic for the costate and give its def-inition. On this basis, we derive the so-called maximum principle for this kind of optimal control problem under the condition of the value function to be C2. Meanwhile, based on a family of "needle variations" at scattered points and dense points respectively, we give rigorous proof of maximum principle for optimal control problem on time scales with the prerequisite convexity condition for the scattered points.At last, we investigate applications of maximum principle for optimal control problems on time scales. Firstly, we find a specific time scale such that the system of impulsive/hybrid system is equivalent to a time scale system. Therefore, time-dependent impulsive control problem can be reduce to a time scale version of control problem. Secondly, a time scale version of Ramsey model is established, results implies that the growth rate of consumption for u(C(-))=lnC can fluctuate depending on the time scale because of v(·). However, discrete and continuous time models would show that the growth rate of consumption is constant. Thirdly, from the history and problem itself, a multi-year mosquito-population model should be developed. One needs a way to model continuous and discrete processes at the same time, time scale calculus would be the answer.This thesis has taken a new step again regarding theory construction of optimal control prob-lems governed by dynamic differential systems on arbitrary time scales.