When The Standard Meaning Of The Next-order Linear Evolution Equations And A Class Of Optimal Control Problems

In the actual life, when mathematics method is used to handle some problems in various natural phenomena, we will meet not only the continuous problems but also the straggling problems. Sometimes both the continuous and straggling components will be held in a single problem; sometimes it is even not clear for us whether this problem is the continuous one or the straggling one. Thus, our researches will be put to inconvenience.The calculus of time scales was initiated by Stefan Hilger in his PhD thesis in 1988 (supervised by Bernd Aulbach) in order to create a theory that can unify discrete and continuous analysis. The time scales calculus has a tremendous potential for applications in some mathematical models of real processes and phenomena studied in physics, chemical technology, population dynamics, biotechnology and economics, neural networks, social sciences (see the monographs of Aulbach and Hilger). For example, it can model insect populations that are continuous while in season (and may follow a difference scheme with variable step-size), die out in (say) winter, while their eggs are incubating or dormant, and then hatch in a new season, giving rise to a nonoverlapping population.In classical analysis, optimal control problems have been studies and have good results. However, to our knowledge, the optimal control problems on time scales have not been considered. In this thesis, we consider a lagrange problem of system governed by the first order linear dynamic equations on time scales with quadratic cost functional.In Chapter 2, we briefly recall the time scales calculus, including basic definitions, differentiation, integration, the generalized exponential function, trigonometric functions, polynomials and examples.Chapter 3 contributes to Banach spaces and Arzela-Ascoli theorem. Two important spaces C_rd([a,b],R) and L_T²([a,b],R) are defined and their completeness is shown. The generalized Arzela-Ascoli theorem on time scales is proved.In Chapter 4, we give the definition of classical solution for first order linear dynamic equation and its some example and applications. The weak solution of first order linear dynamic equation on time scales is defined. The pair of adjoint equations and the important relationship between them are given. Finally, existence of optimal control problems on time scales is presented in Chapter 5. In Chapter 6 the work of further research is given.