Research On The Contrast Structure Of Singularly Perturbed Optimal Control Problems

In this dissertation, by means of further investigation of state solution limiting behavior of singularly perturbed optimal control problems, we dis-cuss the existence of contrast structure of singularly perturbed optimal con-trol problems. In recent years, the research of contrast structure is well de-veloped and thus it will provide theoretical basis for the research of contrast structure of singularly perturbed optimal control problems. The contrast structure in singularly perturbed problems is mainly classified as a step-like contrast structure and a spike-like contrast structure, a step-like contrast structure problem is only concerned in this dissertation. Its fundamental characteristic is that there exists t*(or multiple t*) within the domain of in-terest, t*is called as an internal transition point. The discussion at each t*is exactly the same, so we only consider the case that there exists one internal transition point. The position of t*is unknown in advance and it needs to be determined thereafter. In the neighborhood of t*, the solution will have an abrupt structure change and the solution will approach different reduced solutions when the small parameter μâ†'0.The thesis consists of two parts. The first part considers the step-like contrast structure of singularly perturbed optimal control problems. The second part considers the optimal control of singularly perturbed hybrid dy-namical systems.Chapter1introduces the development of singularly perturbed optimal control, gives some basic concepts and lemmas which are relevant to our study, and introduces the main research work and innovative points in this thesis.Chapter2studies the linear singularly perturbed optimal control prob-lem and singularly perturbed optimal control problem with integral boundary conditions. By means of the properties of exponential dichotomies, Fredholm alternatives and k+σ exchange lemma, we prove the existence of step-like contrast structure solution. Meanwhile, based on the structure of the so-lution, we construct the uniformly valid formal asymptotic solution by the boundary layer function method and direct scheme method.Chapter3is devoted to investigate the step-like contrast structure of nonlinear singularly perturbed optimal control problem. The existence of step-like contrast structure solution is proved by equivalence, which is based on the necessary optimality conditions. Meanwhile, the uniformly valid for-mal asymptotic solution is constructed by the direct scheme method.Chapter4considers the step-like contrast structure of higher dimen-sional singularly perturbed optimal control problem. The existence of step-like contrast structure solution is proved by the k+σ exchange lemma. Using the direct scheme method, we construct the uniformly valid formal asymp-totic solution for the singularly perturbed optimal control problem.In chapter5, we focus on the asymptotic solution for optimal control of singularly perturbed hybrid dynamical systems. The variational method is used to obtain optimality conditions of hybrid dynamical systems and the for-mal asymptotic solution is constructed by boundary layer function method. By sewing orbit, the existence of solution is shown and the asymptotic solu-tion is proved to be uniformly valid in the whole interval.