Control Improvement Methods Based On The Extension Principle

Over the past half a century,the theory of optimal control has become a broad research area that studies various classes of optimal control problems,control objects and solution methods.The continuing complication of control objects inevitably results in the complication of their mathematical models and solution methods,both exact and approximate.As the complexity of investigated objects and the mathematical formulation and calculation become more complex,analytical solutions have become difficult,or even impossible,to find.Hence,approximate methods,such as numerical methods,are implemented to find approximate solutions to the problems.Iterative procedures are commonly used in approximate methods.However,a desired solution can only be guaranteed if the procedure starts with a good initial approximation.Since universal computational procedures and general recommendations for the good choice of the initial approximation have not yet developed,investigations in this direction are of great importance in order to increase the computational efficiency of the iterative procedure.In this dissertation,we apply the extension and localization principles,Krotov's sufficient optimality conditions,the theory of degenerate problems,and Krotov's minimax principle for approximate investigation of optimal control problems.Specifically,we develop new effective iterative control improvement methods and a general approach to approximate studies of optimal control problems.The whole dissertation consists of six chapters.Chapter 1 introduces the background,motivation,and the main results of this dissertation.In Chapter 2,we propose an abstract scheme of iterative optimization based on the extension and localization principles.We apply the proposed scheme to a continuous optimal control problem and formulate the general sufficient optimality conditions.Moreover,we derive a successive control improvement algorithm with consideration of some modifications that are useful in practice.Our results extend and complement other researches that deal with sufficient optimality conditions.In Chapter 3,we derive an improvement method for the discrete time control systems.We present an iterative algorithm analogous to the continuous case.Additionally,we develop contraction-type transformations that constrain solution search to a set of piecewise controls with given basic functions.Such transformations convert the original problems to discrete-continuous optimal control problems whose lower-level(continuous)systems do not contain controls.This chapter develops and extends the corresponding results on approximate optimization methods for discrete control systems.In Chapter 4,we propose a multistage approximate control optimization procedure that uses turnpike solutions known from the theory of degenerate problems.Although turnpike solutions are characteristic for systems with linear controls,the proposed procedure is sufficiently general as we show that a controllable differential system in a general form can be transformed to an equivalent system with linear controls.We develop algorithms that implement this procedure and demonstrate their quality through a model and a real example.The proposed representations and transformations represent an attempt to systematically implement its most important stage,namely search for an approximate globally optimal solution as a possible initial approximation in some iterative procedure.The results have important theoretical and practical significance.In Chapter 5,we provide the theoretical justification for new approaches to iterative control improvement.We consider the properties of iterative processes and their relation to optimality conditions.We pay special attention to new realizations of the general approach to global control improvement.Numerical analyses are presented in this chapter.Finally,in Chapter 6,we present our conclusions and ideas for future research.