A Survey On Optimal Control Problems

The optimal control theory results from the classical variationalmethods, and the optimal control problem is an important part of themodern control theory. Because of the rapid development of computertechnology in the last century, the research on optimal trajectories inaviation applications has led to related researchers focusing on theoptimal control technology. In view of this, here this thesis attempts topresent an overview of the historical development of the optimalcontrol theory and of the status problems of the optimal control theorystudies.And optimal control problem mainly consists of four parts: systemdynamics equation, initial and terminal constraint conditions of thesystem state, system control region and system target functional.Optimal control problems were derived from the optimal control theory.The origin of optimal control theory dates back to the 17th century. In1638, Galileo put forward 'the catenary problem' and 'thebrachistochrone problem', which gave rise to further studies by anumber of outstanding mathematicians such as Newton, Bernoulli,Leibnitz, Euler, Lagrange, Hamilton, etc. The emergence ofPontryagin's maximum principle, Bellman's dynamic programming, thenecessary conditions of Classical variation, Carleman's designtheories and methods of the optimal linear feedback regulator,announced the optimal control theory had been officially formed. Later,several typical methods developed from the theory. The first chapter ofthis paper introduces how the optimal control problems arose anddescribes the optimal control theory researches, and the backgroundof calculation methods.The optimal control problems can be classified in many ways. Theauthor of this thesis attempts to classify the thesis as follows: optimalcontrol problems which are governed by the elliptical equations,optimal control problems which are governed by the parabolicequations, stochastic optimal control problems, time optimal controlproblems and the optimal control problems which are governed bynonlinear discrete system.Among the optimal control problems, the ones that catch the mostattention are the existence and uniqueness of the optimal control, thenecessary conditions and the sufficient conditions of optimal controlproblems. With the advance of studies, the research focus has movedfrom the ordinary differential equations system to the partial differentialequations system and from linear system to nonlinear system.As to the optimal control problems, which are governed byelliptical equations, as its first order necessary conditions and its firstorder sufficient conditions have been studied for a long time. They arerelatively mature. Relevant research literatures on its second order arestill far from sufficient, particularly its second order conditions underthe circumstance of non-convex control sets. Therefore, the thesishighlights the necessary conditions of the optimal control problemswith the non-convex admissible control sets, the necessary conditionsof the optimal control problems with discontinuous state variable,boundary optimal control problems, the finite Approximations todeduce the necessary optimal control conditions and the sufficientconditions of optimal control problems.Over the past 10 years, there have been very active studies onthe optimal control problems, which are governed by parabolicequations. In this aspect, this thesis presents the optimal controlproblems which are governed by parabolic variational inequality andnonlinear hyperbolic semi-variational inequality with state constraint;stabilization, controllability and the optimality necessary conditions ofparabolic & hyperbolic equations system;distributed parameter ofoptimal control problems;nonlinear degenerate equation of optimalcontrol problems;non-well-posed & well-posed of optimal controlproblems and other highlighted research problems.In the optimal control theory, the stochastic optimal controlproblem appears late, yet it has good prospect for research. After theeconomies of East Asia involved themselves in one of the worstfinancial crises of the postwar period in 1997 and after Argentinarefused to pay back its debt in 2001, the world's financial situationbecame unstable. Therefore, stochastic optimal control applications inthe dynamic forecast of world finance aroused the interest of thefinancial research scholars. The thesis focuses on introducing thedevelopment process of the stochastic optimal control, the existence,the research status of necessary and sufficient conditions, and itsapplication status in the fields of economy and finance.Just like the stochastic optimal control problems, the time optimalcontrol problems are also hard nuts to crack. Though such problemshave been studied earlier, there have been no breakthrough results sofar. The main purpose of the research is to explore an optimal controlin a given control sets. This makes the solution of control equationcorresponding to the control can reach a set of control objectives inthe shortest possible time. On the existence of time optimal control, V.Barbu and A. Fattorini have made great efforts in research. To studythe existence of the time optimal control, the key point is to prove theexistence of admissibility control, which is concerned with thecontrollability of some control constraint equations. And this is wherethe difficulty lies. Therefore, in most literatures, researches on the timeoptimal control problems with differential equations are usually carriedout on the supposition of the existence of admissible controls.The nonlinear discrete system governing optimal control problemsare problems people often encounter. To study such issues, peopleoften translate the nonlinear discrete system into a new continuousoptimal control, and they substitute the solution of the new system forthe solution of the former system.Currently, the optimal control theory has a wide range of effectiveapplications in system engineering, economic management anddecision-making, especially in space technology and other fields. Asresearches go further, there have been achievements in the optimalcontrol theory of infinite dimensional system, too. In recent years thefield of nonlinear control issues has become the focus. Usingdifferential geometric methods in studying the nonlinear control is theresult of the development of modern mathematics. It has developedinto a nonlinear system of differential geometric control theory, and ithas become the mainstream research field in the nonlinear controlstudy in the past 20 years. Because of the extreme complexity of thenonlinear control systems, although there have been a lot of results, itis far from enough for the establishment of a complete theoreticalsystem and there is a lot of work to be done. Just like other technicalscientific theories, the birth and development of the optimal controltheory is mainly due to the needs of human beings. Faced with somecomplex systems, it is difficult to solve problems successfully with thetheories and methods in the control field only. Interrelated disciplinesand complementary means should be the mainstream of the optimalcontrol problem studies in the future.