In dieser Arbeit untersuchen wir das Problem der Maximierung bestimmter konkaver Funktionale auf dem Raum der optionalen, zufälligen Maße. Deartige Funktionale treten in der mikroökonomischen Literatur auf, wo ihre Maximierung auf die Bestimmung des optimalen Konsumplans eines ökomischen Agenten hinausläuft. Als Alternative zu den wohlbekannten Methoden der dynamischen Programmierung wird ein neuer Zugang vorgestellt, der es erlaubt, die Struktur der maximierenden Maße in einem über den üblicherweise angenommenen Markovschen Rahmen hinausgehenden, allgemeinen Semimartingalrahmen zu klären. Unser Zugang basiert auf einer unendlichdimensionalen Version des Kuhn-Tucker-Theorems. Die implizierten Bedingungen erster Ordnung erlauben es uns, das Maximierungsproblem auf ein neuartiges Darstellungsproblem für optionale Prozesse zu reduzieren, das damit als ein nicht-Markovsches Substitut für die Hamilton-Jacobi-Bellman Gleichung der dynamischen Programmierung dient. Um dieses Darstellungsproblem im deterministischen Fall zu lösen, führen wir eine zeitinhomogene Verallgemeinerung des Konvexitätsbegriffs ein. Die Lösung im allgemeinen stochastischen Fall ergibt sich über eine enge Beziehung zur Theorie des Gittins-Index der optimalen dynamischen Planung. Unter geeigneten Annahmen gelingt ihre Darstellung in geschlossener Form. Es zeigt sich dabei, daß die maximierenden Maße absolutstetig, diskret und auch singulär sein können, je nach Struktur der dem Problem zugrundeliegenden Stochastik. Im mikroökonomischen Kontext ist es natürlich, daß Problem in einen Gleichgewichtsrahmen einzubetten. Der letzte Teil der Arbeit liefert hierzu ein allgemeines Existenzresultat für ein solches Gleichgewicht. / In this thesis, we study the problem of maximizing certain concave functionals on the space of optional random measures. Such functionals arise in microeconomic theory where their maximization corresponds to finding the optimal consumption plan of some economic agent. As an alternative to the well-known methods of Dynamic Programming, we develop a new approach which allows us to clarify the structure of maximizing measures in a general stochastic setting extending beyond the usually required Markovian framework. Our approach is based on an infinite-dimensional version of the Kuhn-Tucker Theorem. The implied first-order conditions allow us to reduce the maximization problem to a new type of representation problem for optional processes which serves as a non-Markovian substitute for the Hamilton-Jacobi-Bellman equation of Dynamic Programming. In order to solve this representation problem in the deterministic case, we introduce a time-inhomogeneous generalization of convexity. The stochastic case is solved by using an intimate relation to the theory of Gittins-indices in optimal dynamic scheduling. Closed-form solutions are derived under appropriate conditions. Depending on the underlying stochastics, maximizing random measures can be absolutely continuous, discrete, and also singular. In the microeconomic context, it is natural to embed the above maximization problem in an equilibrium framework. In the last part of this thesis, we give a general existence result for such an equilibrium.

<div>Optimal control theory (OCT) exists since the 1950s. However, with the advent of modern computers, the design community delegated the task of solving the optimal control problems (OCPs) largely to computationally intensive direct methods instead of methods that use OCT. Some recent work showed that solvers using OCT could leverage parallel computing resources for faster execution. The need for near real-time, high quality solutions for OCPs has therefore renewed interest in OCT in the design community. However, certain challenges still exist that prohibits its use for solving complex practical aerospace problems, such as landing human-class payloads safely on Mars.</div><div><br></div><div>In order to advance OCT, this thesis introduces Epsilon-Trig regularization method to simply and efficiently solve bang-bang and singular control problems. The Epsilon-Trig method resolves the issues pertaining to the traditional smoothing regularization method. Some benchmark problems from the literature including the Van Der Pol oscillator, the boat problem, and the Goddard rocket problem verified and validated the Epsilon-Trig regularization method using GPOPS-II.</div><div><br></div><div>This study also presents and develops the usage of trigonometry for incorporating control bounds and mixed state-control constraints into OCPs and terms it as Trigonometrization. Results from literature and GPOPS-II verified and validated the Trigonometrization technique using certain benchmark OCPs. Unlike traditional OCT, Trigonometrization converts the constrained OCP into a two-point boundary value problem rather than a multi-point boundary value problem, significantly reducing the computational effort required to formulate and solve it. This work uses Trigonometrization to solve some complex aerospace problems including prompt global strike, noise-minimization for general aviation, shuttle re-entry problem, and the g-load constraint problem for an impactor. Future work for this thesis includes the development of the Trigonometrization technique for OCPs with pure state constraints.</div>