FG Mathematik insbesondere Optimierung
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Institute
In this paper, we consider a class of variational problems on an unbounded interval of the real axis. This type of problems arises, e.g., in quantum mechanics and asymptotic controllability. The problem is treated in a Hilbert space setting with uniformly and non-uniformly weighted Sobolev spaces as state spaces. We provide sufficient conditions that a function from a weighted Sobolev space can be expanded into a Fourier–Laguerre series converging together with its distributional derivative pointwisely and uniformly. With this result, the considered variational problem is transformed into a problem in the sequence space of Fourier–Laguerre coefficients. We develop a Fourier–Laguerre method in these spaces in order to construct a polynomial approximation scheme for the solution of this problem.
Regulator Problems on Unbounded Domains : Stationarity–Optimal Control–Asymptotic Controllability
(2018)
In this paper, we consider a class of infinite horizon variational and control problems arising from economics, quantum mechanics, and stabilization. Herein, we assume that the objective is of regulator type. The problem setting implies a weighted Sobolev space as the state space. For this class of problems, we establish necessary optimality conditions in a form of a Pontryagin type maximum principle. A duality concept of convex analysis is provided and used to find sufficient optimality conditions and to motivate a dual approximation scheme. We apply the theoretical results to find an asymptotically stabilizing control for a linearized Lotka–Volterra type system.
In this paper, a class of infinite horizon optimal control problems with a mixed control-state isoperimetrical constraint, also interpreted as a budget constraint, is considered. The underlying dynamics is assumed to be affine-linear in control. The crucial idea which is followed in this paper is the choice of a weighted Sobolev space as the state space. For this class of problems, we establish an existence result and apply it to a bilinear model of optimal cancer treatment with an isoperimetrical constraint including the overall amount of drugs used during the whole therapy horizon. A numerical analysis of this model is provided by means of open source software package OCMat, which implements a continuation method for solving discounted infinite horizon optimal control problems.
We consider a class of infinite horizon optimal control problems with vector-valued states and controls involving the Lebesgue integral in the objective and a linear dynamics. The key idea is to show that the solutions of the state-equation belong to a weighted Hilbert space, due to a natural growth condition. This allows to introduce the weighted Sobolev space as the state space and the weighted Lebesgue space as the control space in the problem setting. We answer the fundamental question of how one can choose the weight function in dependence of the system dynamics. In the setting of Hilbert spaces, we show a Pontryagin-type maximum principle. Therefore, we use techniques from earlier publications of the authors. In contrast to these publications, we deal with vector-valued states and controls, here. These vector-valued problems arise in many applications. The maximum principle also includes a transversality condition which makes the difference to related publications in the field of necessary optimality conditions. This condition brings advantages for the numerical calculations of the solution by indirect methods, like a pseudospectral method.
In this paper a class of infinite horizon optimal control problems with a mixed control-state isoperimetrical constraint, also interpreted as a budget constraint, is considered. Herein a linear both in the state and in the control dynamics is allowed. The problem setting includes a weighted Sobolev space as the state space. For this class of problems, we establish an existence theorem. The proved theoretical result is applied to a mixed control-state budget constrained advertisement model.
In this paper we consider a class of infinite horizon variational problems resulting from a transformation of singular variational problems. Herein we assume that the objective is convex. The problem setting implies a weighted Sobolev space as state space. For this class of problems we establish necessary optimality conditions in form of a Pontryagin type maximum principle. A duality concept of convex analysis is provided and used to establish sufficient optimality conditions. We apply the theoretical results proven to the problem of the Brachistochrone.
In this paper we consider a class of infinite horizon variational problems resulting from a transformation of singular variational problems. Herein we assume that the objective is convex. The problem setting implies a weighted Sobolev space as state space. For this class of problems we establish necessary optimality
conditions in form of a Pontryagin type maximum principle. A duality concept of convex analysis is provided and used to establish sufficient optimality conditions. We apply the theoretical results proven to the problem of the Brachistochrone.
In this paper a class of infinite horizon optimal control problems with an isoperimetrical constraint, also interpreted as a budget constraint, is considered. Herein a linear both in the state and in the control dynamic is allowed. The problem setting includes a weighted Sobolev space as the state space. For this class of problems, we establish the necessary optimality conditions in form of a Pontryagin Type Maximum Principle including a transversality condition. The proved theoretical result is applied to a linear-quadratic regulator problem.
In recent years route planning has been greatly facilitated by car navigation systems. The desire for resource-efficient traveling has increased, a fact that is reflected by a higher demand for hybrid vehicles. The combination of electric motor and combustion engine adapted to the specific driving situation clearly reduces fuel consumption in comparison to vehicles with a conventional combustion engine. Up to this point, despite the fact that energy consumption and travel time are significantly affected by the drivers individual driving style, the driver has mainly been excluded. The aim of this project is to develop a dynamic model to calculate the optimal power train control—including the route planning. Taking into account the maximum travel time of the entire travel route in combination with energy-minimized driving, this approach allows us to obtain a new powerful concept. We present the model and its formulation as an optimal control problem, as well as some numerical results.