@misc{PickenhainZiemann, author = {Pickenhain, Sabine and Ziemann, Torsten}, title = {First-order necessary optimality conditions for infinite horizon optimal control problems with linear dynamics and convex objective}, series = {Optimization : a Journal of Mathematical Programming and Operations Research}, volume = {67}, journal = {Optimization : a Journal of Mathematical Programming and Operations Research}, number = {6}, issn = {1029-4945}, doi = {10.1080/02331934.2018.1434172}, pages = {757 -- 782}, abstract = {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.}, language = {en} } @misc{Pickenhain, author = {Pickenhain, Sabine}, title = {Infinite Horizon Problems in the Calculus of Variations. The Role of Transformations with an Application to the Brachistochrone Problem}, series = {Set-Valued and Variational Analysis Theory and Applications}, volume = {27}, journal = {Set-Valued and Variational Analysis Theory and Applications}, number = {2}, issn = {1877-0541}, doi = {10.1007/s11228-018-0483-6}, pages = {331 -- 354}, abstract = {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.}, language = {en} } @misc{PickenhainBurtchen, author = {Pickenhain, Sabine and Burtchen, Angie}, title = {Regulator Problems on Unbounded Domains : Stationarity-Optimal Control-Asymptotic Controllability}, series = {Vietnam Journal of Mathematics}, volume = {46}, journal = {Vietnam Journal of Mathematics}, number = {4}, issn = {2305-2228}, doi = {10.1007/s10013-018-0304-0}, pages = {837 -- 861}, abstract = {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.}, language = {en} } @misc{PickenhainBurtchen, author = {Pickenhain, Sabine and Burtchen, Angie}, title = {Problems in the Calculus of Variations on Unbounded Intervals—Fourier-Laguerre Analysis and Approximations}, series = {Vietnam Journal of Mathematics}, volume = {47}, journal = {Vietnam Journal of Mathematics}, number = {3}, issn = {2305-221X}, doi = {10.1007/s10013-019-00349-3}, pages = {617 -- 638}, abstract = {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.}, language = {en} }