arXiv subjects

Nico Tauchnitz

Publications and source records attributed to Nico Tauchnitz.

8 recordsLinked to original sources

arXiv2025

Optimal control problems with free right end point

This paper is dedicated to the elementary proof of Pontryagin's maximum principle for problems with free right end point. The proof for the standard problem is taken from the monography of Ioffe and Tichomirov. We assume piecewise continuous controls and the proof turns out to be very simple. We generalize the concept to the problem of optimal multiprocesses, to control problems with delays and to the control of Volterra integral equations. Furthermore, we discuss the problem on infinite horizon. Moreover, we state Arrow type sufficiency conditions. The optimality conditions are demonstrated on illustrative examples.

math.OC↗

arXiv2020

Notes on Spaces of Functions Converging at Infinity

This preprint concerns Banach spaces of functions converging at infinity. In particular, spaces of continuous functions, Lebesgue spaces and sequence spaces. In each framework we show versions of Riesz's representation theorem.

math.FA↗

arXiv2020

Pontryagin's Maximum Principle for Infinite Horizon Optimal Control Problems with Bounded Processes and with State Constraints

This paper concerns a class of infinite horizon optimal control problems with state constraints. By extending the needle variation method to the infinite horizon case we obtain a complete set of necessary optimality conditions for a strong local minimizer in form of the Pontryagin maximum principle and the validity of several transversality conditions at infinity. To this purpose, we develop an approach in the space of continuous functions converging at infinity. The considerations including new results on optimality and transversality conditions, on Banach spaces of continuous functions, on the needle variation method and on problems with state constraints on the unbounded time horizon.

math.OC↗

arXiv2018

Weak Local Extrema in Infinite Horizon Optimal Control Problems

In this paper we deal with infinite horizon optimal control problems. Basing on weak variations in an extremal problem in weighted function spaces we prove necessary conditions in form of the adjoint equation and a variational inequality. The obtained optimality conditions imply several transversality conditions and the validity of Mangasarian type sufficiency conditions. Arising difficulties in the presence of state constraints are discussed.

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arXiv2018

Optimality Conditions for Weak and Strong Local Extrema in Infinite Horizon Optimal Control Problems

In this paper we summarize our results in infinite horizon optimal control. We present optimality conditions for weak local minimizer in the framework of weighted functions. Moreover we formulate the Pontryagin Maximum Principle for strong local minimizer for trajectories converging at infinity. The considered problems, the requirements and the results depend on the choice of the function space.

math.OC↗

arXiv2016

On Optimality Conditions in Control Theory

We study optimality conditions for various types of control problems like the standard optimal control problem, optimal multiprocesses, problems with infinite horizon or the control of Volterra integral equations. To derive necessary conditions the needle variation method of Ioffe & Tichomirov is the central tool. In the particular control problem with infinite horizon the question of a suitable setting arises. We propose the framework of continuous state trajectories converging at infinity. This requires a version of Riesz' representation theorem and the introduction of regular Borel measures on the extended real number line. The control of Volterra integral equations including an inner and outer time variable. Consequently, we deal with a two-dimensional time set. We extend the needle variation method of Ioffe & Tichomirov to this case. The obtained optimality conditions are demonstrated in illustrative examples.

math.OC↗

arXiv2015

The Pontryagin Maximum Principle for Optimal Multiprocesses with Continuous States

In this paper we give a new approach to introduce switching strategies in a special class of optimal multiprocesses. Defined as the set of partitions of the time interval, switching strategies become the character of a classical control. With this approach we overcome the restrictive comparison of multiprocesses with the same switching strategy. The necessary conditions for strong local optimizer in form of the Pontryagin maximum principle are demonstrated in illustrative examples.

math.OC↗

arXiv2015

The Pontryagin Maximum Principle for Nonlinear Infinite Horizon Optimal Control Problems with State Constraints

The famous proof of the Pontryagin maximum principle for control problems on a finite horizon bases on the needle variation technique, as well as the separability concept of cones created by disturbances of the trajectories. In this preprint, we develop this method for infinite horizon optimal control problems. The results are necessary conditions for a strong local minimizer in form of the Pontryagin maximum principle, Arrow type sufficiency conditions and the validity of diverse transversality conditions.

math.OC↗