@phdthesis{Dikariev2025, author = {Dikariev, Illia}, title = {Relaxierungen von Steuerungsproblemen mit unendlichem Zeithorizont : Young-Maße und Existenz optimaler L{\"o}sungen}, doi = {10.26127/BTUOpen-7107}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:co1-opus4-71078}, school = {BTU Cottbus - Senftenberg}, year = {2025}, abstract = {The focus of this work is on the development of existence results for non-convex optimal control problems over an unbounded time horizon. The work is structured into three parts: - the elaboration of necessary functional-analytic foundations, - the derivation of equivalence relations between different types of relaxations, - existence statements for both compact and unbounded control domains. Particularly, the topology of compact convergence is introduced on the space of locally absolutely continuous functions. Further we develop embedding theorems and a Hardy inequality in relation to weighted reflexive Sobolev spaces, which are proposed as state spaces. The integrand of the objective functional is assumed to be a non-negative normal integrand, and the state equation is assumed to be a Carath{\´e}odory differential equation. The time interval is endowed with measures induced by weights. We present sufficient conditions to establish the equivalence of three pertinent types of relaxations Γ-regularization, generalized control, and sliding mode. To establish the existence results, we focus on the relaxation based on generalized controls (Young measures). In this context, we develop a suitable disintegration theorem and a weakened Prokhorov condition that are especially applicable to the case of generalized controls and non-linear state equations over an unbounded control domain. On the measure spaces, we introduce the vague or tight topology, depending on whether the control domain is compact or not. In the final part of the work we propose growth conditions on the integrand and on the right-hand side of the state equation to enforce closedness and compactness properties of an admissible set. Correspondingly, appropriate lemmas and existence theorems are presented for the cases of a compact resp. unbounded control domain, based on an existence theorem by Weierstrass. Finally, the existence results are illustrated through application examples.}, subject = {Optimal control; Optimale Steuerung; Infinite time horizon; Unendlicher Zeithorizont; Young measures; Young-Maße; Relaxed controls; Verallgemeinerte Steuerungen; Unbounded controls; Unbeschr{\"a}nkte Steuerungen; Optimale Kontrolle; Hardy-Raum; Sobolev-Raum; Carath{\´e}odory-Differentialgleichung; Young-Maß}, language = {de} }