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%0 Thesis
%A Harms, Melanie
%T Invariant zero sets for first and second order nonlinear ODE systems
%I RWTH Aachen University
%V Dissertation
%C Aachen
%M RWTH-2025-09487
%P 1 Online-Ressource
%D 2025
%Z Veröffentlicht auf dem Publikationsserver der RWTH Aachen University
%Z Dissertation, RWTH Aachen University, 2025
%X Studying invariant sets provides insights into the qualitative behaviour of ordinary differential equation (ODE) systems. A subset S of the state space is said to be invariant if any trajectory starting in S remains in S for all times. In this thesis, we investigate invariant zero sets for ODE systems across various function classes. Our first objective is to generalise known results on polynomial systems to extended system classes. For those classes accessible to symbolic computation, we provide constructive characterisations and computational methods – based on the theory of Gröbner bases – to determine the invariance of a prescribed zero set S and to find generators for the module of all vector fields that have S as an invariant set. The second objective is to generalise these concepts from first order to second order ODE systems. While the notion of invariant sets typically addresses subsets of the state space, we aim to extend this concept to subsets of the position or velocity space of a second order ODE system. We focus on varieties and provide conditions to decide whether a variety V is invariant for a second order system. Again, considering system classes accessible to symbolic computation, we turn these conditions into an algorithmic test using symbolic computation. In contrast to the first order case, the set of all polynomial vectors defining a second order system which has V as an invariant set can either be empty or forms an affine module. Particular invariant varieties, namely subspace arrangements, play an important role in studying collision-freeness. An ODE system is called collision-free if any trajectory starting in distinct substates remains in distinct substates for all times. This structural property is found in many common models of particle interactions, arising naturally from their intrinsic symmetries. We discuss this concept for linear, polynomially nonlinear, and general locally Lipschitz continuous systems while addressing the connection between permutation symmetry and collision-freeness. From the perspective of control theory, one seeks to achieve invariance of prescribed sets by implementing suitable feedback laws. A set S is said to be controlled invariant with respect to a control system if there exists a state or output feedback law such that the resulting closed loop system has S as an invariant set. Our results on invariant sets for ODE systems serve as a foundation for investigating controlled invariant zero sets for first and second order input-affine control systems across function classes accessible to symbolic computation. We present constructive characterisations of these sets along with computational methods for deriving all suitable state or output feedbacks.
%F PUB:(DE-HGF)11
%9 Dissertation / PhD Thesis
%R 10.18154/RWTH-2025-09487
%U https://publications.rwth-aachen.de/record/1021128