Approximation of reachable sets using optimal control and support vector machines

We propose and discuss a new computational method for the numerical approximation of reachable sets for nonlinear control systems. It is based on the support vector machine algorithm and represents the set approximation as a sublevel set of a function chosen in a reproducing kernel Hilbert space. In some sense, the method can be considered as an extension to the optimal control algorithm approach recently developed by Baier, Gerdts and Xausa. The convergence of the method is illustrated numerically for selected examples

On minimal invariant sets of certain set-valued dynamical systems: a boundary map approach

We propose the construction of a boundary map to analyse and compute the boundary of an invariant set of a specific set-valued mapping F(M) := B_ε(f(M)) for M ⊂ R^d, d ≥ 2, ε > 0 and some diffeomorphism f. The core of this thesis builds on a result where the outward unit normal bundle of a continuously differentiable boundary of an invariant set is invariant under the boundary map. A similar result applies to the boundary of the dual repeller of an invariant set, where its inward unit normal bundle is invariant under the boundary map. We prove the persistence of the normal bundle of the C^2 smooth boundary of a minimal invariant set under perturbation, provided that it is normally hyperbolic. When the underlying mapping is linear, the compact minimal invariant set has a continuously differentiable boundary, and its normal bundle is globally attractive under the boundary map. Restricting to two-dimensional mappings f and considering a periodic point of the boundary map, we show a special relationship between the eigenvalues of the linearised map on the periodic point. This result provides necessary conditions for a periodic point of the boundary map to lie on the normal bundle of the boundary of an invariant set and its dual repeller respectively. We also construct an adaptive numerical algorithm to approximate the normal bundle of the boundary of an invariant set. The loss of smoothness on the boundary of an invariant set can be detected using a ‘curvature mapping’ by monitoring curvatures on the boundary. In the final chapter, we perform a numerical study on the set-valued mapping F using the boundary map where the underlying mapping f is the Hénon map. In this example, we illustrate the detection of topological bifurcations of the minimal invariant sets by classical bifurcations of the boundary map.Open Acces