arXiv subjects

Mads R. Bisgaard

Publications and source records attributed to Mads R. Bisgaard.

5 recordsLinked to original sources

arXiv2022

A risk analysis framework for real-time control systems

We present a Monte Carlo simulation framework for analysing the risk involved in deploying real-time control systems in safety-critical applications, as well as an algorithm design technique allowing one (in certain situations) to robustify a control algorithm. Both approaches are very general and agnostic to the initial control algorithm. We present examples showing that these techniques can be used to analyse the reliability of implementations of non-linear model predictive control algorithms.

math.OC↗

arXiv2019

Mather theory and symplectic rigidity

Using methods from symplectic topology, we prove existence of invariant variational measures associated to the flow $ϕ_H$ of a Hamiltonian $H\in C^{\infty}(M)$ on a symplectic manifold $(M,ω)$. These measures coincide with Mather measures (from Aubry-Mather theory) in the Tonelli case. We compare properties of the supports of these measures to classical Mather measures and we construct an example showing that their support can be extremely unstable when $H$ fails to be convex, even for nearly integrable $H$. Parts of these results extend work by Viterbo and Vichery. Using ideas due to Entov-Polterovich we also detect interesting invariant measures for $ϕ_H$ by studying a generalization of the symplectic shape of sublevel sets of $H$. This approach differs from the first one in that it works also for $(M,ω)$ in which every compact subset can be displaced. We present applications to Hamiltonian systems on $\mathbb{R}^{2n}$ and twisted cotangent bundles.

math.DS↗

arXiv2017

Invariants of Lagrangian cobordisms via spectral numbers

We extend parts of the Lagrangian spectral invariants package recently developed by Leclercq and Zapolsky to the theory of Lagrangian cobordism developed by Biran and Cornea. This yields a nondegenerate Lagrangian "spectral metric" which bounds the Lagrangian "cobordism metric" (recently introduced by Cornea and Shelukhin) from below. It also yields a new numerical Lagrangian cobordism invariant as well as new ways of computing certain asymptotic Lagrangian spectral invariants explicitly.

math.SG↗

arXiv2017

Topology of (small) Lagrangian cobordisms

We study the following quantitative phenomenon in symplectic topology: In many situations, if a Lagrangian cobordism is sufficiently small (in a sense specified below) then its topology is to a large extend determined by its boundary. This principle allows us to derive several homological uniqueness results for small Lagrangian cobordisms. In particular, under the smallness assumption, we prove homological uniqueness of the class of Lagrangian cobordisms which, by Biran-Cornea's Lagrangian cobordism theory, induces operations on a version of the derived Fukaya category. We also establish a link between our results and Vassilyev's theory of Lagrange characteristic classes. Most currently known constructions of Lagrangian cobordisms yield small Lagrangian cobordisms in many examples.

math.SG↗

arXiv2016

A distance expanding flow on exact Lagrangian cobordism classes

Given an exact Lagrangian $L$ of an exact symplectic manifold $(M,dλ)$, Cornea and Shelukhin recently introduced a remarkable "cobordism metric" $d_c$ on the exact Lagrangian cobordism class $\mathcal{L}(L)$ of $L$. In this note we show that the Liouville flow of $(M,dλ)$ induces a flow on $(\mathcal{L}(L),d_c)$ which expands cobordism distances. In particular we deduce that $(\mathcal{L}(L),d_c)$ has infinite diameter whenever the Liouville flow is complete. We also discuss (Hamiltonian and Lagrangian) Hofer-geometric versions of our result. The proof only uses elementary differential geometry.

math.SG↗