Stability: a search for structure

In this thesis we study stability from several viewpoints. After covering the practical importance, the rich history and the ever-growing list of manifestations of stability, we study the following. (i) (Statistical identification of stable dynamical systems): The simplest case, that of characterizing spectral properties of the least-squares estimator of a linear dynamical system has been largely open. To that end, we propose a principled projection method onto the set of Schur matrices. Leveraging large deviations theory, we show that this projection is optimal in an information-theoretic sense and that the projection can be approximated, by simply adding a LQ optimal feedback term. Going one step beyond stability, we further exploit large deviations theory to identify the topological class. (ii) (Spaces of stable dynamical systems): First, we study the LQR problem and finally provide an operational meaning for cross terms in the stage cost. We show that the topological class of the closed-loop system is invariant under a change of the stage cost, as long as no cross term is introduced. Hence, one can only "tune" the closed-loop behaviour by introducing a cross term. Secondly, motivated by the learning community often employing convex Lyapunov functions to obtain stability certificates, we study the ramifications of the convexity assumption. We show that continuous dynamical systems equipped with convex Lyapunov functions, asserting that the origin is globally asymptotically stable, can always be homotoped to each other such that along this homotopy stability is preserved. (iii) (Numerical stability): We also look at stability of the implementation of certain dynamical systems. Specifically, we look at zeroth-order optimization algorithms. Most zeroth-order optimization algorithms mimic a first-order algorithm, but replace the gradient of the objective with some estimator that can be computed from a number of function evaluations. This estimator is typically constructed randomly, and its expectation matches the gradient of a smooth approximation of the objective whose quality improves as some underlying smoothing parameter, usually a finite difference parameter, is reduced. As such, most zeroth-order algorithms require this smoothing parameter to decay to zero as the algorithm proceeds. While estimators based on just a single function evaluation exist, their variance is unbounded. Then, estimators based on multiple function evaluations, overcome the exploding variance, yet, they suffer from numerical cancellation once the smoothing parameter is sufficiently small. To combat both effects simultaneously, we extend the objective function to the complex domain and leverage the complex-step derivative to construct a new randomized estimator. (iv) (Topological obstructions to stability and stabilization): We study necessary conditions for stability and stabilizability of dynamical control system defined on topological spaces. We provide new insights in multistability and odd-dimensional systems, generalize the Bhat & Bernstein obstruction to submanifolds and we fully characterize when a compact attractor is a strong deformation retract of its domain of attraction. Results of this nature display the synergy between topology and dynamical systems. All of these studies focus on understanding the interplay between underlying structure and desiderata. Naturally emerging future directions close the thesis.