| Publications about 'non-expansive systems' |
| Conference articles |
| In this paper, we study systems of time-invariant ordinary differential equations whose flows are non-expansive with respect to a norm, meaning that the distance between solutions may not increase. Since non-expansiveness (and contractivity) are norm-dependent notions, the topology of $\omega$-limit sets of solutions may depend on the norm. For example, and at least for systems defined by real-analytic vector fields, the only possible $\omega$-limit sets of systems that are non-expansive with respect to polyhedral norms (such as $\ell^p$ norms with $p =1$ or $p=\infty$) are equilibria. In contrast, for non-expansive systems with respect to Euclidean ($\ell^2$) norm, other limit sets may arise (such as multi-dimensional tori): for example linear harmonic oscillators are non-expansive (and even isometric) flows, yet have periodic orbits as $\omega$-limit sets. This paper shows that the Euclidean linear case is what can be expected in general: for flows that are non-expansive with respect to any strictly convex norm (such as $\ell^p$ for any $p ot=1,\infty$), and if there is at least one bounded solution, then the $\omega$-limit set of every trajectory is also an omega limit set of a linear time-invariant system. |
| It is often of interest to know which systems will approach a periodic trajectory when given a periodic input. Results are available for certain classes of systems, such as contracting systems, showing that they always entrain to periodic inputs. In contrast to this, we demonstrate that there exist systems which are globally exponentially stable yet do not entrain to a periodic input. This could be seen as surprising, as it is known that globally exponentially stable systems are in fact contracting with respect to some Riemannian metric. The paper also addresses the broader issue of entrainment when an input is added to a contractive system. |
| This paper adopts a contraction approach to the analysis of the tracking properties of dynamical systems under high gain feedback when subject to inputs with bounded derivatives. It is shown that if the tracking error dynamics are contracting, then the system is input to output stable with respect to the input signal derivatives and the output tracking error. As an application, it iss hown that the negative feedback connection of plants composed of two strictly positive real LTI subsystems in cascade can follow external inputs with tracking errors that can be made arbitrarily small by applying a sufficiently large feedback gain. We utilize this result to design a biomolecular feedback for a synthetic genetic sensor to make it robust to variations in the availability of a cellular resource required for protein production. |
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