System control and rough paths / Terry Lyons and Zhongmin Qian

The authors motivate their work from an application perspective in control theory and signal processing: when complicated data enters a nonlinear vector dynamical system as an input, what in this data characterizes the output? In other words, the goal is to filter away irrelevant information of the input which does not affect the output. Inputs will be irregular (a way of expressing the above-mentioned complexity), thus the word “rough paths” in the title. They can be even random like Brownian motions. Thus the book provides a unified view for integration of ODEs driven by irregular inputs. The development is technical, but the main idea is the following. Higher-order increments are constructed via iterated integrals from the rough paths. For smooth paths, higher-order increments are determined by the first-order increment. In general, complexity can be characterized by a natural number, the one for which all higher-order increments depend only on increments up to this natural number (if it exists). For multidimensional inputs, their complexity will lead to an even more complex output because the order of the events in the inputs will have an impact on the outputs through the Lie brackets of the corresponding input vector fields. So it is necessary to define the iterated integrals with a tensor product (and not a wedge product, which would make the chronology of events irrelevant). The authors consider paths in Banach spaces. (Zentralblatt)

Bibliogr. p. 205-213. Index