System control and rough paths. (English) Zbl 1029.93001

Lyons, Terry; Qian, Zhongmin

Oxford Mathematical Monographs. Oxford: Clarendon Press. x, 216 p. (2002).

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.
Chapter two is devoted to establishing a continuity result for ODEs with Lipschitz controls. Chapter three develops an integration theory for rough paths, the above-mentioned iterated integrals. A corresponding topology is introduced to characterize suitably the nearness of rough paths. Chapter four incorporates inputs as stochastic processes in the above framing theory of rough paths. An association between “canonical geometric rough paths” and a variety of such processes is given. Chapter five develops a path integration along rough paths. The next chapter is a culmination in this work: a continuity result (in the introduced \(p\)-variation topology) for ODEs with rough paths as inputs is obtained. Chapter seven generalizes the result of Maillavin concerning the tangent space to the Wiener space, where Wiener functionals are differentiable, to the more general case considered in this book.
One finds historical comments at the end of the chapters and an extensive bibliography, as well as an index.
This book is very interesting and provides a theoretical platform for further developments.

Reviewer: A.Akutowicz (Berlin)

MSC:

93-02	Research exposition (monographs, survey articles) pertaining to systems and control theory
93E03	Stochastic systems in control theory (general)
60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
60H05	Stochastic integrals

Keywords:

rough paths; Wiener space; complexity; Brownian motions; higher order increments; iterated integrals; continuity result; integration theory; stochastic processes; path integration; tangent space