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Mathematics of Complexity and Dynamical Systems pp 88–101Cite as

Chronological Calculus in Systems and Control Theory

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Article Outline

Glossary

Definition of the Subject

Introduction, History, and Background

Fundamental Notions of the Chronological Calculus

Systems That Are Affine in the Control

Future Directions

Acknowledgments

Bibliography

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Abbreviations

Controllability :

A control system is controllable if for every pair of points (states) p and q there exists an admissible control such that the corresponding solution curve that starts at p ends at q. Local controllability about a point means that all states in some open neighborhood can be reached.

Pontryagin Maximum Principle of optimal control:

Optimality of a control‐trajectory pair geometrically is a property dual to local controllability in the sense that an optimal trajectory (endpoint) lies on the boundary of the reachable sets (after possibly augmenting the state of the system by the running cost). The maximum principle is a necessary condition for optimality. Geometrically it is based on analyzing the effect of families of control variations on the endpoint map. The chronological calculus much facilitates this analysis.

\( \mathcal{E}(M) = C^{\infty}(M) \) :

The algebra of smooth functions on a finite dimensional manifold M, endowed with the topology of uniform convergence of derivatives of all orders on compact sets.

\( \Gamma^{\infty}(M) \) :

The space of smooth vector fields on the manifold M.

Chronological calculus :

An approach to systems theory based on a functional analytic operator calculus, that replaces nonlinear objects such as smooth manifolds by infinite dimensional linear ones, by commutative algebras of smooth functions.

Chronological algebra :

A linear space with a bilinear product ★ that satisfies the identity \( a\star(b\star c)-b\star (a\star c)=(a\star b)\star c-(b\star a)\star c \). This structure arises naturally via the product \( (f \star g)_t=\int_0^t [f_s,g^{\prime}_s]\text{d} s \) of time‐varying vector fields f and g in the chronological calculus. Here \({\left[\,\,,\,\,\right]}\) denotes the Lie bracket.

Zinbiel algebra :

A linear space with a bilinear product that satisfies the identity \( a\ast (b\ast c)=(a\ast b)\ast c+(b\ast a)\ast c \). This structure arises naturally in the special case of affine control systems for the product \( (U\ast V)(t)=\int_0^t U(s)V^{\prime}(s)\text{d} s \) of absolutely continuous scalar valued functions U and V.

The name Zinbiel is Leibniz read backwards, reflecting the duality with Leibniz algebras, a form of noncommutative Lie algebras. There has been some confusion in the literature with Zinbiel algebras incorrectly been called chronological algebras.

\( \mathcal{IIF}(\mathcal{U}^Z) \) :

For a suitable space \( \mathcal{U}\) of time‐varying scalars, e. g. the space of locally absolutely continuous real‐valued functions defined on a fixed time interval, and an indexing set Z, \({\mathcal{IIF}(\mathcal{U}^Z)}\) denotes the space of iterated integral functionals from the space of Z‑tuples with values in \({\mathcal{U}}\) to the space \({\mathcal{U}}\).

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Acknowledgments

This work was partially supported by the National Science Foundation through the grant DMS 05-09030

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  1. Department of Mathematics, Arizona State University, Tempe, USA

    Matthias Kawski

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  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Kawski, M. (2012). Chronological Calculus in Systems and Control Theory. In: Meyers, R. (eds) Mathematics of Complexity and Dynamical Systems. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1806-1_7

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