Extension of chronological calculus for dynamical systems on manifolds. (English) Zbl 1345.58006
Summary: We propose an extension of the Chronological Calculus, developed by A. A. Agrachev and R. V. Gamkrelidze [Mat. Sb., Nov. Ser. 107(149), 467–532 (1978; Zbl 0408.34044)] for the case of \(C^\infty\)-smooth dynamical systems on finite-dimensional \(C^\infty\)-smooth manifolds, to the case of \(C^m\)-smooth dynamical systems and infinite-dimensional \(C^m\)-manifolds. Due to a relaxation in the underlying structure of the calculus, this extension provides a powerful computational tool without recourse to the theory of calculus in Fréchet spaces required by the classical Chronological Calculus. In addition, this extension accounts for flows of vector fields which are merely measurable in time. To demonstrate the utility of this extension, we prove a variant of Chow-Rashevskii theorem for infinite-dimensional manifolds.
MSC:
| 58C20 | Differentiation theory (Gateaux, Fréchet, etc.) on manifolds |
| 58B20 | Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds |
| 37C05 | Dynamical systems involving smooth mappings and diffeomorphisms |
| 93B27 | Geometric methods |
Citations:
Zbl 0408.34044References:
| [1] | Agrachev, A.; Gamkrelidze, R., Exponential representation of flows and a chronological calculus, Mat. Sb. (N.S.), 149 (1978) · Zbl 0408.34044 |
| [2] | Agrachev, A.; Gamkrelidze, R., Chronological algebras and nonstationary vector fields, J. Math. Sci., 17 (1981) · Zbl 0473.58021 |
| [3] | Agrachev, A.; Sachkov, Y., Control Theory from the Geometric Viewpoint (2004), Springer-Verlag: Springer-Verlag Berlin · Zbl 1062.93001 |
| [4] | Monaco, S.; Normand-Cyrot, D.; Califano, C., From chronological calculus to exponential representations of continuous and discrete-time dynamics: a Lie-algebraic approach, IEEE Trans. Automat. Control, 52, 2227-2241 (2007) · Zbl 1366.93337 |
| [5] | Gamkrelidze, R. V.; Agrachëv, A. A.; Vakhrameev, S. A., Ordinary differential equations on vector bundles, and chronological calculus, Current Problems in Mathematics. Newest Results. Current Problems in Mathematics. Newest Results, J. Soviet Math., 55, 4, 1777-1848 (1991), (in Russian); translated in: · Zbl 0727.58042 |
| [6] | Diestel, J.; Uhl, J., Vector Measures (1977), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 0369.46039 |
| [7] | Deimling, K., Ordinary Differential Equations in Banach Spaces (1977), Springer-Verlag: Springer-Verlag Berlin · Zbl 0364.34030 |
| [8] | Lang, S., Fundamentals of Differential Geometry (1999), Springer-Verlag: Springer-Verlag New York, NY · Zbl 0932.53001 |
| [9] | Hormander, L., The Analysis of Linear Partial Differential Operators I. Distributions Theory and Fourier Analysis (1983), Springer-Verlag: Springer-Verlag Berlin · Zbl 0521.35001 |
| [10] | Mauhart, M.; Michor, P., Commutators of flows and fields, (Arch. Math. (Brno), vol. 28 (1992)), 229-236 · Zbl 0784.58051 |
| [11] | Montgomery, R., A Tour of Subriemannian Geometries, Their Geodesics and Applications (2002), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 1044.53022 |
| [12] | Chow, W.-L., Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung, Math. Ann., 117, 98-105 (1939) · JFM 65.0398.01 |
| [13] | Rashevskii, P., About connecting two points of complete nonholonomic space by admissible curve, Uchen. Zapiski Pedag. Inst. K. Libnecht, 2, 83-94 (1938) |
| [14] | Isidori, A., Nonlinear Control Systems V (1989), Springer-Verlag: Springer-Verlag London |
| [15] | Jurdjevic, V., Geometric Control Theory (1997), Cambridge University Press: Cambridge University Press Cambridge, MA · Zbl 0940.93005 |
| [16] | Bloch, A., Nonholonomic Mechanics and Control (2003), Springer-Verlag: Springer-Verlag New York · Zbl 1045.70001 |
| [17] | Coron, J.-M., Control and Nonlinearity, Math. Surveys Monogr., vol. 136 (2007), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 1140.93002 |
| [18] | Kriegl, A.; Michor, P., The Convenient Setting of Global Analysis (1997), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 0889.58001 |
| [19] | Clarke, F.; Ledyaev, Y. S.; Stern, R.; Wolenski, P., Nonsmooth Analysis and Control Theory (1998), Springer-Verlag: Springer-Verlag New York, NY · Zbl 1047.49500 |
| [20] | Schirotzek, W., Nonsmooth Analysis (2007), Springer: Springer Berlin · Zbl 1120.49001 |
| [21] | Ledyaev, Y. S., On an infinite-dimensional variant of the Rashevskiĭ-Chow theorem, Dokl. Akad. Nauk, 398, 735-737 (2004) |
| [22] | Clarke, F.; Ledyaev, Y. S.; Radulescu, M., Approximate invariance and differential inclusions in Hilbert spaces, J. Dyn. Control Syst., 3, 493-518 (1997) · Zbl 0951.49007 |
| [23] | Borwein, J.; Zhu, Q. J., Techniques of Variational Analysis (2005), Springer-Verlag: Springer-Verlag New York, NY · Zbl 1076.49001 |
| [24] | Ledyaev, Y. S.; Zhu, Q., Nonsmooth analysis on smooth manifolds, Trans. Amer. Math. Soc., 359, 3687-3732 (2007) · Zbl 1157.49021 |
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