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I find an interesting paper that mentioned the Definition of curvature of an affine optimal control system. It reminded me that many textbooks on Riemannian geometry only tell us about metrics, geodesics, parallel transport, and curvature tensors and son on, using some special surfaces with constant curvature as examples. So for a general nonlinear control system (eg. with dissipative force or other external forces), how do we calculate its curvature tensor? Are there more references or research on this topic?

Thank you all for your comments and help!

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Robert Gardner, William Shadwick and George Wilkins wrote several papers on the project of a differential geometric description of control theory. The challenge is the usual one of a moving target: control theory involves so many different mathematical models of control systems, that there is no one mathematical theory that contains all of the possible models. However, you might look at

Bryant, Robert L.(1-DUKE); Gardner, Robert B.(1-NC) Control structures.(English summary) Geometry in nonlinear control and differential inclusions (Warsaw, 1993), 111–121. Banach Center Publ., 32 Polish Academy of Sciences, Institute of Mathematics, Warsaw, 1995 Part of Book Collection MR1364415

where you will find a mathematical model of a large class of control systems in terms of $G$-structures, which then have a notion of torsion of all orders, which is a bit like the notion of curvature in Riemannian geometry.

Keep in mind that there is no clear definition of what curvature should mean for a control system.

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