Abstract
A controlled system in phase space is defined as follows: at every point of the space we have not just one velocity vector (as in the usual evolutionary system), but a whole set of vectors called the indicatrix of permissible velocities (Fig. 49).
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Bibliographical Comments
Davydov's classification is presented in his thesis:
A. A. Davydov: Singularities in two-dimensional control systems (in Russian). Moscow State University, 1982, 149 p.
The results are also announced in:
A. A. Davydov: Singularities of the admissibility boundary in two-dimensional control systems. Usp. Mat. Nauk 37:3 (1982), 183–184 English translation: Russ. Math. Surv. 37:3 (1982), 200-201).
The proofs are published (partially) in:
A. A. Davydov: The boundary of attainability of 2-dimensional control systems (in Russian). Usp. Mat. Nauk 37:4 (1982), 129.
A. A. Davydov: The boundary of an attainable set of a multidimensional control system (in Russian). Tr. Tbilis. Univ. 232–233, ser. Mat. Mekh. Astron. 13-14 (1982), 78-96.
A. A. Davydov: Normal forms of a differential equation, not resolved with respect to the derivative, in a neighbourhood of a singular point (in Russian). Funkts. Anal. Prilozh. 19:2 (1985), 1–10 (English translation to appear in Funct. Anal. Appl. 19 (1985)).
V. I. Arnol’d: Ordinary differential equations (3rd edition) (in Russian). Nauka, Moscow 1984, 272 p., pp. 266-267.
Singularities of convex hulls : surfaces in 3-space :
V. M. Zakalyukin: Singularities of convex hulls of smooth manifolds. Funkts. Anal. Prilozh. 11:3 (1977), 76–77 (English translation: Funct. Anal. Appl. 11 (1977), 225-227).
Curves in 3-space :
V. D. Sedykh: Singularities of the convex hull of a curve in IR3. Funkts. Anal. Prilozh. 11:1 (1977), 81–82 (English translation: Funct. Anal. Appl. 11 (1977), 72-73).
The general case:
V. D. Sedykh: Singularities of convex hulls. Sib. Mat. Zh. 24:3 (1983), 158–175 (English translation: Sib. Math. J. 24 (1983), 447-461).
V. D. Sedykh: Functional moduli of singularities of convex hulls of manifolds of codimensions 1 and 2. Mat. Sb., Nov. Ser. 119(161) (1982), 223–247 (English translation: Math. USSR, Sb. 47 (1984), 223-236).
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Arnold, V.I. (1986). Singularities of the Boundary of Attainability. In: Catastrophe Theory. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-96937-9_11
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