Structural stability of simplest dynamical inequalities. (English. Russian original) Zbl 1167.34021

Proc. Steklov Inst. Math. 256, 80-91 (2007); translation from Tr. Mat. Inst. Steklova 256, 89-101 (2007).
Objects of interest in this paper are inequalities of the form
\[ (\dot x(x,y) - a(x,y)^2 + (\dot y(x,y) - b(x,y))^2 \leq f(x,y),\tag{1} \]
where \(v = (a,b)\) is a fixed smooth vector field on the plane \(\mathbb{R}^2\) and \(f : \mathbb{R}^2 \to \mathbb{R}\) is a fixed smooth function. The set of dynamical inequalities is identified with all triples \((a,b,f)\) endowed with the fine Whitney \(C^k\) topology. A set is \(C^k\)-generic if it holds on a dense open set in the fine Whitney \(C^k\) topology.
The set of points on which \(a^2 + b^2 > f > 0\) holds is said to be steep. These are the points at which the controlled object cannot resist drift. In this paper it is assumed that either the set on which \(a^2 + b^2 < f\) or the set on which \(f < 0\) contains all sufficiently distant points, which in turn implies boundedness of the steep domain.
A velocity \((\dot x, \dot y)\) is feasible at a point \((x,y)\) if inequality (1) holds there. A feasible motion is an absolutely continuous mapping from a time interval into the phase space such that the velocity is feasible at each point at which the mapping is differentiable. A point \(A\) is reachable from a point \(B\) is there exists a feasible motion that takes point \(B\) to point \(A\) in finite time. The union of all points that are reachable from a given point (respectively, from which a given point is reachable) is the positive (respectively, negative) orbit of this point. A dynamical inequality is structurally stable if, for any inequality in a neighborhood of it, there exists a near-identity homeomorphism \(h\) of phase space that transforms all families of positive and negative orbits of the points of one inequality into the corresponding families of the other.
In this paper the authors prove the structural stability of the simplest generic smooth dynamical inequality with bounded steep domain.


34D30 Structural stability and analogous concepts of solutions to ordinary differential equations
26D20 Other analytical inequalities
34A40 Differential inequalities involving functions of a single real variable
37C20 Generic properties, structural stability of dynamical systems
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[1] A. A. Andronov and L. S. Pontryagin, ”Structurally Stable Systems,” Dokl. Akad. Nauk SSSR 14(5), 247–250 (1937).
[2] D. V. Anosov, S. Kh. Aranson, I. U. Bronshtein, and V. Z. Grines, ”Smooth Dynamical Systems,” Itogi Nauki Tekh., Ser.: Sovr. Probl. Mat., Fund. Napr. 1, 151–242 (1985); Engl. transl. in Dynamical Systems I (Springer, Berlin, 1988), Encycl. Math. Sci. 1, pp. 149–233. · Zbl 0605.58001
[3] V. I. Arnold, Additional Chapters of the Theory of Ordinary Differential Equations (Nauka, Moscow, 1978) [in Russian].
[4] V. I. Arnold and Yu. S. Il’yashenko, ”Ordinary Differential Equations,” Itogi Nauki Tekh., Ser.: Sovr. Probl. Mat., Fund. Napr. 1, 7–149 (1985); Engl. transl. in Dynamical Systems I (Springer, Berlin, 1988), Encycl. Math. Sci. 1, pp. 1–148.
[5] V. I. Arnold, V. S. Afraimovich, Yu. S. Il’yashenko, and L. P. Shil’nikov, ”Bifurcation Theory,” Itogi Nauki Tekh., Ser.: Sovr. Probl. Mat., Fund. Napr. 5, 5–218 (1986); Engl. transl. in Dynamical Systems V (Springer, Berlin, 1994), Encycl. Math. Sci. 5, pp. 1–205.
[6] V. I. Arnol’d, ”Contact Structure, Relaxational Oscillations and Singular Points of Implicit Differential Equations,” in Global Analysis-Studies and Applications III (Springer, Berlin, 1988), Lect. Notes Math. 1334, pp. 173–179.
[7] R. Garcia and J. Sotomayor, ”Structural Stability of Parabolic Lines and Periodic Asymptotic Lines,” Mat. Contemp. 12, 83–102 (1997). · Zbl 0930.53003
[8] R. Garcia, C. Gutierrez, and J. Sotomayor, ”Structural Stability of Asymptotic Lines on Surfaces Immersed in \(\mathbb{R}\)3,” Bull. Sci. Math. 123, 599–622 (1999).
[9] A. A. Davydov, ”Normal Form of a Differential Equation, Not Solvable for the Derivative, in a Neighborhood of a Singular Point,” Funkts. Anal. Prilozh. 19(2), 1–10 (1985) [Funct. Anal. Appl. 19, 81–89 (1985)]. · Zbl 0576.14045
[10] A. A. Davydov, ”Structural Stability of Control Systems on Orientable Surfaces,” Mat. Sb. 182(1), 3–35 (1991) [Math. USSR, Sb. 72, 1–28 (1992)]. · Zbl 0735.93013
[11] A. A. Davydov, Qualitative Theory of Control Systems (Am. Math. Soc., Providence, RI, 1994), Transl. Math. Monogr. 141. · Zbl 0830.93002
[12] A. A. Davydov, ”Local Controllability of Typical Dynamical Inequalities on Surfaces,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 209, 84–123 (1995) [Proc. Steklov Inst. Math. 209, 73–106 (1995)]. · Zbl 0878.93006
[13] A. A. Davydov and E. Rosales-Gonsales, ”The Complete Classification of Typical Second-Order Linear Partial Differential Equations on the Plane,” Dokl. Akad. Nauk 350(2), 151–154 (1996) [Dokl. Math. 54 (2), 669–672 (1996)]. · Zbl 0909.35006
[14] A. A. Davydov, ”Controllability of Generic Control Systems on Surfaces,” in Geometry of Feedback and Optimal Control, Ed. by B. Jakubczyk and W. Respondek (M. Dekker, New York, 1998), Monogr. Textbooks Pure Appl. Math. 207, pp. 111–163. · Zbl 0965.93013
[15] A. G. Kuz’min, ”On the Behavior of the Characteristics of a Mixed-Type Equation near a Line of Degeneracy,” Diff. Uravn. 17(11), 2052–2063 (1981).
[16] J. Palis, Jr. and W. de Melo, Geometric Theory of Dynamical Systems (Springer, New York, 1982; URSS, Moscow, 1998).
[17] A. D. Piliya and V. I. Fedorov, ”Singularities of an Electromagnetic Wave Field in Cold Plasma with a Two-Dimensional Inhomogeneity,” Zh. Eksp. Teor. Fiz. 60(1), 389–399 (1971).
[18] A. V. Pkhakadze and A. A. Shestakov, ”Classification of Singular Points of a Differential Equation Not Solved with Respect to the Derivative,” Mat. Sb. 49(1), 3–12 (1959). · Zbl 0086.28401
[19] F. Takens, ”Constrained Equations: A Study of Implicit Differential Equations and Their Discontinuous Solutions,” in Structural Stability, the Theory of Catastrophes, and Applications in the Sciences: Proc. Conf. Seattle, WA, 1975 (Springer, Berlin, 1976), Lect. Notes Math. 525, pp. 143–234.
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