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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.

MSC:

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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