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Generalizations of L. S. Pontryagin’s lemma concerning squares. (English. Russian original) Zbl 0574.90103
Differ. Equations 20, 1118-1123 (1984); translation from Differ. Uravn. 20, No. 9, 1548-1555 (1984).
The aim of this note is to extend the following result called Pontryagin’s lemma concerning squares. Let \(\phi_ i(.)\) \((i=1,...,m)\) be a linearly independent system of real scalar functions, defined and analytic on [0,1]; \(\Sigma =\{\phi (.)|\) \(\phi (.)=\sum^{m}_{i=1}c_ i\phi_ i(.)\), \(c_ i\in {\mathbb{R}}^ 1\}\) is the family of functions spanned by the \(\phi_ i(.)\); \(\Pi =\{(x_ 1,x_ 2)|\) \(a\leq x_ 1\leq a+d\), \(b\leq x\leq b+d\}\) is the given square in \({\mathbb{R}}^ 2\); \(\psi (t)=[\psi_ 1(t)t^{-k_ 1}\), \(\psi_ 2(t)t^{-k_ 2}]\), \(t\in (0,1]\), is a curve in \({\mathbb{R}}^ 2\), where \(\psi_ 1(\cdot)\), \(\psi_ 2(\cdot)\in \Sigma\); \(k_ 1,k_ 2\) are non-negative integers. Then, there is a constant \(d_ 1>0\) such that there is a square \(\Pi_ 1\subset \Pi\) with side \(d_ 1\) which the curve \(\psi\) (t), \(t\in (0,1]\), does not intersect.
The author extends this result and applies it to special classes of nonanalytic functions, namely to the case when components \(\psi_ 1(t)\) and \(\psi_ 2(t)\) of the curve \(\psi\) (t), \(t\in (0,1]\), are solutions of a differential equation of a special type.
Reviewer: O.I.Nikonov
91A23 Differential games (aspects of game theory)