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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
MSC:
 91A23 Differential games (aspects of game theory)