The property of so-called uniform covering is proved for a family of linear operators generated

by a differential equation linear with respect to functional parameters. Using this fact, a uniform

estimate of the distance to the level set of a nonlinear operator is obtained on a broader set than

a neighborhood of the examined point in a Banach space. This allows to prove an approximation

theorem for a system with sliding mode controls and endpoint state constraints.

Key words:

nonlinear operator, covering, weak-* topology, sliding modes.

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