The paper proposes method of quadratic Lyapunov functional construction for stability analysis of quasilinear functional differential equations. The approach is based on analysis of linear operator semigroup, which is acting in the partially ordered space of countable additive symmetric matrix-valued measures. The weak infinitesimal operator of this semigroup helps to find a positive matrix measure for construction of a quadratic functional, that defines necessary and sufficient condition of exponential stability for linear functional differential equation. It is shown how this quadratic functional can be successfully applied for stability analysis of quasilinear equation.

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