Geodesic
Conditions for the~non-negativity of integral quadratic forms with constant
Sbornik. Mathematics, Tome 193 (2002) no. 4, pp. 531-557

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An integral quadratic functional with constant coefficients on a half-axis is considered. A necessary and sufficient condition for its non-negativity at all square integrable pairs of functions related by a linear ODE is proposed, which is based on the Hamilton–Jacobi inequality. A connection between this condition and the well-known frequency criterion is established. 
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     author = {A. A. Milyutin},
     title = {Conditions for the~non-negativity of integral quadratic forms with constant},
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A. A. Milyutin. Conditions for the~non-negativity of integral quadratic forms with constant. Sbornik. Mathematics, Tome 193 (2002) no. 4, pp. 531-557. http://geodesic.mathdoc.fr/item/SM_2002_193_4_a3/