Second Variation Condition and Quadratic Integral Inequalities with Higher Order Derivatives

The positivity of quadratic integrals involving variable coefficients and derivatives of any order is studied. The result is determined by the solution of an initial value problem for a system of first order nonlinear differential equations. The system is identified as the matrix Riccati differential equation in control theory. A complete conclusion is reached by considering the cases when the solution is bounded and when the solution is unbounded.