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On the maximum possible negativity margin for the first derivative (first difference) of a quadratic Lyapunov function. (English. Russian original) Zbl 1065.93030
Differ. Equ. 39, No. 11, 1645-1647 (2003); translation from Differ. Uravn. 39, No. 11, 1562-1563 (2003).
Consider the linear system $\dot{x} = Ax$ and the quadratic Lyapunov function $V(x) = x^TPx$ such that $$A$$ is Hurwitz and $$P>0$$. If the derivative of $$V$$ along the system, i.e. $$W(x) = x^T(A^TP + PA)x$$, is considered, it is stated that its maximal value on the level surface $$V(x) = V_0$$ is not less than $$2(\max_i\{\operatorname{Re}\,\lambda_i\})V_0$$ where $$\lambda_i$$ are the eigenvalues of $$A$$. A discrete-time analogue is also stated.

##### MSC:
 93D30 Lyapunov and storage functions 93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, $$L^p, l^p$$, etc.) in control theory 15A06 Linear equations (linear algebraic aspects) 15A18 Eigenvalues, singular values, and eigenvectors 34D08 Characteristic and Lyapunov exponents of ordinary differential equations
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