On the maximum possible negativity margin for the first derivative (first difference) of a quadratic Lyapunov function. (English. Russian original) Zbl 1065.93030

Antonovskaya, O. G.

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.

Reviewer: Vladimir Răsvan (Compiègne)

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

Keywords:

linear system; quadratic Lyapunov function; negativity margin; level surface; eigenvalues