Lyapunov functions for a class of large-scale systems

The behavior of solutions of a class of large-scale systems with the bounded coefficients of the form [1] has been studied. The function
V_{i}(x_{i},t)=A_{i}(t)x_{i}^{2}
, whose coefficient
A_{i}(t)
is bounded for all
t \geq 0
, gives the conditions on the Lyapunov function such that the system (1) is either asymptotically stable or unstable. Since the function of the form
e^{-t}x^{2}
is not positive definite in the Lyapunov sense, a Lyapunov function of the form
V_{i}(x_{i},t)=e^{-Q_{i}(t)}x_{i}^{2}
where
Q_{i}(t)
is bounded is constructed which is positive definite in the Lyapunov sense. A systematic procedure for the construction of such a function is given. The approach is quite general. The method is inspired from the papers of Grujic and Siljak [1], Siljak [2], [3], Weissenberger [4], Bailey [5], and Michel et al. [7], [8].