Dissipativity of some class of uncertain systems

Abstract

We consider the system

$$ \dot x = A\left( \cdot \right)x + B\left( \cdot \right)u, u = S\left( \cdot \right)x, t \geqslant t_0 , $$

where A(·) ∈ ℝ^n×n, B(·) ℝ^n×p, and S(·) ∈ ℝ^p×n. The entries of matrices A(·), B(·), and S(·) are arbitrary bounded functionals. We consider the problem of constructing a matrix H > 0 and finding relations between the entries of the matrices B(·) and S(·) such that for a given constant matrix R the inequality

$$ V\left( {x\left( t \right)} \right) < V\left( {x\left( {t_0 } \right)} \right) + \int\limits_{t_0 }^t {x*\left( \tau \right)Rx\left( \tau \right)d\tau ,} $$

where V(x) = x*Hx, is satisfied. This problem is solved for the cases where matrix A(·) has p sign-definite entries on the upper part of some subdiagonal or on the lower part of some superdiagonal. It is assumed also that all entries located to the left (or to the right) of the sign-definite entries are equal to zero.