Optimal ellipsoidal bounds of the reachable set of linear systems with an uncertain matrix
The linear dynamical system \[ \dot x = (A_0 (t) + A(t))x + w(t), \quad w(t) \in E_1, \quad x_0\in E_0 , \] \[ A(t) \in M(t) = \{ A \subset M_n : | | A| | _{\alpha} \leq h \} , \] is considered. Here \(E_0 = E(a_0, Q_0)\) and \(E_1 = E(c,B)\) are ellipsoids in \(R^n\) and \(| | A| | _{\alpha}\) is a norm in the space \(M_n\) of \(n\times n\) -matrices. The author finds the upper ellipsoidal bounds \(E(a(t),Q(t))\), \[ E(a(t),Q(t)) \supset D(t), \] of the attainable sets \(D(t)\) of the system for some types of norms \(| | A| | _{\alpha}\). The differential equations for the centers \(a(t)\) and for the matrices \(Q(t)\) of the ellipsoids that define the upper bounds are given.
scientific article; zbMATH DE number 1158011