Locally minimax and minimax controls of linear discrete systems (Q1290675)

This paper mainly considers the following linear discrete system: \[ x_{t+ 1}= Ax_t+ Bu_t+ Fv_t, \] where \(x_t\in \mathbb{R}^m\) is a state, \(u_t\in \mathbb{R}^k\) is a control and \(v_t\in \mathbb{R}^l\) is an \(l^2\)-perturbation and \(A\), \(B\) and \(F\) are some matrices. The cost functional is \[ J_1(u,v)= \sum^\infty_0(x^T_t Q_u x_t+ u^T_t u_t- \gamma^2 v^T_t v_t) \] with \(Q_u= Q^T_u\geq 0\) and \(\gamma\neq 0\). The author proves that the worst disturbances and the minimax control for this functional are always the locally worst disturbance and the locally minimax control respectively, but the converse is true if and only if some frequency condition holds. It is shown that the set of all stable state-dependent feedbacks are optimal controls without disturbances contains a subset that corresponds to minimax controls under disturbances. Some conditions, under which a given feasible disturbance presented as a linear feedback is the worst one for some integral quadratic functional with a nonnegative-definite weight matrix, are found.