Let \(M\) be a smooth manifold and \(x=(x_1, \dots, x_n)^\top\) a system of local coordinates. One considers a nonlinear feedback control system with disturbances of the following form: \[ \dot x=f(x) +g(x)u +k(x)d,\;z^\top =\bigl(h(x)^\top, u^\top \bigr),\;f(x_0) =0,\;h(x_0) =0. \] Here \(u\in \mathbb{R}^m\) is the control input, \(d\in \mathbb{R}^r\) the disturbance and \(z\in \mathbb{R}^s\) the penalty variable. The functions \(f,g,k\), and \(h\) are assumed to be sufficiently smooth. Given a nonnegative constant \(\gamma\) the \({\mathcal H}_\infty\)-suboptimal control problem consists in finding a feedback control \(u= \alpha(x)\) with \(\alpha (x+0)=0\) such that for each \(x\in M\) there exists a constant \(K(x)\) with \[ \int^T_0 \bigl|z(t) \bigr|^2 dt\leq \gamma^2 \int^T_0\bigl |d(t) \bigr|^2 dt+K \bigl(x(0) \bigr) \] for all \(T\geq 0\) and all square integrable \(d\) on \([0,T]\).
The authors consider such a control problem on the manifold \(M=SO(3)\) modelling a rigid spacecraft. Their penalty function involves the kinetic energy of the spacecraft and the geodesic distance from a reference point on \(M\). Using the Euler angles as coordinates they succeed to give an explicit global solution for the case \(\gamma >1\).