A continuation theorem on periodic solutions of regular nonlinear systems and its application to the exact tracking problem for the inverted spherical pendulum
The authors present a continuation method to obtain a family of \(T\)-periodic solutions \(\{x_s(\cdot)\}_{0\leq s < \delta}\) associated to a family of \(T\)-periodic systems \[ \dot{x}_s(t) = F(t,s,x_s(t))\quad \forall s \in [0,\delta), \] such that \[ x_0(\cdot) = \tilde{x}(\cdot), \] where \(\tilde{x}\) is an assigned \(T\)-periodic solution. The authors apply these conditions to the exact tracking problem for the inverted spherical pendulum. They consider the following problem: given an arbitrary \(T\)-periodic curve \(\gamma\in C^3({\mathbb R}, {\mathbb R}^3)\), find a control force \(f\in C({\mathbb R}, {\mathbb R}^3)\), such that, if \(x(0) = \gamma(0)\), then \(x(t) = \gamma(t)\) \(\forall t \geq 0\).
scientific article; zbMATH DE number 5810935