Controlled invariant varieties for second order polynomial control systems

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We investigate controlled invariant varieties for second order input-affine control systems with polynomial nonlinearity. For autonomous, first order systems, a variety V is said to be invariant if any trajectory starting in V remains in V for all times. We discuss how to adapt this notion in the second order case and give conditions to decide whether a variety is invariant for a second order system using symbolic computation. This results serve as a foundation to characterise controlled invariant varieties, i.e., varieties that can be rendered invariant by a polynomial state feedback.