Polynomial Control Systems: Invariant Sets given by Algebraic Equations/Inequations

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Consider a nonlinear input-affine control system x(t) = f(x(t)) + g(x(t))u(t), y(t) = h(x(t)), where f, g, h are polynomial functions. Let S be a set given by algebraic equations and inequations (in the sense of =). Such sets appear, for instance, in the theory of the Thomas decomposition, which is used to write a variety as a disjoint union of simpler subsets. The set S is called controlled invariant if there exists a polynomial state feedback law u(t) = α(x(t)) such that S is an invariant set of the closed loop system x = (f + gα)(x). If it is possible to achieve this goal with a polynomial output feedback law u(t) = β(y(t)), then S is called controlled and conditioned invariant. These properties are discussed and algebraically characterized, and algorithms are provided for checking them with symbolic computation methods.