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EE291E Lecture Notes 9. Controllers for Safety for Continuous Systems

where x ∈ X where X = R, u ∈ U is the set of control inputs and d ∈ D is the set of disturbance inputs, where U andD are convex and compact subsets of R and R respectively; f is a vector field, assumed to be globally Lipschitz in x and continuous in u and d; and the initial state x(0) ∈ Init where Init ⊆ R. We will use U to denote the set of piecewise continuous functions from R to U and D to denote the set of piecewise continuous functions from R to D. Consider a set of states F ⊆ R. Try to establish the maximal controlled invariant subset of F , i.e. the largest set of initial states for which there exists a controller that manages to keep all executions inside F . Somewhat informally this set can be characterized as: