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  1. We present a coordinate-free version of Fefferman’s solution of Whitney’s extension problem in the space Cm−1,1(Rn). While the original argument relies on an elaborate induction on collections of partial derivatives, our proof uses the language of ideals and translation-invariant subspaces in the ring of polynomials. We emphasize the role of compactness in the proof, first in the familiar sense of topological compactness, but also in the sense of finiteness theorems arising in logic and semialgebraic geometry. These techniques may be relevant to the study of Whitney-type extension problems on sub-Riemannian manifolds where global coordinates are generally unavailable. 
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  2. This paper introduces a strategy for satisfying basic control objectives for systems whose dynamics are almost entirely unknown. This setting is motivated by a scenario where a system undergoes a critical failure, thus significantly changing its dynamics. In such a case, retaining the ability to satisfy basic control objectives such as reach-avoid is imperative. To deal with significant restrictions on our knowledge of system dynamics, we develop a theory of myopic control. The primary goal of myopic control is to, at any given time, optimize the current direction of the system trajectory, given solely the limited information obtained about the system until that time. Building upon this notion, we propose a control algorithm which simultaneously uses small perturbations in the control effort to learn local system dynamics while moving in the direction which seems to be optimal based on previously obtained knowledge. We show that the algorithm results in a trajectory that is nearly optimal in the myopic sense, i.e., it is moving in a direction that seems to be nearly the best at the given time, and provide formal bounds for suboptimality. We demonstrate the usefulness of the proposed algorithm on a high-fidelity simulation of a damaged Boeing 747 seeking to remain in level flight. 
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  3. We consider the problem of safety assessment of a dynamical system for which no model and just limited data on the states is available. That is, given samples of the state {x(t i )} N i=1 at time instances t 1 ≤ t 2 ≤ ··· ≤ t N and some other side information in terms of the regularity of the state evolutions, we are interested in checking whether x(T) ∉ Xu, where T > t N and Xu ⊂ R n (the unsafe set) are pre-specified. To this end, we use piecewise-polynomial approximations of the trajectories based on the data along with the regularity side information to formulate a data-driven differential inclusion model. For these classes of data-driven differential inclusions, we propose a safety assessment theorem based on barrier certificates. The barrier certificates are then found using polynomial optimization. The method is illustrated by two examples. 
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