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Most of the spaces of random variables are non-Hausdorff linear topological spaces, whereas a nuinber of useful results of the theory of vector topologies require the Hausdorff separation axiom for the underlying spaces. Therefore we review and complete the machinery of non-Hausdorff linear topological spaces and the techniques for their representation with suitable Hausdorff linear topological spaces, and then we work out some examples, which emphasize the ideas involved in a consistent application of the topological point of view. The final part of the paper is devoted to the discussion of the pathology, which arises when we try to treat the special case of almost everywhere convergence in the same framework.

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We consider controlled invariance for cones, translated cones, polyhedra and various special polyhedral structures. For any polyhedron, if controlled invariance occurs, then all and nothing but the admissible controls can be obtained by an inequative feedback controller. For each special polyhedral structure we compare this feedback controller against piecewise affine, linear and affine feedback controller.Abstract-An observer design is presented which makes use of bounds on the slope of system nonlinearities. Necessary and sufficient conditions are derived for the feasibility of the design. A class of state feedback control laws are characterized which, when implemented with the observer states, ensure global asymptotic stability. One such certainty-equivalence design is illustrated on an active magnetic bearing example.Index Terms-Certainty-equivalence, linear matrix inequalities, nonlinear observers, output-feedback control. P. Kokotović is with the