1990

Given a 2-coloring of the vertices of a regular n-gon P, how many parallel lines are needed to separate the vertices into monochromatic subsets? We prove that floor(n/2) is a tight upper bound, and also provide an O(n log n) time algorithm to determine the direction that gives the minimum number of lines. If the polygon is a non-regular convex polygon, then n-3 lines may be necessary, while n-2 lines always suffice. This problem arises in machine learning and has implications about the representational capabilities of some neural networks.

We describe a speedup technique that uses extrapolatory methods to predict the weights in a Neural Network using Back Propagation (BP) learning. The method is based on empirical observations of the way the weights change as a function of time. We use numerical function fitting techniques to determine the parameters of an extrapolation function and then use this function to project weights into the future. Significant computational savings result by using the extrapolated weights to jump over many iterations of the standard algorithm, achieving comparable performance with fewer iterations.

This paper presents a geometric study of controllability for discrete-time nonlinear systems. Various accessibility properties are characterized in terms of Lie algebras of vector fields. Some of the results obtained are parallel to analogous ones in continuous-time, but in many respects the theory is substantially different and many new phenomena appear.

The second edition (1998) is now online; please follow that link.

We prove that for any family of n-dimensional controllable linear systems, continuously parameterized by up to three parameters, and for any continuous selection of n eigenvalues (in complex conjugate pairs), there is some dynamic controller of dimension 3n which is itself continuously parameterized and for which the closed-loop eigenvalues are these same eigenvalues, each counted 4 times. An analogous result holds also for smooth parameterizations.

Previous results about input to state stabilizability are shown to hold even for systems which are not linear in controls, provided that a more general type of feedback be allowed. Applications to certain stabilization problems and coprime factorizations, as well as comparisons to other results on input to state stability, are also briefly discussed.d local minima may occur, if the data are not separable and sigmoids are used.

Results are given on the integrability of certain distributions which arise from smoothly parametrized families of diffeomorphisms acting on manifolds. Applications to control problems and in particular to the problem of sampling are discussed. Pages 42-50 apply the results to the control of continuous time systems; this is an exposition of some of the basic results of the Lie algebraic accessibility theory.

This paper shows the existence of (nonlinear) smooth dynamic feedback stabilizers for linear time invariant systems under input constraints, assuming only that open-loop asymptotic controllability and detectability hold.

This paper surveys some well-known facts as well as some recent developments on the topic of stabilization of nonlinear systems. (NOTE: figures are not included in file; they were pasted-in.)