Nelly Villamizar

A computational method for the synthesis of time-optimal feedback control laws for linear nilpotent systems is proposed. The method is based on the use of the bang-bang theorem, which leads to a characterization of the time-optimal trajectory as a parameter-dependent polynomial system for the control switching sequence. A deflated Newton’s method is then applied to exhaust all the real roots of the polynomial system. The root-finding procedure is informed by the Hermite quadratic form, which provides a sharp estimate of the number of real roots to be found. In the second part of this paper, the polynomial systems are sampled and solved to generate a synthetic dataset for the construction of a time-optimal deep neural network (NN)—interpreted as a binary classifier—via supervised learning. Numerical tests in integrators of increasing dimension assess the accuracy, robustness, and real-time control capabilities of the approximate control law.

A computational method for the synthesis of time-optimal feedback control laws for linear nilpotent systems is proposed. The method is based on the use of the bang-bang theorem, which leads to a characterization of the time-optimal trajectory as a parameter-dependent polynomial system for the control switching sequence. A deflated Newton's method is then applied to exhaust all the real roots of the polynomial system. The root-finding procedure is informed by the Hermite quadratic form, which provides a sharp estimate on the number of real roots to be found. In the second part of the paper, the polynomial systems are sampled and solved to generate a synthetic dataset for the construction of a time-optimal deep neural network -- interpreted as a binary classifier -- via supervised learning. Numerical tests in integrators of increasing dimension assess the accuracy, robustness, and real-time-control capabilities of the approximate control law.

Geometrically continuous splines are piecewise polynomial functions defined on a collection of patches which are stitched together through transition maps. They are called G r G^{r} -splines if, after composition with the transition maps, they are continuously differentiable functions to order r r on each pair of patches with stitched boundaries. This type of spline has been used to represent smooth shapes with complex topology for which (parametric) spline functions on fixed partitions are not sufficient. In this article, we develop new algebraic tools to analyze G r G^r -spline spaces. We define G r G^{r} -domains and transition maps using an algebraic approach, and establish an algebraic criterion to determine whether a piecewise function is G r G^r -continuous on the given domain. In the proposed framework, we construct a chain complex whose top homology is isomorphic to the G r G^{r} -spline space. This complex generalizes Billera-Schenck-Stillman homological complex used to study parametric splines. Additionally, we show how previous constructions of G r G^r -splines fit into this new algebraic framework, and present an algorithm to construct a basis for G r G^r -spline spaces. We illustrate how our algebraic approach works with concrete examples and prove a dimension formula for the G r G^r -spline space in terms of invariants to the chain complex. In some special cases, explicit dimension formulas in terms of the degree of splines are also given.

We construct new families of Artinian Gorenstein graded $K$-algebras of arbitrary codimension having binomial Macaulay dual generators and satisfying the weak or the strong Lefschetz property. This is a companion paper to \cite{ADFMMSV}, which studies codimension three algebras having binomial Macaulay dual generators in great depth, establishing in particular that they enjoy the strong Lefschetz property.

We present a generalized formulation for reweighted least squares approximations. The goal of this article is twofold: firstly, to prove that the solution of such problem can be expressed as a convex combination of certain interpolants when the solution is sought in any finite-dimensional vector space; secondly, to provide a general strategy to iteratively update the weights according to the approximation error and apply it to the spline fitting problem. In the experiments, we provide numerical examples for the case of polynomials and splines spaces. Subsequently, we evaluate the performance of our fitting scheme for spline curve and surface approximation, including adaptive spline constructions.

We consider spaces of multivariate splines defined on a particular type of simplicial partitions that we call (generalized) oranges. Such partitions are composed of a finite number of maximal faces with exactly one shared medial face. We reduce the problem of finding the dimension of splines on oranges to computing dimensions of splines on simpler, lower-dimensional partitions that we call projected oranges. We use both algebraic and Bernstein-Bézier tools.