Discrete time optimal control with frequency constraints for non-smooth systems

We present a Pontryagin maximum principle for discrete time optimal control problems with (a) pointwise constraints on the control actions and the states, (b) frequency constraints on the control and the state trajectories, and (c) nonsmooth dynamical systems. Pointwise constraints on the states and the control actions represent desired and/or physical limitations on the states and the control values; such constraints are important and are widely present in the optimal control literature. Constraints of the type (b), while less standard in the literature, effectively serve the purpose of describing important spectral properties of inertial actuators and systems. The conjunction of constraints of the type (a) and (b) is a relatively new phenomenon in optimal control but are important for the synthesis control trajectories with a high degree of fidelity. The maximum principle established here provides first order necessary conditions for optimality that serve as a starting point for the synthesis of control trajectories corresponding to a large class of constrained motion planning problems that have high accuracy in a computationally tractable fashion. Moreover, the ability to handle a reasonably large class of nonsmooth dynamical systems that arise in practice ensures broad applicability our theory, and we include several illustrations of our results on standard problems

Algorithmic construction of Lyapunov functions for continuous vector fields via convex semi-infinite programs

This article presents a novel numerically tractable technique for synthesizing Lyapunov functions for equilibria of nonlinear vector fields. In broad strokes, corresponding to an isolated equilibrium point of a given vector field, a selection is made of a compact neighborhood of the equilibrium and a dictionary of functions in which a Lyapunov function is expected to lie. Then an algorithmic procedure based on the recent work [DACC22] is deployed on the preceding neighborhood-dictionary pair and charged with the task of finding a function satisfying a compact family of inequalities that defines the behavior of a Lyapunov function on the selected neighborhood. The technique applies to continuous nonlinear vector fields without special algebraic structures and does not even require their analytical expressions to proceed. Several numerical examples are presented to illustrate our results.Comment: 26 pages. Submitte

A numerical algorithm for attaining the Chebyshev bound in optimal learning

Given a compact subset of a Banach space, the Chebyshev center problem consists of finding a minimal circumscribing ball containing the set. In this article we establish a numerically tractable algorithm for solving the Chebyshev center problem in the context of optimal learning from a finite set of data points. For a hypothesis space realized as a compact but not necessarily convex subset of a finite-dimensional subspace of some underlying Banach space, this algorithm computes the Chebyshev radius and the Chebyshev center of the hypothesis space, thereby solving the problem of optimal recovery of functions from data. The algorithm itself is based on, and significantly extends, recent results for near-optimal solutions of convex semi-infinite problems by means of targeted sampling, and it is of independent interest. Several examples of numerical computations of Chebyshev centers are included in order to illustrate the effectiveness of the algorithm.Comment: 22 pages, 16 figure