Pradyumna Paruchuri

Shruti Kotpalliwar, Pradyumna Paruchuri, Karmvir Singh Phogat et al.

In this article we present a geometric discrete-time Pontryagin maximum principle (PMP) on matrix Lie groups that incorporates frequency constraints on the controls in addition to pointwise constraints on the states and control actions directly at the stage of the problem formulation. This PMP gives first order necessary conditions for optimality, and leads to two-point boundary value problems that may be solved by shooting techniques to arrive at optimal trajectories. We validate our theoretical results with a numerical experiment on the attitude control of a spacecraft on the Lie group SO(3).

Pradyumna Paruchuri, Debasish Chatterjee

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.

Pradyumna Paruchuri, Debasish Chatterjee

We establish a Pontryagin maximum principle for discrete time optimal control problems under the following three types of constraints: a) constraints on the states pointwise in time, b) constraints on the control actions pointwise in time, and c) constraints on the frequency spectrum of the optimal control trajectories. While the first two types of constraints are already included in the existing versions of the Pontryagin maximum principle, it turns out that the third type of constraints cannot be recast in any of the standard forms of the existing results for the original control system. We provide two different proofs of our Pontryagin maximum principle in this article, and include several special cases fine-tuned to control-affine nonlinear and linear system models. In particular, for minimization of quadratic cost functions and linear time invariant control systems, we provide tight conditions under which the optimal controls under frequency constraints are either normal or abnormal.