Polynomial Turnpike Property for a Class of Infinite-Dimensional Oscillating Systems

TL;DR

This paper proves a polynomial turnpike estimate for an infinite-dimensional system of oscillators, using spectral analysis and Riesz basis construction, with a novel example involving a rotating body-beam system.

Contribution

It introduces the first polynomial (not exponential) pointwise turnpike estimate for infinite-dimensional oscillating systems.

Findings

Established polynomial turnpike estimate for infinite-dimensional oscillators

Constructed a Riesz basis for spectral analysis

Provided a concrete rotating body-beam example

Abstract

We establish a polynomial turnpike estimate for an optimal control problem consisting of a system of infinitely many controlled oscillators, considered as an abstract differential equation in a Hilbert space, with a quadratic cost. Our proof relies on spectral considerations and on the construction of a Riesz basis. A concrete example is given, which involves a rotating bodybeam system. To our knowledge, this is the first example of a pointwise turnpike estimate around a steady-state that is polynomial but not exponential.