A flatness proof of the exponential turnpike phenomenon for linear-quadratic optimal control problems

TL;DR

This paper presents a flatness-based analysis of the exponential turnpike phenomenon in linear-quadratic optimal control, offering a new perspective that simplifies understanding of the turnpike mechanism via differential flatness and polynomial differential equations.

Contribution

It introduces a flatness viewpoint to analyze the turnpike phenomenon, connecting it to the spectral properties of reduced Euler-Lagrange equations and clarifying effects of endpoint constraints.

Findings

Turnpike behavior arises from stable-unstable splitting in the flat equation.

Optimal trajectories exhibit boundary layers and interior arcs exponentially close to the static optimum.

The flatness approach clarifies effects of semidefinite weights and endpoint constraints.

Abstract

We revisit finite-dimensional linear-quadratic optimal control from the viewpoint of differential flatness. If the pair (A, B) is controllable, then the linear control system is flat, and every trajectory can be parametrized by a flat output and finitely many of its derivatives. Once this parametrization is inserted into the quadratic functional, the Euler-Lagrange condition becomes a linear differential equation with constant coefficients, or more generally a polynomial matrix differential equation. After reduction to Smith normal form, this equation decouples into scalar constant-coefficient equations, and its solutions are exponentialpolynomials. This yields a viewpoint on the turnpike phenomenon that is quite different from the classical Hamiltonian-Riccati analysis: the turnpike mechanism appears directly from the stable-unstable splitting of the reduced flat equation. In particular, when the reduced Euler-Lagrange operator has no purely imaginary characteristic roots and when the endpoint constraints act nondegenerately on the stable and unstable modes, the optimal trajectory consists of a left-boundary layer, a right-boundary layer, and a long interior arc exponentially close to the static optimum. The same viewpoint also clarifies what changes when some weights are only semidefinite: the order of the reduced equation may drop, some endpoint conditions may become incompatible, and polynomial or oscillatory modes may destroy the exponential turnpike. It also gives a natural meaning to certain endpoint constraints on the control and on finitely many derivatives of the control: such traces are not defined on the ambient L 2 control space, but they are meaningful on the smooth extremals selected by the reduced Euler-Lagrange equation. We formulate this principle as a general theorem and illustrate it in detail on the double integrator.