On the turnpike phenomenon for optimal boundary control problems with hyperbolic systems

TL;DR

This paper investigates the turnpike phenomenon in optimal boundary control problems governed by hyperbolic PDEs, showing that for large time horizons, dynamic solutions approximate steady states with convergence rates and extending results to integer-constrained controls, supported by numerical examples.

Contribution

It demonstrates the turnpike phenomenon for hyperbolic systems with boundary controls, including cases with integer constraints, and provides convergence rates and numerical validation.

Findings

Solution approximates steady state as T increases

Convergence rate of 1/T in L^2 norm

Integer control solutions align with static problem for large T

Abstract

We study problems of optimal boundary control with systems governed by linear hyperbolic partial differential equations. The objective function is quadratic and given by an integral over the finite time interval $(0,\, T)$ that depends on the boundary traces of the solution. If the time horizon $T$ is sufficiently large, the solution of the dynamic optimal boundary control problem can be approximated by the solution of a steady state optimization problem. We show that for $T\rightarrow\infty$ the approximation error converges to zero in the sense of the norm in the Hilbert space $L^2(0,\,1)$ with the rate $1/T$, if the time interval $(0,\, T)$ is transformed to the fixed interval $(0,\, 1)$. Moreover, we show that also for optimal boundary control problems with integer constraints for the controls the turnpike phenomenon occurs. In this case the steady state optimization problem also has the integer constraints. If $T$ is sufficiently large, the integer part of each solution of the dynamic optimal boundary control problem with integer constraints is equal to the integer part of a solution of the static problem. A numerical verification is given for a control problem in gas pipeline operations.