The turnpike property in semilinear control

TL;DR

This paper proves an exponential turnpike property for semilinear control problems, demonstrating convergence of optimal controls and states to steady solutions, with numerical simulations supporting the theoretical results.

Contribution

It establishes the exponential turnpike property for semilinear control problems with small targets and extends results to large targets with control acting everywhere.

Findings

Exponential turnpike property proven for small targets.

Convergence of the averaged functional to the steady one for large targets.

Numerical simulations validate theoretical results.

Abstract

An exponential turnpike property for a semilinear control problem is proved. The state-target is assumed to be small, whereas the initial datum can be arbitrary. Turnpike results are also obtained for large targets, requiring that the control acts everywhere. In this case, we prove the convergence of the infimum of the averaged time-evolution functional towards the steady one. Numerical simulations are performed.