Exponential turnpike conjecture for semilinear control with local control

Consider the control problem defined by the state equation and cost functional referred to as

$$-$$

, with control domain $\omega\subsetneq\Omega$. Let $y_0\in L^{\infty}(\Omega)$ be an initial datum and let $z\in L^{\infty}(\omega_0)$ be a target. For a time horizon $T$, let $u^T$ be a minimizer of the finite-horizon functional, and write $y^T$ for its optimal state. An optimal pair $(\overline{u},\overline{y})$ for the corresponding steady control problem exists.

Exponential turnpike conjecture. There are constants $K>0$ and $\mu>0$, independent of $T$, such that

$$\|u^T(t)-\overline{u}\|_{L^{\infty}(\omega)}+\|y^T(t)-\overline{y}\|_{L^{\infty}(\Omega)}\leq K\left[\exp(-\mu t)+\exp(-\mu(T-t))\right],\qquad \forall t\in[0,T].$$

This conjecture would extend the exponential turnpike property from the everywhere-controlled setting to arbitrary targets with a local control domain. The paper proves convergence of time averages for arbitrary targets when the control acts everywhere; establishing the displayed uniform exponential estimate for $\omega\subsetneq\Omega$ remains open.