Prior compact-manifold control under GCC and unique continuation

A smooth manifold without boundary is a topological manifold equipped with a smooth structure, as in Smooth manifolds and their smooth charts. Compactness means that every open cover has a finite subcover, as in Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right. These describe the compact-manifold setting of the comparison; the exterior domain studied in the article is not thereby assumed compact.

In the local terminology of Reachability and observation vocabulary for controlled evolutions, internal control acts through a source in a specified region of the manifold, with any boundary data prescribed independently of the control. In the endpoint language of Reachability and observation vocabulary for controlled evolutions, exact controllability means that each allowed pair of initial and terminal states can be joined by an admissible controlled solution. Semi-global describes a result for arbitrarily prescribed bounded sets of states, with parameters and hypotheses allowed to depend on the chosen bound; it does not mean an unqualified global result with uniform constants. Stabilization concerns decay toward a specified equilibrium under a specified damped or feedback evolution. The cited language defines exponential stabilization when an exponential rate is actually asserted; this remark does not prescribe a decay rate.

For the compact comparison, supply a smooth Riemannian metric $g$ on the Hausdorff second-countable smooth manifold $M$ without boundary, in the sense of Riemannian metric and riemannian manifold. Its Levi–Civita connection $\mathrm{\nabla }$ is the metric-compatible torsion-free affine connection of Levi civita connection; existence and uniqueness for the supplied metric follow from Fundamental theorem of riemannian geometry. A geodesic is affinely parametrized for this connection: $D_t{\gamma }^{\mathrm{\prime }}(t)=0$, as in Geodesic of an affine connection. Unit speed means $\mathrm{\mid }{\gamma }^{\mathrm{\prime }}(t){\mathrm{\mid }}_g=\sqrt{g_{\gamma (t)}({\gamma }^{\mathrm{\prime }}(t),{\gamma }^{\mathrm{\prime }}(t))}=1$, as in Riemannian speed and length. Using the boundaryless complete-geodesic convention recorded in the External input of Independent generalized reflected-ray and GCC convention, for a specified open control region $\omega \subseteq M$, GCC means that there is a common $T>0$ such that every complete unit-speed geodesic $\gamma :\mathbb{R}\to M$ satisfies $\gamma (t)\in \omega$ for some $t\in (0,T)$. This uses only that supplier’s boundaryless convention, not its reflected-ray structures for Euclidean domains. In the local terminology of Reachability and observation vocabulary for controlled evolutions, a unique continuation assumption is an additional implication that a solution in a specified class which vanishes on the relevant observation region must vanish throughout the relevant domain. These local explanations do not identify the detailed continuation hypotheses in the cited papers. The precise equations, solution classes, observation intervals and hypotheses belong to the individual results; neither condition is proved by this remark.