Nonlinear control for trapping obstacle configurations

PDF p. 22: §5.2(E)

Assume the Axiom of Choice. Let $n\in \{2,3\}$ and consider a compact smooth obstacle $\mathrm{\Theta }\subset {\mathbb{R}}^n$ with connected exterior $\mathrm{\Omega }={\mathbb{R}}^n\setminus \mathrm{\Theta }$, allowing trapping. In particular, one may consider suitable configurations of disjoint strictly convex obstacles; strict convexity means that the open segment between distinct points of a body lies in its interior. For $R_0$ with $\mathrm{\Theta }\subset B_{R_0}(0)$, set ${\mathrm{\Omega }}_0=B_{R_0}(0)\setminus \mathrm{\Theta }$, ${\mathrm{\Gamma }}_1=\mathrm{\partial }{B}_{R_0}(0)$ and ${\mathrm{\Gamma }}_2=\mathrm{\partial }\mathrm{\Theta }$.

Can nonlinear control results analogous to those in the article be established for suitable trapping configurations? For example, for $T>0$ and $s\in \{1,2\}$, determine conditions under which sufficiently small initial and target states in $H^s({\mathrm{\Omega }}_0)$ with zero trace on ${\mathrm{\Gamma }}_2$ can be joined by a solution in $C([0,T];H^s({\mathrm{\Omega }}_0))$ of the cubic Dirichlet problem, with control $u{\mathrm{\mid }}_{{\mathrm{\Gamma }}_1}\in C([0,T];H^{s-1\mathrm{/}2}({\mathrm{\Gamma }}_1))$. The cubic equation is $i{u}_t+\mathrm{\Delta }u+\lambda \mathrm{\mid }u{\mathrm{\mid }}^{2}u=0$, with real nonzero $\lambda$; the equation is understood distributionally and boundary values as traces. An analogous question concerns the quintic nonlinearity $\lambda \mathrm{\mid }u{\mathrm{\mid }}^{4}u$.

Here $H^s({\mathrm{\Omega }}_0)$ uses complex scalars and the $L^2$ Bessel-potential restriction space with its quotient norm. For $j\in \{1,2\}$, $H^r({\mathrm{\Gamma }}_j)$, $r\in \{1\mathrm{/}2,3\mathrm{/}2\}$, uses the sum of squared whole-space $H^r({\mathbb{R}}^{n-1})$ norms of localized coordinate representatives in a finite smooth atlas and partition; for multiple boundary components the sum also runs over the components. The value trace is the bounded map ${\gamma }_j:H^s({\mathrm{\Omega }}_0)\to H^{s-1\mathrm{/}2}({\mathrm{\Gamma }}_j)$ extending smooth restriction. Thus the inner condition is ${\gamma }_{2}u(t)=0$ and the outer control is ${\gamma }_{1}u(t)$; initial and terminal states are values of the norm-continuous path at $0$ and $T$. The notation $C([0,T];X)$ means norm-continuous $X$-valued maps with the supremum norm.

For the boundary question put $Q=(0,T)\times {\mathrm{\Omega }}_0$. The distributional solution condition requires measurable representatives with $u,\mathrm{\mid }u{\mathrm{\mid }}^{m-1}u\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1(Q)$ and the identity ${\int }_Q(-iu{\mathrm{\partial }}_t\eta +u{\mathrm{\Delta }}_x\eta +\lambda \mathrm{\mid }u{\mathrm{\mid }}^{m-1}u\eta )dtdx=0(\eta \in C_c^{\mathrm{\infty }}(Q;\mathbb{C})),$ where $m=3$ for the cubic question and $m=5$ for its quintic variant. Pairings are complex bilinear, without conjugating $\eta$. The initial, terminal and boundary conditions above are imposed separately from this interior identity.

Another possible extension asks for small-data exact internal control of the corresponding exterior equations, with a forcing $\phi h$ supported away from the obstacle, in suitable state and control spaces. Exact control here means finding a control and an admissible solution joining each sufficiently small initial and target pair in the stated spaces. These are questions about suitable geometries and functional settings, not assertions that every trapping obstacle admits control or that the same spaces must suffice.

In §5.2(E), the article presents the extension to trapping obstacles as an open question. It reports linear outer-boundary controllability with a loss of derivatives for some strictly convex trapping configurations and identifies handling that loss in the nonlinear perturbation argument as the issue for the proposed extension. The article's cited linear results are reported here as its background attribution, not imported as independently verified premises. This item records the status in the frozen article; it does not certify the present state of the literature or prove either controllability or its failure.