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Proposition 3.10: order-two local control

PDF p. 16: Proposition 3.10

Statement

Assume the Axiom of Choice and (H): $n \in {2, 3}$ , the nonempty compact obstacle $Θ \subset R^{n}$ has smooth boundary and connected exterior $Ω = R^{n} ∖ Θ$ , and its generalized reflected rays leave every compact set in finite time. Let $T > 0$ , $m \in {3, 5}$ and $λ \in R ∖ {0}$ . Let $φ$ be the control cutoff of Control region and nested real cutoffs. Put $H = H_{D}^{2} (Ω)$ and $X_{T} = C ([0, T]; H)$ with its supremum norm. The space $H$ is the order-two Dirichlet spectral space; it equals $H^{2} (Ω) \cap H_{0}^{1} (Ω)$ with equivalent norms, where $H_{0}^{1}$ is the $H^{1}$ closure of compactly supported smooth functions.

The three hypotheses of Abstract nonlinear control-space hypotheses hold at $s = 2$ for this $X_{T}$ . Consequently there is $δ > 0$ such that any $v_{0}, v_{T} \in H^{2} (Ω) \cap H_{0}^{1} (Ω)$ with both spectral $H$ norms at most $δ$ can be joined by an integral solution $v \in C ([0, T]; H)$ of $i \partial_{t} v + Δ_{D} v + λ ∣ v ∣^{m - 1} v = φ h$ , with $v (0) = v_{0}$ , $v (T) = v_{T}$ and a control $h \in C ([0, T]; H)$ . Thus the system is locally exactly controllable in $H^{2} (Ω) \cap H_{0}^{1} (Ω)$ ; equivalence of norms allows the smallness condition to be expressed in the ordinary $H^{2}$ norm. The integral solution is understood through the variation-of-constants formulation in Mild formulation of the controlled power NLS, with $σ = 2$ .

Facts & Assumptions

[F1]

The geometry and cutoff are those of Exterior and truncated obstacle domains and Control region and nested real cutoffs. The spectral scale Dirichlet spectral Sobolev scale and duality is complete, and Exterior H2 algebra and polynomial nonlinear estimates identifies $H$ with $H^{2} \cap H_{0}^{1}$ with equivalent norms. The weak Sobolev and closure conventions are those of Complex integer-order Sobolev notation.

[F2]

Under Choice, Scale isometries and free Schrödinger evolution gives a strongly continuous unitary group $W (t) = e^{i t Δ_{D}}$ on $H$ . Choice is assumed as in The Axiom of Choice.

[F3]

Exterior H2 algebra and polynomial nonlinear estimates gives, for $g (z) = λ ∣ z ∣^{m - 1} z$ and $u, v \in X_{T}$ , Bochner-integrability of $g (u), g (v)$ and the bounds $∥ g (v) ∥_{L^{1} H} \leq C T ∥ v ∥_{X_{T}}^{m}$ and $∥ g (v) - g (u) ∥_{L^{1} H} \leq C T (∥ v ∥_{X_{T}}^{m - 1} + ∥ u ∥_{X_{T}}^{m - 1}) ∥ v - u ∥_{X_{T}}$ .

[F4]

Continuous and mixed time-space function spaces fixes norm-continuous paths, strongly measurable Bochner $L^{1}$ functions and their norms. The Bochner integrability criterion, integral norm bound and commutation with bounded linear operators are supplied by Bochner integrability criterion, Bochner integral norm inequality and Bounded linear maps commute with Bochner integration. Scalar integrals of integrable functions are absolutely continuous by Absolute continuity of the integral.

[F5]

If (Y,d) is complete then Y^X is complete in the uniform metric, and so is C(X,Y) gives completeness of the continuous-function space for a complete target in the uniform metric. Abstract nonlinear control-space hypotheses specifies the three bounds required for the application of Theorem 3.7: abstract small-data perturbation result.

[F6]

Mild formulation of the controlled power NLS specifies the integral-solution concept and its equivalent variation-of-constants formulation on $H_{D}^{σ} (Ω)$ ; here $σ = 2$ , with the same assumed Choice principle, free group, nonlinearity and control force.

Proof

Given: The assumptions, spaces and parameters in the Statement.

Goal: Verify the three abstract perturbation hypotheses and apply the nonlinear control theorem.

1.1F1F4F5

The target $H$ is complete by F1. F5 makes the continuous path space complete in the truncated uniform metric; for distances below one that metric agrees with the supremum distance. Hence every supremum-norm Cauchy sequence converges in supremum norm. With the pointwise vector operations and supremum norm of F4, $X_{T}$ is a Banach space. Its inclusion into $C ([0, T]; H)$ is the identity and has norm one.

1.2F2F4

Let $f \in L^{1} (0, T; H)$ . Strong measurability of $r \mapsto W (- r) f (r)$ follows by approximating $f$ almost everywhere by finite-range measurable functions: for each fixed value, $r \mapsto W (- r) x$ is continuous, and its restriction to the compact interval is a uniform limit of finite-range measurable functions. Unitarity preserves the norm, so F4 gives its Bochner integrability. Set $F (t) = \int_{0}^{t} W (- r) f (r) d r$ . The integral norm bound gives $∥ F (t) - F (s) ∥_{H} \leq \int_{\min (s, t)}^{\max (s, t)} ∥ f (r) ∥_{H} d r$ , so F4 makes $F$ continuous. Strong continuity and unitarity imply that $W (t) F (t)$ is continuous: split its increment into $W (t) (F (t) - F (s))$ and $(W (t) - W (s)) F (s)$ .

2.1F2step 1.1

By F2, $W (\cdot) v_{0}$ is a continuous $H$ -valued path and $∥ W (\cdot) v_{0} ∥_{X_{T}} = ∥ v_{0} ∥_{H}$ . This proves the homogeneous hypothesis.

2.2F2F4step 1.2

By the group law and commutation of a bounded linear map with the Bochner integral, $W (t) F (t) = \int_{0}^{t} W (t - r) f (r) d r = : D f (t)$ . The integral norm bound and unitarity give $∥ D f ∥_{X_{T}} \leq ∥ f ∥_{L^{1} (0, T; H)}$ . This proves the retarded-forcing hypothesis, including existence and path continuity.

3.1F3F5step 2.1step 2.2

F3 gives the required nonlinear membership and both power bounds. Taking one constant $C_{0} \geq 1$ large enough for the constants in steps 2.1 and 2.2 and F3 verifies all three conditions of F5 for the same $X_{T}$ .

4.1F1F5F6step 3.1∎

The dimension, geometry, time, exponent, coefficient and cutoff meet the hypotheses of Theorem 3.7: abstract small-data perturbation result, and $s = 2$ lies in its stated range. Applying it with the space verified in step 3.1 gives the stated $δ$ , control and integral solution with the prescribed endpoints. The identification and norm equivalence in F1 give the stated formulation in $H^{2} \cap H_{0}^{1}$ and permit an ordinary- $H^{2}$ smallness radius. This proves the proposition.

Source notes

The frozen article is cited for attribution of Proposition 3.10. The deduction uses the separately stated control theorem and its mathematical prerequisites; the article is not used as a proof of those prerequisites.

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Sources

Nonlinear Schrödinger equation in an exterior domain, v2 · PDF page 16, Proposition 3.10