Linear damping decay and observability comparison

PDF p. 5: end of §1.4(i); PDF p. 6: continuation citing [25]

Assume the Axiom of Choice (The Axiom of Choice) and use the exterior Dirichlet realization and $L^2$ Hilbert space of Dirichlet Laplacian and positive shift. Let $W(t)=e^{it{\mathrm{\Delta }}_D}$ be the unitary group in Scale isometries and free Schrödinger evolution, and let $a\in L^{\mathrm{\infty }}(\mathrm{\Omega })$ be nonnegative. For the linear damped equation $i{u}_t+{\mathrm{\Delta }}_{D}u+ia(x)u=0$, uniform exponential stabilization in $L^2$ is equivalent to the existence of some $T,C>0$ such that $\mathrm{\parallel }{v}_0{\mathrm{\parallel }}_2^2\le C{\int }_0^T{\int }_{\mathrm{\Omega }}a(x)\mathrm{\mid }W(t)v_0(x){\mathrm{\mid }}^{2}dxdt(v_0\in L^2(\mathrm{\Omega }))\mathrm{.}$ This is the bounded-operator equivalence imported below, applied to the actual multiplication damping. Exponential stabilization means uniform constants $M,\gamma >0$ with $\mathrm{\parallel }u(t){\mathrm{\parallel }}_2\le M{e}^{-\gamma t}\mathrm{\parallel }u(0){\mathrm{\parallel }}_2$ for every initial state and all $t\ge 0$; observability has the norm convention of Reachability and observation vocabulary for controlled evolutions, and the time integrals have the convention of Continuous and mixed time-space function spaces.

In particular, for $\omega =\{x\in \mathrm{\Omega }:\mathrm{\mid }x\mathrm{\mid }>R\}$ of Control region and nested real cutoffs, the admissible coefficient $a={\mathbf{1}}_{\omega }$ gives exactly the unweighted observation ${\int }_0^T\mathrm{\parallel }W(t)v_0{\mathrm{\parallel }}_{L^2(\omega )}^{2}dt$. Bortot–Cavalcanti's exterior theorem covers this coefficient in its stated smooth compact connected nontrapping-obstacle setting, with nontrapping as defined in Non-trapping obstacle: their only coefficient conditions are $a\in L^{\mathrm{\infty }}$, $a\ge 0$ and $a\ge a_0>0$ outside an enclosing ball. Linear damping is their choice $g(z)=z$. The same result permits $a={\phi }^2$ for the smooth cutoff in Control region and nested real cutoffs, using the enclosing radius $R+1$; its observation is ${\int }_0^T\mathrm{\parallel }\phi W(t)v_0{\mathrm{\parallel }}_2^{2}dt$. Thus both the unweighted exterior observation and the smooth-cutoff observation have exact bounded-damping realizations.

For a general coefficient allowed by that theorem, the independent equivalence observes $a^{1\mathrm{/}2}v$, not automatically ${\mathbf{1}}_{\omega }v$. A lower bound $a\ge a_0$ on $\omega$ gives $a_0\mathrm{\parallel }v{\mathrm{\parallel }}_{L^2(\omega )}^2\le \int a\mathrm{\mid }v{\mathrm{\mid }}^2$; a two-sided comparison with the unweighted observation additionally holds when $a=0$ off $\omega$. The indicator specialization above needs no such extra inference.

The original article attributes the historical stabilization/observability comparison to Bortot–Cavalcanti [6] and Liu [25]. It contrasts that comparison at $L^2$ and some positive time with its own stated observability range $H_D^{\sigma }$ of Dirichlet spectral Sobolev scale and duality, $-2\le \sigma \le 2$, at every positive time. This last sentence records the article's scope and attribution; those stronger article estimates are not imported from the earlier stabilization theorem.

Bortot–Cavalcanti input. Theorem 1.1 and the exterior hypotheses in §3, especially Assumptions 3.1–3.2, give the stated eventual uniform-on-bounded-sets $L^2$ decay for their equation. Equation (4.19) supplies its dissipation identity. We use only the linear specialization and exactly the coefficient conditions stated above. The exterior branch describes a compact connected smooth nontrapping obstacle, $n\ge 2$, with nontrapping as defined in Non-trapping obstacle; its full manifold branch is not being substituted for this case. This source supplies the preceding historical decay input, independently of the article under study.

Bounded-operator equivalence input. Trélat, §5.3.4, Theorem 31: for a densely defined skew-adjoint $A$ and bounded self-adjoint nonnegative $B$ on a Hilbert space, observability of $\dot{v}+Av=0$ through $B^{1\mathrm{/}2}$ for some positive time is equivalent to uniform exponential decay of the energy $\mathrm{\parallel }u(t){\mathrm{\parallel }}^2\mathrm{/}2$ for $\dot{u}+Au+Bu=0$. This is an explicit external mathematical premise, not a locally reconstructed equivalence theorem. Its boundedness and square-root observation are retained in the application.

[F1] The self-adjoint nonpositive ${\mathrm{\Delta }}_D$ and its domain are given by Dirichlet Laplacian and positive shift, and Scale isometries and free Schrödinger evolution supplies its strongly continuous unitary $W(t)$. The adjoint-domain convention is Adjoint of a densely defined operator.

[F2] Every bounded positive operator on a nonzero complex Hilbert space has a unique bounded positive square root under Choice (Positive square root, The Axiom of Choice).

[F3] The cutoff facts $0\le \phi \le 1$, $\phi =0$ on $\mathrm{\mid }x\mathrm{\mid }\le R$, and $\phi =1$ on $\mathrm{\mid }x\mathrm{\mid }\ge R+1$ are Control region and nested real cutoffs. The observation/stabilization and time-norm meanings are Reachability and observation vocabulary for controlled evolutions and Continuous and mixed time-space function spaces.

[F4] The bounded real-linear Volterra construction in Bounded real-linear perturbations preserve solution estimates, applied with constant perturbation $P=-M_a$, supplies unique linear mild solutions on every finite interval. For this complex-linear $P$, uniqueness also gives complex linearity. This identifies the bounded-damping evolution with the usual mild solution and supplies the linearity needed for normalization of its initial data.

Bortot–Cavalcanti input. Theorem 1.1 and the exterior hypotheses in §3, especially Assumptions 3.1–3.2, give the stated eventual uniform-on-bounded-sets $L^2$ decay for their equation. Equation (4.19) supplies its dissipation identity. We use only the linear specialization and exactly the coefficient conditions stated above. The exterior branch describes a compact connected smooth nontrapping obstacle, $n\ge 2$, with nontrapping as defined in Non-trapping obstacle; its full manifold branch is not being substituted for this case. This source supplies the preceding historical decay input, independently of the article under study.

Bounded-operator equivalence input. Trélat, §5.3.4, Theorem 31: for a densely defined skew-adjoint $A$ and bounded self-adjoint nonnegative $B$ on a Hilbert space, observability of $\dot{v}+Av=0$ through $B^{1\mathrm{/}2}$ for some positive time is equivalent to uniform exponential decay of the energy $\mathrm{\parallel }u(t){\mathrm{\parallel }}^2\mathrm{/}2$ for $\dot{u}+Au+Bu=0$. This is an explicit external mathematical premise, not a locally reconstructed equivalence theorem. Its boundedness and square-root observation are retained in the application.

Given: The actual Dirichlet flow, nonnegative bounded multiplication coefficient, and the two independent external inputs with their recorded hypotheses.

Goal: Match the generator, damping, weighted/unweighted observation and uniform decay constants, without treating the original article as a proof of its prerequisites.

1.1 Put $A=-i{\mathrm{\Delta }}_D$ on $D({\mathrm{\Delta }}_D)$. Self-adjointness in [F1] and the adjoint definition give $D(A^{\ast })=D({\mathrm{\Delta }}_D)$ and $A^{\ast }=i{\mathrm{\Delta }}_D=-A$, so the conservative equation $v_t+Av=0$ has solution $W(t)v_0$. Put $B=M_a$, multiplication by $a$. Its bound is $\mathrm{\parallel }Bv{\mathrm{\parallel }}_2\le \mathrm{\parallel }a{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\parallel }v{\mathrm{\parallel }}_2$, its adjoint is itself by the $L^2$ pairing with real $a$, and $⟨Bv,v⟩=\int a\mathrm{\mid }v{\mathrm{\mid }}^2\ge 0$. Multiplication by $\sqrt{a}$ is likewise bounded, self-adjoint and nonnegative, and its square is $B$; uniqueness in [F2] identifies it with $B^{1\mathrm{/}2}$. Therefore $\mathrm{\parallel }{B}^{1\mathrm{/}2}W(t)v_0{\mathrm{\parallel }}_2^2=\int a\mathrm{\mid }W(t)v_0{\mathrm{\mid }}^2$, and the damped equation $u_t+Au+Bu=0$ is precisely $i{u}_t+{\mathrm{\Delta }}_{D}u+iau=0$. The imported equivalence applies to these exact operators, proving the first assertion. [F1, F2, F4, given]

1.2 In Bortot–Cavalcanti's Assumption 3.1, $g(z)=z$ is continuous, zero fixing and monotone since $\mathrm{Re}((z-w)\overline{z-w})=\mathrm{\mid }z-w{\mathrm{\mid }}^2$. Its imaginary energy pairing vanishes and its two quadratic constants may both be $1$. Their Assumption 3.2 admits the bounded nonnegative measurable coefficient ${\mathbf{1}}_{\omega }$, with lower bound $1$ outside the radius $R$; the difference between $\mathrm{\mid }x\mathrm{\mid }>R$ and $\mathrm{\mid }x\mathrm{\mid }\ge R$ is a null sphere. It also admits ${\phi }^2$, with lower bound $1$ outside the enclosing radius $R+1$, by [F3]. Thus the two observations asserted in the Statement are the exact square-root observations of legitimate linear instances of that independent decay theorem. This check uses no support assumption on an arbitrary coefficient $a$ and uses the nontrapping convention of Non-trapping obstacle. [F3, given, algebra]

2.1 The source theorem states its decay for $t\ge T_0$ uniformly on each bounded set of initial data. In its linear instance choose its constants on the unit ball. For nonzero $u_0$, apply that estimate to $u_0\mathrm{/}\mathrm{\parallel }{u}_0{\mathrm{\parallel }}_2$ and use the solution linearity in [F4]; since energy is quadratic, it gives the same constants for every $u_0$. The zero datum is immediate. For $0\le t<T_0$, the source dissipation identity with $g(z)=z$ gives $E(t)\le E(0)$; therefore enlarging the prefactor to $\max (C_0,e^{{\delta }_0{T}_0})$ makes $E(t)\le C{e}^{-{\delta }_{0}t}E(0)$ valid for all $t\ge 0$. Taking square roots converts this energy estimate to the norm convention in [F3]. Hence it has the uniformity required by the imported bounded-operator equivalence. [F3, F4, step 1.2, given]

3.1 Finally, if $a\ge a_0>0$ on $\omega$, integrate the pointwise bound $a_0{\mathbf{1}}_{\omega }\le a$ against $\mathrm{\mid }v{\mathrm{\mid }}^2$ to obtain the stated lower comparison. If also $a=0$ off $\omega$, the upper bound is $a\le \mathrm{\parallel }a{\mathrm{\parallel }}_{\mathrm{\infty }}{\mathbf{1}}_{\omega }$, giving the converse comparison. For the indicator, both comparisons are identities; for the cutoff, the square root is exactly $\phi$ because it is nonnegative. These direct applications finish the mathematical comparison. The article's wider spectral and time-range description remains attribution, as stated. ∎ [F2, F3, step 1.1, step 1.2, step 2.1, algebra] The independent Bortot–Cavalcanti PDF was read through its tool-extracted text at the listed theorem, coefficient/nonlinearity assumptions, equation and dissipation identity. A local binary request was denied, so no retained binary or visual reading of that PDF is claimed. The Trélat author PDF was downloaded; its Theorem 31 on pp.119–120 was read and the rendered pages inspected. The full proof was not reconstructed. The original article cites Liu for the comparison; Liu's paper was not freshly read here, and the independent theorem contract used here is Trélat's exact bounded-operator statement.

The independent Bortot–Cavalcanti PDF was read through its tool-extracted text at the listed theorem, coefficient/nonlinearity assumptions, equation and dissipation identity. A local binary request was denied, so no retained binary or visual reading of that PDF is claimed. The Trélat author PDF was downloaded; its Theorem 31 on pp.119–120 was read and the rendered pages inspected. The full proof was not reconstructed. The original article cites Liu for the comparison; Liu's paper was not freshly read here, and the independent theorem contract used here is Trélat's exact bounded-operator statement.