2608.09648-Exact boundary controllability of a singular/degenerate wave equation via singular Sturm–Liouville theory

Introduction to the show: ident: Paper Radio. Generated commentary on the latest Artificial Intelligence papers.

Tom: Next we'll be talking about the paper "Exact boundary controllability of a singular/degenerate wave equation via singular Sturm–Liouville theory".

Jane: The paper was written by Marcos López-García from Universidad Nacional Autónoma de México.

Tom: Stay tuned as we take you through the paper and discuss its implications.

Paper summary: Tom: We've got a paper today that solves a problem people have been circling for a while: moving the boundary control to the singular endpoint of a degenerate wave equation.

Jane: And not just for one special equation — for a whole family of them, all at once. The equation is u_tt minus the derivative of x^α u_x, with a drift term β x^{α−1} u_x and a singular potential μ x^{α−2} u, on the interval (0,1), with α between zero and two, and a control applied exactly at x=0.

Lu: What makes that hard is that x=0 is where the equation degenerates or becomes singular. The usual boundary traces don't exist there, so you need Sturm-Liouville theory to decide what boundary condition is even meaningful.

Meng: And then they diagonalize the operator using Bessel functions, and use Ingham-type inequalities to turn the spectral information into observability. It's a very clean machinery.

Lalam: The headline result is exact controllability in any time T greater than 4/(2−α), under the condition (1−α−β)^2 ≥ 4μ. That threshold is natural, because 2−α controls the speed of propagation near zero.

Jane: The really nice part is that the same proof covers three regimes: subcritical, the critical logarithmic case, and the limit-point case. Earlier work had to handle those separately, if at all.

Tom: And unlike previous results where the control sat at the regular endpoint, here it acts at the singular endpoint itself. That's the gap.

Meng: It also recovers the classical wave equation when α, β, and μ are all zero, so the framework generalizes something familiar instead of being exotic.

Lu: I like that the boundary observation operator isn't invented by hand; it comes out of the Lagrange bracket of the Sturm-Liouville expression. The mathematics tells you which trace to use.

Lalam: In control theory, that's a big deal. If you pick the wrong boundary condition at a singular point, you can get a completely different system, maybe one that isn't controllable at all.

Jane: So the paper gives both a result and a method. The spectral approach is constructive enough that you could compute controls from the eigenfunction expansion.

Tom: We'll walk through how it all fits together, starting from the introduction.

Page 1 of the paper: Tom: So on the very first page, the paper positions itself against two earlier lines of work.

Jane: Gueye proved exact boundary controllability for weakly degenerate hyperbolic equations with the control at the degenerate endpoint. That was a landmark, but it didn't cover singular potentials.

Lu: Then Fragnelli, Mugnai, and Sbai handled degenerate and singular hyperbolic equations with drift and singular potentials, but they put the control at the nondegenerate, nonsingular endpoint. So there was a missing case.

Meng: The paper also mentions higher-order degenerate equations, where the control usually goes through the regular boundary. That's a sign of how hard the singular endpoint is.

Lalam: The authors say the formulation depends on choosing the right self-adjoint realization of the differential operator. Formal integration by parts can mislead you at a singular point, because the boundary terms may not make sense without knowing the admissible traces.

Tom: Exactly. The boundary control operator is not imposed ad hoc; it's determined by the Lagrange bracket associated with the Sturm-Liouville expression.

Jane: The control itself is a function f in L^2(0,T), with a Dirichlet condition at x=1 and a weighted boundary condition at x=0. For ν=0 you get a critical logarithmic trace, and for ν>0 a power-type weighted trace.

Meng: The abstract says the proof covers subcritical, critical logarithmic, and limit-point regimes all at once. That's genuinely new.

Lu: I find the remark on page one already useful: in a certain range, the boundary form can be written as a weighted Neumann trace, so you can interpret it as a derivative condition rather than a ratio of functions.

Lalam: But the punchline on page one is the theorem: exact controllability in any time bigger than 4/(2−α), for every initial and final state in the fractional energy space X = H^{ν+1/2} × H^{ν−1/2}.

Jane: That theorem is stated before the machinery, which is a bold structure for a paper. It says, here is the result, now let's build the tools to prove it.

Tom: Next, the paper stops talking about boundary traces in the abstract and actually constructs the operator.

Page 2 of the paper: Tom: The functional setting starts with a weighted Lebesgue space: L^2_β(0,1) means square-integrable with respect to x^β dx.

Jane: The differential expression is M u = −(p u_x)_x + q u, with p = x^{α+β}, q = −μ x^{−2+α+β}, and w = x^β. Dividing by w gives exactly the operator in the wave equation.

Lu: So the space is tailored to the degeneracy: the weight x^β compensates for the growth of solutions near zero. And p and w are positive on (0,1), with 1/p, q, and w locally integrable, so the equation is regular at x=1 and singular only at zero.

Meng: They introduce three parameters: κ_α = (2−α)/2, the discriminant Δ = (1−α−β)^2 − 4μ, and σ = (1−α−β)/(2κ_α). A third parameter ν comes out of Δ and κ_α and controls everything.

Lalam: The maximal domain D_max consists of functions where both u and p u_x are locally absolutely continuous and u and A u belong to the weighted space. That's the natural class of candidates.

Tom: Then they define the Lagrange bracket

u,v: (x) = u(x)p(x)v'(x) − v(x)p(x)u'(x). That bracket is the key to boundary terms at the singular endpoint, because it has a limit even when u and v don't have ordinary traces.

Jane: The minimal operator is the closure of the operator on compactly supported functions, and its adjoint is the maximal operator. That's standard Sturm-Liouville theory, but now with weights.

Lu: The point is that the singular endpoint can be either limit-circle or limit-point depending on ν, and that distinction determines whether you need a boundary condition at zero.

Meng: In the limit-circle case, there are two square-integrable solutions of A u = 0, so the operator is not essentially self-adjoint and you have to choose a self-adjoint extension. The paper chooses the Friedrichs extension.

Tom: That's a natural choice because the operator is bounded below, and the Friedrichs domain usually encodes the right physical boundary condition.

Jane: And in the limit-point case, there's only one square-integrable solution, so the boundary condition at x=1 alone gives a self-adjoint operator.

Lalam: This is the point where a purely formal calculation could lead you astray. The Lagrange bracket tells you which linear combination of boundary values actually has a finite limit.

Tom: Next, the paper works through those cases one by one and writes down the domains explicitly.

Page 3 of the paper: Tom: Now the paper separates three cases. In the subcritical case 0<ν<1, the endpoint is limit-circle and non-oscillatory.

Jane: Two solutions of A u = 0 are φ_−(x) = Δ^{−1/2} x^{σ−√Δ/2} and φ_+(x) = x^{σ+√Δ/2}. The plus one grows faster near zero, which makes it the principal solution; the minus one is non-principal.

Lu: The Friedrichs domain is D(S_F) = {u ∈ D_max :

u,φ_+: (0)=0 and u(1)=0}. The bracket condition means the coefficient of φ_+ in the asymptotic expansion of u is zero.

Meng: They also show that

u,φ_+: (0) equals the limit of u/φ_− as x→0+, because the bracket with the non-principal solution gives the coefficient of the principal component. It's a clever rearrangement.

Tom: In the critical case ν=0, the two solutions become y_+ = x^σ and y_− = −x^σ ln x. The logarithm appears exactly when you'd otherwise see a resonance. The boundary condition is still

u,y_+: (0)=0, which becomes the limit of u/y_− vanishing.

Jane: So instead of a power decay condition, you get a condition on u divided by x^σ ln(1/x). That's what the abstract called the critical logarithmic regime.

Lu: For ν≥1, the endpoint is limit-point. The non-principal solution φ_− is no longer square-integrable, so there is no boundary condition at zero at all. The operator is determined by u(1)=0 alone.

Meng: There's a nice remark: for 0<ν<1 with a nonzero discriminant term, the condition

u,φ_+: (0)=0 can be rewritten as the limit of x^{1−σ+√Δ/2} u'(x) being zero. So it really is a weighted Neumann condition.

Lalam: Those explicit domains are what make the later observation operator meaningful. You can't define a boundary control until you know which traces have limits.

Jane: And the proof shows all these cases fit into one framework instead of requiring separate ad hoc arguments.

Tom: With the domains settled, the next step is to diagonalize the operator.

Page 4 of the paper: Tom: The spectral decomposition begins with a proposition that packages the domains we just discussed into one operator A.

Jane: For ν=0 the domain is those functions where the limit of x^{−σ} u(x)/ln(1/x) is zero; for 0<ν<1 it's the limit of x^{Δ/2−σ} u(x) being zero; for ν≥1 it's just u(1)=0. All of these are the Friedrichs extension in the limit-circle cases, and the unique self-adjoint realization in the limit-point case.

Lu: Then comes the key calculation: if u = x^σ v(x^{κ_α}), the eigenvalue equation A u = λ u becomes Bessel's equation in the variable z = x^{κ_α}. That's why Bessel functions appear.

Meng: The eigenfunctions are Φ_k(x) = C_k x^σ J_ν(j_{ν,k} x^{κ_α}), where j_{ν,k} are the positive zeros of J_ν, normalized by J'_ν(j_{ν,k}). The eigenvalues are λ_k = κ_α^2 j_{ν,k}^2.

Tom: The boundary condition at x=1 forces the argument at x=1 to be a zero of J_ν, so the spectrum is discrete, a sequence of numbers growing like k^2.

Jane: And at x=0, the Bessel J_ν behaves like z^ν, so the eigenfunctions have exactly the power behavior allowed by the domain. The Y_ν terms are ruled out because they don't satisfy the boundary condition — they either blow up too fast or aren't square-integrable.

Lu: The orthogonality of the eigenfunctions comes from the standard L^2 orthogonality of Bessel functions after the change of variables y = x^{κ_α}. The normalization is chosen so the family is orthonormal in L^2_β.

Meng: They cite Hochstadt's theorem that the Fourier-Bessel series converges in the mean, so these functions form an orthonormal basis of the weighted space.

Lalam: That's a very strong tool. An orthonormal basis means you can identify the Hilbert space with ℓ^2 via the coefficients in this basis, which is exactly what you need for Ingham inequalities later.

Tom: And because the operator is self-adjoint and discrete, the wave group is just a superposition of oscillating modes.

Jane: Right, so next the paper builds the fractional spaces and the wave dynamics on top of that basis.

Page 5 of the paper: Tom: So now we have an orthonormal basis of eigenfunctions, and the paper uses it to define a scale of fractional spaces H^s for any real s.

Jane: H^s consists of expansions whose coefficients squared, weighted by λ_k^s, have finite sum. So H^0 is the weighted L^2 space, H^1 is the form domain, H^2 is the operator domain, and negative exponents give dual spaces.

Lu: The energy space for the wave is X_0 = H^{1/2} × H, but the paper actually works in X_s = H^{s+1} × H^s, and later picks s = ν−1/2.

Meng: The abstract wave equation is written as a first-order system with generator A =

[0,I: ,

−A,0: ]. That matrix is skew-adjoint, so it generates a unitary group on each X_s.

Tom: Because A is diagonalizable with eigenvectors built from the Φ_k, the group T(t) acts as multiplication by e^{iγ_k t} in the basis, with γ_k = κ_α j_{ν,k} and negative frequencies for the conjugate modes.

Jane: The spectral representation of a solution is w(x,t) = Σ (b_k e^{iγ_k t} + b_{−k} e^{−iγ_k t}) Φ_k(x), with coefficients determined by the initial data. That's the formula that makes everything else possible.

Lu: Proposition 6 states that for every initial data in X_s there is a unique mild solution with the expected regularity, and the energy — the sum of the H^{s+1} norm of w and the H^s norm of w_t — is conserved.

Meng: Conservation is automatic because the generator is skew-adjoint and the group is unitary. But it's nice to see it stated explicitly for this singular problem.

Lalam: Now you can see why the fractional index matters. The observation operator will sit at a specific regularity level, and s = ν−1/2 is chosen so that the boundary trace at zero is a bounded linear functional on the state space.

Tom: That's the crucial step. Next, the paper defines the observation operator using the Lagrange bracket.

Page 6 of the paper: Tom: The observation operator B*_ν is defined directly from the boundary forms we saw earlier.

Jane: For ν=0, it sends (w_0,w_1) to −

w_0,y_−: (0), which is the coefficient of the principal component y_+ in a certain expansion. For ν>0, it's −√Δ

w_0,φ_−: (0), equivalently a weighted trace O_{−σ−√Δ/2}(w_0).

Lu: So the observation reads off the term that would be controlled at the singular endpoint. It's not an arbitrary point evaluation; it's the resonant coefficient of the asymptotic expansion.

Meng: Proposition 7 shows B*_ν is bounded from X_1 to C. The proof uses the eigenfunction expansion and the asymptotic estimates for Bessel functions near zero.

Tom: Then Lemma 8 establishes admissibility: for every initial state, the L^2(0,T) norm of the observation along the trajectory is bounded by a constant times the energy. That's what you need to make the control map well-defined.

Jane: The real work is Proposition 10, exact observability. The frequencies γ_k are uniformly separated, and the upper Beurling density D^+ of the frequency set is 1/(κ_α π). Then 2πD^+ equals 4/(2−α), so the condition T > 4/(2−α) is exactly the threshold for Ingham's lower inequality.

Meng: The lower bound on the observation coefficients |d_k|^2 is at least c λ_k^{ν+1/2}, which is essential for the reverse inequality. Those estimates come from the asymptotics of Bessel functions and their zeros.

Lalam: So observability is a spectral gap plus a density condition. The frequencies are like the tones of a drum that degenerates near zero; as long as you listen long enough, you can reconstruct the whole state.

Tom: That gives exact observability, and by duality, exact controllability. The paper then needs to make the weak solution rigorous.

Page 7 of the paper: Tom: Before proving controllability, the paper makes sure the weak formulation actually makes sense with a control at a singular point.

Jane: They take smooth solutions of the homogeneous wave equation and compute the time derivative of E(τ) = ⟨u_t,w⟩_β − ⟨u,w_t⟩_β. Green's identity gives E'(τ) =

u(·,τ),w(·,τ): (0). So the boundary contribution is exactly the Lagrange bracket.

Lu: Then they expand a controlled solution near zero as u ~ a_u(t) y_+ + b_u(t) y_− in the critical case, or a_u(t) φ_+ + b_u(t) φ_− in the subcritical and limit-point cases. The coefficients are brackets with the solutions.

Meng: Those brackets identify the trace: for instance, O_{−σ}(w_0) = −

w_0,y_−: (0), which is the coefficient of the principal component. So the observation operator really is the coefficient of the singular part.

Tom: Definition 11 gives the weak solution by duality: for every test state W = (w_0,w_1), the identity holds with the control appearing through f(τ) B*_ν W. That's the transposition formulation.

Jane: The input-state mapping Φ_τ integrates T(τ−s) B_ν f(s) ds, which is the Duhamel formula for the controlled system. Proposition 13 guarantees a unique solution in the extrapolation space X_{−1}, and it actually lands in X for every time.

Lu: So the control is admissible: despite being applied at a point where the PDE degenerates, its effect on the state is bounded in the energy space.

Meng: The regularity statement is nice too — u is continuous in H^{ν+1/2}, its time derivative in H^{ν−1/2}, and it has a second time derivative in an even weaker space.

Lalam: That's enough to state the controllability problem cleanly and prove it by duality. The paper is very careful to separate the abstract semigroup formalism from the singular boundary analysis.

Tom: Now we reach the final proof, where all the pieces click together.

Page 8 of the paper: Tom: The final section is short. Definition 14 says the pair (A,B_ν) is exactly controllable in time T if the input map Φ_T maps L^2(0,T) onto the whole state space X.

Jane: The proof of Theorem 1 is then one line conceptually: by the duality theorem from Tucsnak and Weiss, exact controllability is equivalent to exact observability of the adjoint pair. Since we already proved exact observability for any T > 4/(2−α), the controllability follows.

Lu: The appendix fills in the Bessel function toolkit: the series definition, the asymptotic J_ν(x) ~ (x/2)^ν / Γ(ν+1), the recurrence formula, and the asymptotics of Y_ν near zero.

Meng: They also need the zeros of J_ν. Lemma A.2 says the differences between consecutive zeros converge to π, and Lemma A.3 gives |J'_ν(j_{ν,k})| behaves like sqrt(2/(π j_{ν,k})). Those are exactly the estimates used in the lower bounds on d_k.

Lalam: The Beurling density theorem is stated at the end: for a uniformly separated sequence of frequencies, the exponentials satisfy Ingham's inequalities on intervals longer than 2π times the upper density. That's the engine of the observability proof.

Tom: And the Friedrichs extension definition is included, so the paper is self-contained for anyone who wants to check the functional analytic setup.

Jane: It's elegant how the proof rests on facts about Bessel functions that were known for a century, plus a modern control theory duality.

Meng: The time threshold is not an artifact; it's dictated by the frequency density. If T were smaller, the observability inequality would fail.

Lalam: That's a satisfying explanation for a condition that might otherwise look mysterious.

Tom: So the paper gives a complete, rigorous, and unified answer for this class of singular wave equations.

Conclusion: Tom: We've reached the end of the paper, and it really does deliver on its promise.

Jane: It proves exact boundary controllability for a whole family of singular and degenerate wave equations, with the control acting at the singular endpoint, and it does so in a unified framework covering the subcritical, critical logarithmic, and limit-point regimes.

Lu: The method is as important as the result. Singular Sturm-Liouville theory tells you which boundary traces are legitimate, the Bessel eigenbasis gives you a spectral representation, and Ingham-type inequalities convert frequency separation into observability.

Meng: And everything is quantitative: the controllability time T > 4/(2−α) comes directly from the density of the eigenfrequencies, and the fractional energy space H^{ν+1/2} × H^{ν−1/2} appears naturally from the boundary trace regularity.

Lalam: For control theory, this is a template. The same combination of operator theory and nonharmonic Fourier series could be applied to other degenerate systems where the controlled endpoint is singular.

Tom: It also connects classical analysis to modern control in a way that feels very satisfying. Bessel functions and Sturm-Liouville theory aren't just background; they're doing the heavy lifting.

Jane: And the paper is careful about rigor. Every boundary term is justified through Lagrange brackets, not formal integration by parts, which is exactly where singular problems usually hide their traps.

Lu: I think the most exciting part is that the control is at the degenerate point itself. That's harder, but it's also the situation that matters in applications where a sensor or actuator really has to sit at the singular location.

Meng: The authors also kindly dedicate the paper to the memory of a sister, which is a small reminder that mathematical papers come from real people.

Tom: That's a nice note to end on. We'll say goodbye to this paper and get ready for the next one.

Jane: Thanks for listening, and we'll be back soon with another discussion from the arXiv.