Controlling the Uncontrollable — Null Controllability for Degenerate Coupled Parabolic Equations | AI Trend Blend

Applied Mathematics · Journal of Mathematical Analysis and Applications 563 (2026) · UERJ, UPB, UFF & FAU Erlangen · 16 min read

Controlling the Uncontrollable — How Mathematicians Are Steering Degenerate Heat Equations That Change Over Time

A research team spanning Brazil, Bolivia, and Germany has proved that you can steer a degenerate, non-autonomous coupled heat system to a complete rest in finite time — no matter where it starts — using a control signal that acts on only one of the two equations and only on a tiny portion of the spatial domain.

Carleman Estimates Null Controllability Degenerate Parabolic Equations Non-Autonomous Systems Coupled PDEs Control Theory Stackelberg-Nash Climate Modeling

Imagine a heat-like process unfolding inside a thin rod — one where the conductivity at the left endpoint completely vanishes, time itself is warping the physics, and two separate quantities are tangled together through coupling terms. Can you inject a signal at just one location, acting on just one of the two quantities, and bring the entire system to a dead stop at a prescribed future time? The answer, this paper proves, is yes — and the mathematical machinery used to get there is as elegant as the result itself.

Why Degenerate and Non-Autonomous Together Is So Difficult

If you have studied basic heat equations, you know the standard picture: heat spreads smoothly through a rod, governed by a diffusion coefficient that stays positive and bounded everywhere. Control theory for those clean equations is mature and well understood. The real mathematical frontier sits in the degenerate regime — where the diffusion coefficient actually touches zero at a boundary point.

Degeneracy at the boundary changes everything. The equation loses the uniform ellipticity that most standard tools assume. Boundary conditions need to be reinterpreted, function spaces need to be redesigned with weights, and the Carleman estimates — the weighted integral inequalities that are the backbone of controllability proofs — have to be rebuilt from scratch.

Now add a second complication: the diffusion itself depends on time. The system studied in this paper contains a factor $b(t)$ multiplying the diffusion operator, where $b(t)$ can vary freely as long as it stays bounded away from zero and belongs to the Sobolev space $W^{1,\infty}(0,T)$. This non-autonomy is not just a technical curiosity. It is precisely what appears in physically motivated problems — for instance, when a degenerate equation defined on a moving spatial domain is pulled back to a fixed reference domain via a change of variables. The original autonomous equation becomes non-autonomous on the fixed domain, so any controllability result for the moving-domain problem genuinely requires a non-autonomous theory.

And then there is the coupling. Rather than one equation, the paper treats two equations for two unknowns $u$ and $v$, linked through terms $b_{12}v$ and $b_{21}u$ that appear in both equations. The key structural challenge is that the control $h$ acts only in the first equation, on a small open sub-interval $\omega$ of the domain. The question is whether the coupling is strong enough — specifically, whether the coefficient $b_{21}$ is bounded away from zero somewhere inside $\omega$ — to propagate controllability from the first equation to the second.

The Central Challenge in Plain Language

You have a degenerate heat process with time-varying conductivity, described by two interacting quantities. You are allowed to apply a control force to just one of the two quantities, only inside a small region of the domain. The question is: can you drive both quantities to zero simultaneously at a prescribed future time? This paper proves you can — and derives the precise mathematical conditions under which this works.

Setting Up the System — What the Equations Actually Say

The paper studies the following system on the interval $\Omega = (0,1)$ over the time window $(0,T)$:

Main System $$\begin{cases} u_t – b(t)(a(x)u_x)_x + d_1(x,t)\sqrt{a}\,u_x + b_{11}u + b_{12}v = h\mathbf{1}_\omega + H_1 \\ v_t – b(t)(a(x)v_x)_x + d_2(x,t)\sqrt{a}\,v_x + b_{21}u + b_{22}v = H_2 \\ u(0,t)=u(1,t)=v(0,t)=v(1,t)=0, \quad u(\cdot,0)=u_0,\; v(\cdot,0)=v_0 \end{cases}$$

Every piece of this system has a precise role. The coefficient $a(x)$ is the spatial degeneracy: it satisfies $a(0) = 0$, is positive on $(0,1]$, is non-decreasing, and obeys the structural bound $xa'(x) \leq K\,a(x)$ for some $K \in [0,1)$. This is called the weakly degenerate condition, and it means $a$ behaves like $x^\alpha$ near the left endpoint, with $\alpha \in (0,1)$. The coefficient $b(t)$ wraps around the entire diffusion operator and depends only on time, giving the non-autonomy.

The lower-order terms $d_i(x,t)\sqrt{a}\,u_x$ are gradient terms with a specific structure: the factor $\sqrt{a}$ in front of the spatial derivative is essential for the function spaces to work out correctly in the degenerate setting. The coupling coefficients $b_{ij}$ are bounded functions on the space-time cylinder. The control $h$ enters only in the first equation, supported on the sub-interval $\omega \subset \Omega$.

The goal is null controllability at time $T$: find $h \in L^2(\omega \times (0,T))$ such that $u(\cdot,T) = 0$ throughout $\Omega$. Remarkably, it turns out that $v(\cdot,T) = 0$ follows automatically from the coupling, provided $b_{21}$ is bounded away from zero on some sub-interval of $\omega$.

The Carleman Estimate — The Engine of the Proof

Every controllability result for parabolic equations of this type ultimately rests on a Carleman estimate — an integral inequality for solutions of the adjoint (backward-in-time) problem. The estimate shows that the weighted energy of the adjoint solution over the whole domain is controlled by its weighted energy restricted to the tiny observation region $\omega$. Once you have that, duality arguments convert it into the existence of a control that drives the forward system to zero.

What makes the Carleman estimate for this system non-trivial is the combination of three features: the vanishing diffusion coefficient $a$ at $x=0$, the time-dependent multiplier $b(t)$, and the coupling between two equations. Each of these individually would make the standard Carleman approach require modification. Together, they force a careful redesign of the weight functions and a new set of technical lemmas.

The weights used in the paper are built from the function

Carleman Weights $$\Psi(x) = \begin{cases} \displaystyle\int_0^x \frac{s}{a(s)}\,ds, & x \in [0,\alpha’), \\ -\displaystyle\int_{\beta’}^x \frac{s}{a(s)}\,ds, & x \in [\beta’,1], \end{cases}$$ $$\theta(t) = \frac{1}{(t(T-t))^4}, \quad \sigma(x,t) = \theta(t)\,e^{\lambda(|\Psi|_\infty + \Psi)}, \quad \phi(x,t) = \theta(t)\!\left(e^{\lambda(|\Psi|_\infty+\Psi)} – e^{3\lambda|\Psi|_\infty}\right)$$

where $\omega’ = (\alpha’,\beta’) \subset\subset \omega$ is a slightly smaller interval nested inside the control region. The function $\theta(t)$ blows up at both $t = 0$ and $t = T$, which is what forces the exponential weight to suppress any non-zero behavior at those endpoints. The spatial weight $\Psi$ is adapted to the degenerate operator: near the left boundary, it involves the integral of $x/a(x)$, which diverges as $x \to 0^+$ precisely when $a$ vanishes — this is the mechanism that handles the degenerate boundary correctly.

The main Carleman estimate for a single adjoint equation states:

Theorem 1 — Carleman Estimate for the Adjoint Equation

There exist constants $C > 0$ and $\lambda_0, s_0 > 0$ such that every solution $w$ of the adjoint equation satisfies, for all $s \geq s_0$ and $\lambda \geq \lambda_0$,

$$\int_0^T\!\int_0^1 e^{2s\phi}\!\left((s\lambda)\sigma b(t)^2 a(x) w_x^2 + (s\lambda)^2\sigma^2 b(t)^2 w^2\right) \leq C\!\left(\int_0^T\!\int_0^1 e^{2s\phi}|F|^2 + (s\lambda)^3\int_0^T\!\int_\omega e^{2s\phi}\sigma^3 w^2\right)$$

The left-hand side controls the full weighted $H^1_a$-norm of $w$ across the whole domain. The right-hand side involves only the forcing $F$ and the restriction of $w$ to the small control region $\omega$.

This estimate is the non-autonomous version of the Carleman inequality proved by Demarque, Límaco and Viana for the autonomous degenerate case. The key distinction is the presence of $b(t)$ inside the integrands on both sides, and the need to handle cross-terms involving the time derivative $\dot{b}(t)$ that appear when computing the inner product $\langle L^+ w, L^- w \rangle$ after splitting the adjoint operator into its symmetric and skew parts.

“The factor $b(t)$ inside the diffusion operator introduces new terms in the Carleman computation — particularly $\dot{b}(t)$ terms from integration by parts in time — that are not present in the autonomous case. Controlling these terms requires using the ratio $|\dot{b}(t)/b(t)| \leq B$ rather than just the boundedness of $b$ itself.” — Gamboa, Limaco, Yapu · J. Math. Anal. Appl. 563 (2026)

The proof unfolds through twelve lemmas, each estimating one piece of the inner product decomposition. Several of these — particularly the estimates for the time-derivative terms and the boundary terms at $x=0$ and $x=1$ — require new arguments compared to the autonomous case. The structure near $x=0$ is especially delicate: the Hardy-Poincaré inequality for the weighted space $H^1_a(0,1)$ plays a crucial role in absorbing the degenerate boundary contribution.

From One Equation to Two — The System Carleman Estimate

With the single-equation Carleman estimate in hand, the paper extends it to the full coupled adjoint system. This is where the coupling coefficient $b_{21}$ enters the argument in an essential way.

The adjoint of the coupled forward system is itself a coupled backward system for two unknowns $(\varphi, \psi)$, with terminal conditions $\varphi(\cdot, T) = \varphi_T$ and $\psi(\cdot, T) = 0$. Applying the single-equation Carleman estimate to each equation separately and adding the results, you get a combined inequality that controls the weighted energy of both $\varphi$ and $\psi$ — but with a troublesome term involving $\int e^{2s\phi}|z|^2$ over the control region still on the right-hand side.

Getting rid of this excess term requires multiplying the first equation of the adjoint system by the product $e^{2s\phi}\lambda^3 s^3 \sigma^3 \chi z$ — where $\chi$ is a smooth cutoff supported in $\omega$ and equal to 1 in a slightly smaller interval $\omega_1 \subset\subset \omega$ — and integrating over the full space-time cylinder. The positivity of $b_{21}$ on $\omega_1$ is then used to absorb the $z$-term and express it entirely in terms of the observation of $\varphi$ on $\omega$.

Proposition 3 — Carleman Estimate for the Coupled Adjoint System

There exist constants $C, \lambda_0, s_0 > 0$ such that for any $\varphi_T \in L^2(\Omega)$, the solution $(\varphi, \psi)$ of the adjoint coupled system satisfies

$$I(\varphi) + I(\psi) \leq C\!\left(\int_Q e^{2s\phi}s^4\lambda^4\sigma^4\!(|F_1|^2 + |F_2|^2)\,dxdt + \int_{\omega\times(0,T)} e^{2s\phi}s^8\lambda^8\sigma^8\varphi^2\,dxdt\right)$$

where $I(\cdot)$ denotes the full Carleman left-hand side functional. Only the observation of $\varphi$ — the component corresponding to the controlled equation — appears on the right-hand side.

The higher power $s^8\lambda^8\sigma^8$ on the observation term, compared to the $s^3\lambda^3\sigma^3$ in the single-equation estimate, is the price of the cascade argument: each application of Young’s inequality to absorb the cross-coupling term inflates the power by one level. This is standard in cascade controllability arguments, and the resulting observability inequality — which bounds the initial energy of $\varphi$ by the observed energy of $\varphi$ on $\omega$ — is still sufficient for the null controllability conclusion.

The Function Spaces — Why Standard Sobolev Spaces Are Not Enough

One of the subtler aspects of this paper is the function space framework. Because the diffusion coefficient $a(x)$ vanishes at $x = 0$, the natural energy space for the degenerate equation is not the standard Sobolev space $H^1_0(0,1)$ but a weighted version adapted to the degeneracy:

Weighted Sobolev Space $$H^1_a(0,1) = \left\{u \in L^2(0,1) : u \text{ absolutely continuous on } (0,1],\; \sqrt{a}\,u_x \in L^2(0,1),\; u(0)=u(1)=0\right\}$$

The inner product on this space uses $\sqrt{a}\,u_x$ in the gradient term rather than $u_x$ itself. This is the right space precisely because the energy dissipated by the degenerate operator involves $\int a(x)|u_x|^2\,dx$ rather than $\int |u_x|^2\,dx$. Working in $H^1_a$ ensures that the energy estimate for the forward system, and all subsequent a priori estimates, are compatible with the degenerate structure of the equation.

Proving well-posedness in this space — that the coupled system has a unique solution with the right regularity — requires a Galerkin approximation argument using an orthonormal basis of $H^1_a(0,1)$ consisting of eigenfunctions of the degenerate operator. The paper carries out three separate energy estimates at increasing regularity levels (estimates I, II, and III in the appendix), ultimately showing that the approximate solutions are bounded in $L^2(0,T; L^2) \cap H^1(0,T; H^2_a)$, independently of the approximation parameter. Compactness via the Aubin-Lions theorem then gives the existence of a weak solution, and uniqueness follows from standard Gronwall arguments.

Why the Space Matters

Using the wrong function space for a degenerate equation is not just an inconvenience — it leads to estimates that break down at the degenerate boundary and Carleman weights that do not correctly handle the behavior near $x = 0$. The weighted space $H^1_a(0,1)$ is tailor-made for this problem, and the entire proof architecture is built around it.

The Null Controllability Theorem — What the Paper Actually Delivers

The main result synthesizes everything: the system Carleman estimate, the observability inequality derived from it, and a duality argument based on the Lax-Milgram theorem. It reads as follows:

Theorem 2 — Global Null Controllability

Under the hypotheses on $a$, $b$, $d_i$, and $b_{ij}$, with $\inf\{b_{21}(x,t) : (x,t) \in \omega_1 \times [0,T]\} > 0$, and with $\rho_2 H_1, \rho_2 H_2 \in L^2(Q)$ for a specific weight $\rho_2$ that blows up at $t = T$: for any $T > 0$ and any initial data $u_0, v_0 \in H^1_a(\Omega)$, there exists a control $h \in L^2(\omega \times (0,T))$ such that the solution of the coupled system satisfies

$$u(x,T) = 0 \quad \text{and} \quad v(x,T) = 0 \quad \text{for all } x \in [0,1].$$

Moreover, the control $h$, the state $u$, and the state $v$ all satisfy explicit weighted $L^2$ bounds controlled by the initial data and the source terms $H_1$, $H_2$.

The duality argument proceeds via the bilinear form

Lax-Milgram Setup $$\mathcal{B}((\tilde\varphi,\tilde\psi),(\varphi,\psi)) = \int_Q \rho_0^{-2}\,\mathcal{L}^*(\tilde\varphi,\tilde\psi)\cdot\mathcal{L}^*(\varphi,\psi)\,dxdt + \int_{\omega\times(0,T)} \rho_1^{-2}\,\tilde\varphi\varphi\,dxdt$$

where $\mathcal{L}^*$ denotes the adjoint operator pair and $\rho_0, \rho_1$ are specific exponential weights derived from the Carleman estimate. The observability inequality guarantees that this bilinear form is coercive, so Lax-Milgram gives a unique minimizer $(\hat\varphi, \hat\psi)$. Setting $h = -\rho_1^{-2}\hat\varphi\,\mathbf{1}_\omega$ then produces the desired control, and the weighted energy bound (27) in the paper follows from the coercivity estimate combined with Cauchy-Schwarz.

The Comparison: Autonomous vs. Non-Autonomous Weights

A careful comparison with the earlier work of Akil, Fragnelli, and Ismail on non-autonomous degenerate equations reveals a meaningful technical difference. Their Carleman weights produce a term $x^2/a$ in the second summand on the left-hand side of their estimate, while the approach in this paper, following Demarque-Límaco-Viana, does not carry that factor. This difference has practical consequences for future applications.

Akil-Fragnelli-Ismail (2025)

Carleman estimate includes a $x^2/a$ factor in the second left-hand side term. Constants $s$ and $\lambda$ are linked, reducing flexibility. Works for both divergent and non-divergent degenerate operators.

Gamboa-Limaco-Yapu (2026)

No $x^2/a$ factor in the analogous term. The parameters $s$ and $\lambda$ can be chosen independently and made large, matching the standard non-degenerate Carleman framework. Greater freedom for future cascade and multi-control applications.

The freedom to choose $s$ and $\lambda$ independently is not just an aesthetic preference. In cascade controllability arguments — where the Carleman estimate is applied to each component of a multi-equation system sequentially — each application typically requires inflating one of these parameters to absorb cross-terms. Having both parameters free means the argument can be iterated without running into constraints between them.

Physical Motivations and Real-World Connections

This paper is not purely abstract. The motivations explicitly cited by the authors connect to several active areas of applied mathematics and mathematical physics.

The most direct motivation is the control of equations on moving domains. Consider a degenerate heat equation defined on a spatial interval $(0, \ell(t))$ whose right endpoint moves with time. After a diffeomorphism pulls the problem back to the fixed interval $(0,1)$, the transformed equation is non-autonomous — it picks up a term involving $\dot\ell(t)/\ell(t)$ that multiplies the diffusion operator. Controllability of the moving-domain problem is therefore equivalent to controllability of a non-autonomous fixed-domain problem, and the Carleman estimate in this paper is the key ingredient needed to carry that argument through.

A second application involves the Stackelberg-Nash control strategy for hierarchical systems, where a leader control and two follower controls play a Nash equilibrium game. After computing the optimality conditions for the followers (which produce a system of three coupled equations), and after the change of variables to a fixed domain, the resulting system has time-varying coefficients — again requiring the non-autonomous Carleman framework. The authors have developed this application in detail in a companion preprint.

The third connection is to climate modeling. The Budyko-Sellers energy balance model describes the surface temperature of the Earth averaged over latitude bands, with a diffusion operator that degenerates at the poles (where the latitude-dependent coefficient $a$ vanishes). Non-autonomous variations of this model — where seasonal forcing makes the coefficients time-dependent — are precisely the class studied in this paper. Proving controllability for such models provides a rigorous foundation for asking questions about optimal intervention in simplified climate systems.

Connecting Math to the Real World

Degenerate, non-autonomous, coupled PDEs are not a theoretical curiosity. They show up naturally when you try to control processes on moving domains, implement hierarchical control strategies in multi-agent systems, or model geophysical phenomena where the physics varies with both position and season. This paper provides the mathematical guarantee that controllability survives all three of these complications simultaneously.

Open Problems and What Comes Next

The authors close the paper with a thoughtful discussion of open problems, each representing a natural extension of the current work.

The first open problem is local controllability for semilinear coupled systems. When nonlinear terms $F_1(u_1, u_2, u_{1x})$ and $F_2(u_1, u_2, u_{2x})$ — possibly depending on the gradient — are added to the coupled system, local null controllability near the zero solution should follow from the linear theory via a linearization and inverse function theorem argument. But this demands additional $L^\infty$ estimates on the states and gradients beyond what the current weighted $L^2$ bounds provide, making it a genuinely harder problem.

The second is null controllability for quasilinear equations, where the diffusion coefficient itself depends on the unknown: systems of the form $u_t – b(t)(B(u)\,a(x)\,u_x)_x + F(u,u_x) = h\mathbf{1}_\omega$. Here $B(u)$ introduces a fully nonlinear structure into the diffusion, and the Carleman estimates cannot be applied directly to the nonlinear equation — a linearization strategy followed by a fixed-point argument would be needed.

The third is a fourth-order problem: null controllability for the degenerate beam equation $u_t – b(t)(a(x)u_{xx})_{xx} = h\mathbf{1}_\omega$ on a moving domain. Fourth-order parabolic equations are technically harder than second-order ones because the boundary conditions need to specify both the function and its second derivative at both endpoints, and the corresponding Carleman estimates involve more complex weight calculations.

What This Paper Means for PDE Control Theory

This paper makes a clear and clean contribution to an active field. It does not introduce a fundamentally new proof technique — the De Giorgi-style iteration, Carleman-weight framework, and Lax-Milgram duality argument are all classical. What it does is carry those techniques into a genuinely harder setting: degenerate diffusion, time-varying coefficients, and two-equation coupling, all at once.

The result on well-posedness in the weighted space $H^1_a$ is complete and self-contained. The Carleman estimate for the single non-autonomous degenerate equation extends the autonomous case cleanly, with the key new ingredient being the control of $\dot{b}(t)/b(t)$ terms. The cascade argument for the coupled system follows the pattern of Demarque-Límaco-Viana, adapted to the non-autonomous setting. And the null controllability theorem is both sharp in its hypotheses — the condition $\inf b_{21} > 0$ on a sub-interval of $\omega$ is essentially necessary for cascade controllability — and concrete in its conclusion.

For researchers working on controllability of degenerate parabolic systems — particularly those interested in moving-domain problems, hierarchical control strategies, or geophysical models — this paper provides both a finished result and a set of tools that can be adapted to nearby problems. The companion preprints on Stackelberg-Nash controllability for degenerate equations in non-cylindrical domains, cited in the paper, show that the authors are already putting those tools to work.

Read the Full Paper

Published in the Journal of Mathematical Analysis and Applications, Volume 563 (2026). All proofs, lemmas, and technical appendices are available via the journal’s website.

📄 Read the Paper (DOI) 🔗 JMAA Journal Page

Academic Citation:
Alfredo S. Gamboa, Juan Limaco, Luis P. Yapu. Controllability of a system of non-autonomous degenerate coupled parabolic equations. Journal of Mathematical Analysis and Applications, 563 (2026) 130777. https://doi.org/10.1016/j.jmaa.2026.130777

This article is an independent editorial analysis of a peer-reviewed paper. Mathematical statements paraphrase the original results; for complete proofs and precise formulations, consult the published paper. The paper was received September 28, 2025, and made available online May 6, 2026.

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