A Verification Theorem for an Optimal Control Problem Governed by the Convective Brinkman--Forchheimer Equations

Source: arXiv:2606.27312 · Published 2026-06-25 · By Sagar Gautam, Manil T. Mohan

TL;DR

This paper addresses an optimal control problem governed by the convective Brinkman-Forchheimer (CBF) equations defined on a d-dimensional torus, for d = 2 or 3. These equations model incompressible fluid flow through porous media, incorporating viscous diffusion, inertial convection, linear Darcy drag, and nonlinear Forchheimer drag. Classical Navier-Stokes equations lack the ability to capture flows in porous media where nonlinear resistance materializes, motivating the study of CBF equations. The key contribution is establishing a rigorous verification theorem for the associated Hamilton-Jacobi-Bellman (HJB) equation and deriving a feedback characterization of the optimal controls using the Pontryagin maximum principle (PMP). This is notably the first such verification framework for CBF equations including the physically relevant 3D supercritical regime (r in (3, 5]) and the critical regime r=3 under a specified damping condition (2βμ ≥ 1).

The work overcomes significant analytical challenges that differentiate the CBF equations from Navier-Stokes, including the presence of nonlinear absorption terms, lack of Poincaré inequality in periodic setting, and the need to work with strong rather than weak solutions for three-dimensional cases. New estimates in negative-order Sobolev spaces and careful continuous dependence analysis are developed to enable the dynamic programming approach to verification. The authors derive a complete theory for existence, uniqueness, and regularity of strong solutions, linearize the system to handle adjoint variables, and show the differentiability and local Lipschitz regularity of an auxiliary functional used to characterize feedback controls. These advance prior results on optimal control for NSE and porous media flows and significantly extend verification and feedback synthesis results to porous flow models in both two and three dimensions.

Key findings

• Global existence and uniqueness of strong solutions hold for CBF system on a torus in 2D with r in (1, ∞), and in 3D for supercritical r in (3, 5] and critical r=3 with 2βμ ≥ 1.
• A continuous dependence estimate in the negative Sobolev norm ∥(A + I)⁻¹/²(Z₁ - Z₂)∥_H is established (Proposition 3.3), crucial for showing local Lipschitz continuity of the value function.
• The Pontryagin maximum principle is rigorously derived from the HJB viscosity solution framework under strong solution regularity, connecting optimal controls to adjoint variables and the Hamiltonian.
• A new verification theorem (Theorem 7.3) confirms that the viscosity solution of the HJB equation equals the value function and yields an optimal feedback control characterized by the superdifferential of the value function.
• Novel negative-order energy estimates for the linearized CBF system allow differentiability properties of the auxiliary functional W to be established (Proposition 6.1 and 6.4).
• The nonlinear absorption term |u|^{r-1}u, absent in NSE, provides enhanced dissipation that enables global strong regularity even in 3D supercritical regime.
• The feedback characterization and verification framework applies uniformly to the entire considered range of r and dimensionalities (Table 1).
• New algebraic inequalities for bilinear and nonlinear operators B and C extend prior 2D NSE estimates to the critical and supercritical 3D CBF setting.

Methodology — deep read

1. Threat Model & Assumptions: The study considers an optimal control problem for incompressible fluid flow through porous media modeled by the CBF equations on a d-dimensional torus, with d ∈ {2,3}. Controls are L2 forcing terms u(t,x) influencing the system dynamics. The goal is to minimize a cost functional over admissible controls, subject to the PDE constraints. The adversary analogy corresponds to nature's unknown disturbances represented implicitly by nonlinearities and the need for feedback under uncertainty. The analysis assumes strong solutions exist globally in time, requiring sufficient dissipation (e.g., 2βμ ≥ 1 in 3D critical case).

2. Data: The PDE is posed on the torus Td with periodic boundary conditions, eliminating boundary effects and enabling spectral analysis. Initial data z belong to V (divergence-free, H1-regular vector fields). Controls u and external forcing f are in L2 spaces. The absorption exponent r controls nonlinear damping strength and is a key parameter in regimes considered.

3. Architecture / Algorithm: The CBF system is first projected onto divergence-free fields via the Helmholtz projection to get an abstract evolution equation in H: dZ/dt = -μ A Z - B(Z) - α Z - β C(Z) + u(t) where A is the Stokes operator, B is the bilinear convective operator, and C is the nonlinear absorption operator. The nonlinear operators B and C and their Fréchet derivatives are carefully analyzed, with new algebraic inequalities derived to handle absorption nonlinearity. An auxiliary functional W on V is introduced (eqn 6.2) whose differentiability links the HJB equation's superdifferential with adjoint variables.

4. Training Regime: The analysis involves deriving energy and regularity estimates for weak and strong solutions, with particular attention to continuous dependence estimates in negative Sobolev norms (∥(A + I)^(-1/2)(Z1 - Z2)∥_H). The linearized and adjoint equations are studied to enable Pontryagin maximum principle derivations. Results use functional inequalities like Agmon, Ladyzhenskaya, Sobolev embeddings tuned to r and dimension. The proofs are constructive and quantitative but no data-driven training is involved.

5. Evaluation Protocol: Verification theorem (Theorem 7.3) is proved under the established regularity framework. The value function is shown to be a viscosity solution of the HJB equation. The PMP characterizes optimal controls as minimizing the Hamiltonian via feedback characterized by elements of the superdifferential of the value function. Comparison to NSE is provided, emphasizing that 3D NSE lacks global strong solvability and thus does not admit such verification results.

6. Reproducibility: The study is rigorous and self-contained with detailed proofs and estimates. However, it is a theoretical PDE/control analysis without code or data release. The analysis is constructive but no numerical schemes or experimental validations are provided. The paper extensively references previous well-posedness theory for the CBF equations.

In one concrete example, given initial data z in V and an admissible control u in L2, the system solution Z(t; z,u) is shown to depend continuously in negative norm on z and u. The auxiliary functional W involving integrations over the torus is shown to be differentiable. Using the linearized adjoint system, the PMP provides a characterization of the optimal control feedback law minimizing the Hamiltonian pointwise, and the verification theorem confirms this control is optimal by equality between value and viscosity solution.

Technical innovations

• Development of a verification theorem and dynamic programming framework for the convective Brinkman-Forchheimer equations with nonlinear absorption, extending beyond classical Navier-Stokes settings.
• Derivation of new negative-order Sobolev norm continuous dependence estimates for strong solutions in 3D supercritical regime (r in (3,5]) and critical regime r=3 with 2βμ≥1.
• Introduction and detailed analysis of an auxiliary functional W whose differentiability properties enable feedback synthesis and Pontryagin maximum principle characterization in infinite-dimensional porous media flows.
• Extension of Pontryagin maximum principle derivation to infinite-dimensional, nonlinear PDE systems with nonlinear damping terms, using strong solution theory instead of weaker approaches.
• Rigorous feedback characterization of optimal controls via superdifferential of viscosity solution to Hamilton-Jacobi-Bellman equation in a three-dimensional porous media flow context.

Baselines vs proposed

• 2D Navier-Stokes verification results: not available in 3D due to lack of strong solution theory; CBF system extends verification to 3D with global strong solutions for r > 3 and critical cases under damping
• Energy estimate for differences of two solutions in negative Sobolev norm (3.5): sup_{s∈[t,T]}∥(A+I)^(-1/2)(Z1 - Z2)(s)∥_H^2 ≤ C ∥(A+I)^(-1/2)(z1 - z2)∥_H^2
• Energy estimate for strong solutions (3.3) extends Navier-Stokes estimates by incorporating damping from nonlinear Forchheimer term.
• Verification theorem (Theorem 7.3): value function V equals viscosity solution of HJB; optimal control characterized by minimization of the Hamiltonian involving derivatives of V.

Limitations

• Analysis restricted to periodic boundary conditions on the torus; extension to bounded domains with boundaries is nontrivial due to loss of commutativity of Helmholtz projection and Laplacian.
• Focus on strong solution regimes requiring nonlinear damping parameters (e.g., 2βμ≥1 in critical 3D case); subcritical regimes with weaker damping and possibly blow-up remain open.
• No numerical examples, simulations, or empirical validations supporting the theoretical verification results.
• Results depend on rather technical functional analytic assumptions and parameter ranges, making practical implementation and computation of feedback controls challenging.
• The verification theorem and PMP are established in an infinite-dimensional PDE framework but do not address robustness to model uncertainty or stochastic forcing.

Open questions / follow-ons

• Extension of verification and feedback theory to CBF equations on bounded domains with non-periodic boundary conditions and boundary layers.
• Developing analogous verification results for the subcritical regime r < 3 and critical regime with insufficient damping where global strong solutions are not guaranteed.
• Numerical methods and approximation schemes to compute the value function and implement the feedback controls arising from the verification theorem.
• Incorporation of stochastic forcing or random perturbations into the CBF optimal control framework and corresponding verification results.

Why it matters for bot defense

This work is a rigorous mathematical contribution in infinite-dimensional optimal control for fluid models with nonlinear damping, which does not directly address bot detection or CAPTCHA design. However, the techniques employed—such as establishing verification theorems for complex nonlinear PDE-controlled systems, deriving Pontryagin maximum principles with strong regularity frameworks, and characterizing feedback controls via viscosity solutions—may inspire analogous theoretical frameworks in bot-defense research where control or dynamic adaptation of system states governed by nonlinear dynamics is studied.

In particular, the paper demonstrates advanced methods to overcome challenges from nonlinear terms and lack of compactness that could be analogous to difficulties in modeling or controlling sophisticated adversarial interactions in bot defense. The emphasis on feedback synthesis from infinite-dimensional systems also aligns with needs in adaptive and reactive security mechanisms. Nonetheless, direct applications would require transferring this PDE-control framework to systems more germane to bot interaction or CAPTCHA response modeling.

Cite

@article{arxiv2606_27312,
  title={ A Verification Theorem for an Optimal Control Problem Governed by the Convective Brinkman--Forchheimer Equations },
  author={ Sagar Gautam and Manil T. Mohan },
  journal={arXiv preprint arXiv:2606.27312},
  year={ 2026 },
  url={https://arxiv.org/abs/2606.27312}
}

Read the full paper

• arXiv:2606.27312 (abstract)
• arXiv:2606.27312 (PDF)