Optimal Bilinear control restricted to the three-dimensional chemo-repulsion model with potential production

1. Problem Statement

The paper investigates a three-dimensional parabolic-parabolic chemorepulsion system with nonlinear production and bilinear control. The system models the movement of cells ($u$) repelled by a chemical signal ($v$) produced by the cells themselves. The control function $f$ acts bilinearly on the chemical equation within a subdomain $\Omega_c \subseteq \Omega$.

The mathematical model is defined on $Q = (0, T) \times \Omega$ with non-flux boundary conditions:

Non-Logistic System (1.1):
$$\begin{cases} \partial_t u - \Delta u = \nabla \cdot (u \nabla v), \\ \partial_t v - \Delta v + v = u^p + f v \mathbb{1}_{\Omega_c}, \end{cases}$$

Logistic System (1.2):
$$\begin{cases} \partial_t u - \Delta u = \nabla \cdot (u \nabla v) + r u - \mu u^p, \\ \partial_t v - \Delta v + v = u^p + f v \mathbb{1}_{\Omega_c}, \end{cases}$$

Key Parameters:

• $p > 1$: Exponent of nonlinear chemical production.
• $r, \mu \geq 0$: Logistic growth and competition coefficients.
• $f \in L^{5/2}(0, T; L^{5/2}(\Omega_c))$: The bilinear control (acting as a proliferation/degradation coefficient).
• Goal: To establish the existence of global weak and strong solutions and to solve an optimal control problem minimizing a cost functional $J$ that tracks desired states $(u_d, v_d)$ and penalizes the control effort.

2. Methodology

The authors employ a multi-stage analytical approach combining PDE theory, functional analysis, and optimization techniques:

A. Existence of Weak Solutions

• Regularization: The authors introduce a regularized system with an artificial diffusion parameter $\epsilon > 0$ for the variable $v$ (solving an elliptic problem $v_\epsilon - \epsilon \Delta v_\epsilon = w_\epsilon$) to handle the nonlinear coupling.
• Energy Estimates: They derive crucial energy inequalities by testing the equations with specific powers of $u$ and $v$ (e.g., $u^{p-1}$ and $-\Delta v$).
  • Case $1 < p \leq 5/3$: The diffusion term alone is sufficient to control the nonlinear production term $u^p$ via Gagliardo-Nirenberg interpolation.
  • Case $p > 5/3$: Diffusion is insufficient. The logistic term ($ru - \mu u^p$) is strictly necessary to bound the energy and prevent blow-up.
• Compactness: Using the Aubin-Lions lemma and weak convergence arguments, they pass to the limit as $\epsilon \to 0$ to prove the existence of global weak solutions.

B. Regularity and Strong Solutions

• Regularity Criterion: A "Serrin-type" criterion is established. If a weak solution satisfies $u \in L^{5p/2}(Q) \cap L^{10/3}(Q)$, it becomes a strong solution.
• Bootstrap Argument: The authors use a non-standard bootstrapping argument involving linear diffusion-convection-reaction equations with non-regular coefficients.
• Control Regularity: They require the control $f$ to be in $L^{5/2}(0, T; L^{5/2+}(\Omega_c))$ to ensure the state variables achieve the necessary regularity ($u, \nabla v \in L^5(Q)$) for the control-to-state map to be differentiable.

C. Optimal Control Theory

• Control-to-State Map: The authors define the operator $S$ mapping controls to states. They prove this map is Fréchet differentiable using the Implicit Function Theorem in Banach spaces. This requires proving the well-posedness of the linearized system.
• Adjoint System: To compute the gradient of the reduced cost functional, they introduce an adjoint system $(\sigma, \eta)$. The well-posedness of this system is established via a duality argument in reflexive Banach spaces.
• Optimality Conditions: By identifying the gradient of the reduced cost, they derive first-order necessary optimality conditions for local optimal solutions.

3. Key Contributions

1. Existence in 3D with Nonlinear Production:

   • Proved global existence of weak solutions for the 3D chemorepulsion model with nonlinear production $u^p$.
   • Identified the critical threshold $p = 5/3$: For $p \leq 5/3$, diffusion suffices; for $p > 5/3$, the logistic competition term is mathematically necessary to ensure global existence.

2. Regularity Criterion for Strong Solutions:

   • Established a specific regularity criterion ($u \in L^{5p/2} \cap L^{10/3}$) that upgrades weak solutions to strong solutions.
   • Proved uniqueness of these strong solutions under the derived regularity conditions.

3. Optimal Control Framework:

   • First 3D Result: This is one of the first works to address optimal control for 3D chemorepulsion models with nonlinear production ($p > 1$). Previous works were limited to 2D or linear production ($p=1$).
   • Differentiability: Successfully proved the Fréchet differentiability of the control-to-state operator, avoiding generic Lagrange multiplier arguments in favor of the Implicit Function Theorem.
   • Gradient Identification: Explicitly derived the gradient of the reduced cost functional using the adjoint state, facilitating potential numerical implementations via gradient descent.

4. Main Results

• Theorem 2.3 (Existence): Global weak solutions exist for system (1.1) if $1 < p \leq 5/3$, and for system (1.2) (logistic) if $p > 1$.
• Theorem 2.4 (Regularity): If the control $f$ and initial data satisfy specific regularity conditions, any weak solution satisfying the criterion $u \in L^{5p/2} \cap L^{10/3}$ is a unique strong solution in the space $X \times X_{5/2}$.
• Theorem 2.8 (Optimal Existence): There exists at least one global optimal solution to the control problem.
• Theorem 2.9 & Corollary 2.10 (Optimality Conditions): The reduced cost functional is Fréchet differentiable. The first-order necessary condition for a local optimal control $f^*$ is given by the variational inequality:
  $$\int_0^T \int_{\Omega_c} \left( \gamma_f \frac{\text{sgn}(f^*)|f^*|^{3/2}}{\|f^*\|_{L^{5/2+}}^{0+}} + v^* \eta \right) (f - f^*) \geq 0, \quad \forall f \in F$$
  where $(\sigma, \eta)$ solves the adjoint system.

5. Significance and Impact

• Mathematical Rigor in 3D: The paper overcomes significant analytical challenges associated with 3D chemotaxis systems, specifically the lack of global regularity for nonlinear production terms without logistic damping.
• Bridging Theory and Control: It provides a rigorous theoretical foundation for controlling complex biological processes (like cancer invasion or wound healing) modeled by chemorepulsion in 3D, a setting previously unexplored for nonlinear production.
• Numerical Implications: By explicitly identifying the gradient of the cost functional via the adjoint state, the authors provide the necessary theoretical tool to develop gradient-based numerical optimization schemes (e.g., steepest descent or conjugate gradient methods) for these complex PDE-constrained problems.
• Methodological Innovation: The use of the Implicit Function Theorem combined with duality arguments for the adjoint system offers a robust alternative to standard Lagrange multiplier approaches, particularly useful when dealing with non-smooth or highly nonlinear state constraints.

In summary, this work fills a critical gap in the literature by extending optimal control theory for chemotaxis models from 2D/linear cases to the more complex and biologically relevant 3D/nonlinear setting, providing both existence proofs and the necessary conditions for optimization.