Stackelberg-Nash controllability for a multi-objective Stefan problem

1. Problem Statement

The paper investigates a hierarchical multi-objective control problem for a one-dimensional Stefan system, which models phase transitions (melting/solidification) with a moving boundary.

• The System: The state is described by the temperature $y(x,t)$ in a time-dependent domain $Q_\ell = \{(x,t) : 0 < x < \ell(t), 0 < t < T\}$. The evolution is governed by a parabolic equation with a moving boundary $\ell(t)$ determined by the Stefan condition:
  $$\ell'(t) = -\frac{1}{\beta} y_x(\ell(t), t)$$
  where $\beta$ is the latent heat coefficient.
• The Controls: The system involves three control functions:
  1. Leader ($f$): Acts on a subdomain $\mathcal{O}$. Its objective is null controllability (driving the temperature $y$ to zero at time $T$).
  2. Followers ($v_1, v_2$): Act on subdomains $\mathcal{O}_1$ and $\mathcal{O}_2$. Their objectives are tracking: minimizing the deviation of $y$ from desired target profiles $y_{i,d}$ in subregions $\mathcal{O}_{i,d}$, while penalizing control effort.
• The Game Structure: The problem adopts a Stackelberg-Nash framework:
  • Stackelberg: $f$ is the "leader" who chooses a control first, anticipating the reaction of the followers.
  • Nash: The followers $v_1$ and $v_2$ react non-cooperatively. Given $f$, they seek a Nash equilibrium where neither can improve their cost functional by unilaterally changing their control.
• The Goal: Prove the existence of a leader control $f$ such that the system achieves null controllability at time $T$, subject to the constraint that $(v_1, v_2)$ form a Nash equilibrium.

2. Methodology

The authors employ a rigorous analytical approach combining optimal control theory, game theory, and partial differential equations (PDEs) on non-cylindrical domains. The proof strategy is divided into three main steps:

Step 1: Reduction to an Optimality System

The authors first characterize the Nash equilibrium for the followers. By analyzing the Gâteaux derivatives of the convex cost functionals $J_1$ and $J_2$, they show that a Nash equilibrium $(v_1, v_2)$ exists and is unique for sufficiently large penalty parameters $\mu_i$.
Crucially, the equilibrium controls are expressed explicitly in terms of the adjoint states $\phi_1, \phi_2$:
$$v_i = -\frac{1}{\mu_i} \phi_i \mathbb{1}_{\mathcal{O}_i}$$
Substituting these back into the state equation transforms the original multi-objective problem into a nonlinear optimality system (a coupled system of forward and backward parabolic equations) where the leader's control $f$ must drive the state $y$ to zero.

Step 2: Linearization and Approximate Controllability

To handle the nonlinearity and the moving boundary, the authors linearize the optimality system. They establish approximate controllability for this linearized system using a duality argument.

• Adjoint System: They derive the adjoint system associated with the linearized optimality system.
• Observability Inequality: The core of the proof relies on establishing a global observability inequality for the adjoint system. This inequality bounds the initial energy of the adjoint state by the observation of the state on the control region $\mathcal{O}$.
• Carleman Estimates: To prove the observability inequality, the authors derive Carleman estimates tailored for the moving boundary.
  • They construct specific Fursikov-type weight functions that adapt to the moving boundary $\ell(t)$.
  • They handle two geometric configurations for the control and observation domains:
    1. Case (G1): The observation domains for followers coincide ($\mathcal{O}_{1,d} = \mathcal{O}_{2,d}$).
    2. Case (G2): The observation domains differ ($\mathcal{O}_{1,d} \neq \mathcal{O}_{2,d}$).
  • The construction involves matching two different weight functions to handle the coupling in the adjoint system, a significant technical challenge when the domain is non-cylindrical.

Step 3: Fixed-Point Argument for the Nonlinear System

Once approximate controllability is established for the linearized system, the authors use a Schauder fixed-point theorem to prove the controllability of the original nonlinear Stefan problem.

• They define an operator $\Lambda_\epsilon$ that maps a candidate moving boundary $\ell$ to a new boundary generated by the solution of the linearized system with an approximate control.
• Compactness: A major technical hurdle is proving the compactness of this operator. The authors utilize parabolic Hölder regularity estimates for the normal derivative $y_x$ at the moving boundary. They show that for small initial data and targets, the operator maps a bounded, closed, convex set into itself and is continuous.
• Limit Process: By taking the limit as the approximation parameter $\epsilon \to 0$, they obtain a solution that satisfies the exact null controllability condition $y(\cdot, T) = 0$.

3. Key Contributions

1. First Stackelberg-Nash Treatment of Stefan Problems: This is the first work to apply hierarchical Stackelberg-Nash strategies to a free-boundary Stefan problem, extending previous results on fixed-domain parabolic equations.
2. Novel Weight Functions for Moving Boundaries: The authors construct new Fursikov weight functions that are compatible with the moving boundary condition. This allows for the derivation of Carleman estimates in non-cylindrical domains, overcoming the limitations of existing weights designed for fixed domains.
3. Handling Multi-Objective Conflicts: The paper successfully resolves the conflict between the leader's goal (null controllability) and the followers' goals (tracking) by reducing the problem to the null controllability of a coupled optimality system.
4. Geometric Flexibility: The results hold under two distinct geometric configurations regarding the overlap of control and observation regions, providing robustness to the domain setup.

4. Main Results

• Theorem 1.1 (Main Result): Under suitable geometric conditions on the control regions and assuming small initial data and target functions, there exists a leader control $f$ and a unique Nash equilibrium $(v_1, v_2)$ such that the temperature $y$ reaches zero at time $T$.
• Theorem 1.2: Establishes the null controllability of the nonlinear optimality system (the core mathematical engine of the problem).
• Theorem 1.3 & 1.4: Provide the approximate controllability of the linearized system and the crucial observability inequality derived via Carleman estimates.

5. Significance and Future Directions

• Theoretical Advancement: The work bridges the gap between game-theoretic control strategies (Stackelberg-Nash) and free-boundary problems, a class of problems known for their mathematical complexity due to the moving interface.
• Applications: The framework is applicable to various physical and biological processes involving phase changes, such as metal casting, ice melting, and tumor growth modeling, where multiple stakeholders (e.g., different heating zones) have competing objectives.
• Open Questions: The authors highlight several open problems, including:
  • Extending the results to quasi-linear and degenerate parabolic equations.
  • Investigating the case where the leader and follower roles are inverted.
  • Addressing the control of the Stefan problem in star-shaped domains or higher dimensions.

In summary, this paper provides a rigorous mathematical foundation for controlling complex, multi-agent systems governed by moving-boundary PDEs, utilizing advanced techniques in Carleman estimates and fixed-point theory.