Stackelberg-Nash controllability for a multi-objective Stefan problem

Abstract: We investigate a hierarchical control problem for a one-dimensional Stefan system with localized distributed controls. The setting combines a Stackelberg strategy with a Nash equilibrium among multiple followers, yielding a multi-objective free-boundary problem. The interaction between the hierarchical control and the moving interface results in a nonlinear optimality system, and we show that the original problem reduces to the null controllability of this optimality system. Under suitable geometric conditions on the control regions, we establish a local null controllability result. The proof relies on an observability inequality for a linearized system, obtained through Carleman estimates adapted to the presence of a moving boundary. These results constitute, to the best of our knowledge, the first treatment of a Stefan system within a Stackelberg-Nash framework.

• The paper establishes local null controllability for a one-dimensional Stefan system with one leader and two Nash-equilibrium followers under two geometric control-observation configurations.
• It derives a coupled forward–backward optimality system, proves uniform observability through Carleman weights adapted to the moving interface, and obtains unique follower equilibria when penalty parameters are sufficiently large.
• The result closes the nonlinear free-boundary problem with a Schauder fixed-point argument, but requires small initial data and tracking targets and leaves quasilinear, degenerate, and global-control extensions open.

Problem and contribution

The paper studies a hierarchical, multi-objective control problem for the one-dimensional one-phase Stefan system on the non-cylindrical domain $Q_\ell = \{(x,t): 0<x<\ell(t),\ 0<t<T\}$:

$y_t - y_{xx} + a(x,t)y = f\mathds{1}_{\mathcal{O}} + v_1\mathds{1}_{\mathcal{O}_1} + v_2\mathds{1}_{\mathcal{O}_2}, \qquad \ell'(t) = -\tfrac{1}{\beta}y_x(\ell(t),t),$

with homogeneous Dirichlet conditions at $x=0$ and at the moving interface. Three controls act on localized subdomains of $(0,\ell^*)$, with $\ell^* < \ell(t) < B$. The leader control $f$ must drive the state to zero at time $T$, while two follower controls $v_1, v_2$ minimize tracking functionals toward targets $y_{i,d}$ in observation sets $\mathcal{O}_{i,d}$. The followers are required to form a Nash equilibrium, producing a Stackelberg–Nash configuration in the sense introduced by Lions and Díaz–Lions. The authors state that this is the first treatment of a Stefan system within a Stackelberg–Nash framework; prior work on Stefan controllability (e.g., Fernández-Cara–Limaco–de Menezes, Araújo et al.) addressed only single-objective null controllability.

Main result

Under one of two geometric configurations — $y_t - y_{xx} + a(x,t)y = f\mathds{1}_{\mathcal{O}} + v_1\mathds{1}_{\mathcal{O}_1} + v_2\mathds{1}_{\mathcal{O}_2}, \qquad \ell'(t) = -\tfrac{1}{\beta}y_x(\ell(t),t),$0: $y_t - y_{xx} + a(x,t)y = f\mathds{1}_{\mathcal{O}} + v_1\mathds{1}_{\mathcal{O}_1} + v_2\mathds{1}_{\mathcal{O}_2}, \qquad \ell'(t) = -\tfrac{1}{\beta}y_x(\ell(t),t),$1 with $y_t - y_{xx} + a(x,t)y = f\mathds{1}_{\mathcal{O}} + v_1\mathds{1}_{\mathcal{O}_1} + v_2\mathds{1}_{\mathcal{O}_2}, \qquad \ell'(t) = -\tfrac{1}{\beta}y_x(\ell(t),t),$2; or $y_t - y_{xx} + a(x,t)y = f\mathds{1}_{\mathcal{O}} + v_1\mathds{1}_{\mathcal{O}_1} + v_2\mathds{1}_{\mathcal{O}_2}, \qquad \ell'(t) = -\tfrac{1}{\beta}y_x(\ell(t),t),$3: $y_t - y_{xx} + a(x,t)y = f\mathds{1}_{\mathcal{O}} + v_1\mathds{1}_{\mathcal{O}_1} + v_2\mathds{1}_{\mathcal{O}_2}, \qquad \ell'(t) = -\tfrac{1}{\beta}y_x(\ell(t),t),$4 with $y_t - y_{xx} + a(x,t)y = f\mathds{1}_{\mathcal{O}} + v_1\mathds{1}_{\mathcal{O}_1} + v_2\mathds{1}_{\mathcal{O}_2}, \qquad \ell'(t) = -\tfrac{1}{\beta}y_x(\ell(t),t),$5 — the main theorem establishes local null controllability: there exist $y_t - y_{xx} + a(x,t)y = f\mathds{1}_{\mathcal{O}} + v_1\mathds{1}_{\mathcal{O}_1} + v_2\mathds{1}_{\mathcal{O}_2}, \qquad \ell'(t) = -\tfrac{1}{\beta}y_x(\ell(t),t),$6 and a weight $y_t - y_{xx} + a(x,t)y = f\mathds{1}_{\mathcal{O}} + v_1\mathds{1}_{\mathcal{O}_1} + v_2\mathds{1}_{\mathcal{O}_2}, \qquad \ell'(t) = -\tfrac{1}{\beta}y_x(\ell(t),t),$7 blowing up at $y_t - y_{xx} + a(x,t)y = f\mathds{1}_{\mathcal{O}} + v_1\mathds{1}_{\mathcal{O}_1} + v_2\mathds{1}_{\mathcal{O}_2}, \qquad \ell'(t) = -\tfrac{1}{\beta}y_x(\ell(t),t),$8 such that if $y_t - y_{xx} + a(x,t)y = f\mathds{1}_{\mathcal{O}} + v_1\mathds{1}_{\mathcal{O}_1} + v_2\mathds{1}_{\mathcal{O}_2}, \qquad \ell'(t) = -\tfrac{1}{\beta}y_x(\ell(t),t),$9 and

$x=0$0

then there exists a quintuple $x=0$1 where $x=0$2 is the unique Nash equilibrium and $x=0$3 in $x=0$4. The result is local in both data and boundary: smallness of initial condition and targets is essential, and the interface is confined to the class $x=0$5.

Reduction to the optimality system

Since each follower functional $x=0$6 is convex, the Nash equilibrium is characterized by vanishing Gâteaux derivatives, which via adjoint states $x=0$7 solving backward adjoint equations forced by $x=0$8 yields the explicit feedback $x=0$9 on $(0,\ell^*)$0. Substituting into the state equation produces a coupled forward–backward optimality system in $(0,\ell^*)$1 closed by the Stefan condition. Existence and uniqueness of the Nash equilibrium for each triple $(0,\ell^*)$2 follows from a Lax–Milgram argument once $(0,\ell^*)$3 exceeds a threshold; consequently, the original constrained problem reduces exactly to the null controllability of this nonlinear optimality system.

Carleman estimates and observability

The core analytic difficulty is an observability inequality for the linearized adjoint system coupling $(0,\ell^*)$4 (adjoint of $(0,\ell^*)$5) with $(0,\ell^*)$6 (adjoints of $(0,\ell^*)$7):

$(0,\ell^*)$8

with $(0,\ell^*)$9 uniform over all admissible interfaces $\ell^* < \ell(t) < B$0. The proof combines three ingredients. First, a global Carleman estimate for a scalar parabolic equation on $\ell^* < \ell(t) < B$1 with source terms in both $\ell^* < \ell(t) < B$2 and $\ell^* < \ell(t) < B$3 is established using Fursikov-type weights $\ell^* < \ell(t) < B$4 built from auxiliary functions $\ell^* < \ell(t) < B$5 adapted to the moving boundary. Second, a new constructive lemma builds these weights under both geometric configurations; notably, in case $\ell^* < \ell(t) < B$6 the construction enforces $\ell^* < \ell(t) < B$7 and coincidence outside $\ell^* < \ell(t) < B$8, which allows the two weighted estimates to be combined as in Araruna–Fernández-Cara–Guerrero–Santos, but with weights compatible with the free boundary — the weights of the cylindrical theory do not transfer directly. Third, an energy lemma localizes the estimate near $\ell^* < \ell(t) < B$9: because the adjoint system is dissipative on a short interval $f$0 with $f$1 depending only on $f$2 and $f$3, the trace $f$4 is controlled by the Carleman-weighted norm, while a Gronwall argument with the blow-up weight $f$5 controls the $f$6 components.

A technical point worth noting: in case $f$7 the proof splits into three sub-scenarios depending on how $f$8 and $f$9 intersect the opposite target sets, handled by binary coefficients $T$0 ensuring $T$1 on the relevant regions.

From observability, approximate controllability of the linearized optimality system follows by the standard Fenchel–Rockafellar dual functional $T$2, whose coercivity rests on the observability inequality; the resulting controls satisfy a bound uniform in $T$3 and in $T$4, which is what makes the subsequent fixed point possible.

Nonlinear closure via Schauder

Passage from approximate to exact null controllability of the nonlinear optimality system uses Schauder's theorem applied to the map $T$5. Compactness comes from a Hölder estimate: since all controls are supported in $T$6, the states enjoy $T$7 regularity on $T$8, uniformly bounded by the data norms. This yields $T$9 provided $v_1, v_2$0 is chosen as the minimum of three explicit quantities involving $v_1, v_2$1, $v_1, v_2$2, $v_1, v_2$3, and the gaps $v_1, v_2$4, $v_1, v_2$5, $v_1, v_2$6 — making transparent that the smallness assumption is tied to keeping the interface inside its prescribed corridor. Continuity of $v_1, v_2$7 requires two convergence steps: weak-to-strong identification of minimizers of the dual functionals as $v_1, v_2$8 in $v_1, v_2$9, then propagation to the normal derivative traces at the moving boundary using the difference system and the uniform Hölder bounds. A final compactness argument in $y_{i,d}$0 as $y_{i,d}$1 delivers the null-controllable solution.

Limitations and open questions

Several restrictions are intrinsic to the approach. The result is local: it requires smallness of $y_{i,d}$2 and of the weighted targets, and large penalty parameters $y_{i,d}$3; global results or small $y_{i,d}$4 are not addressed. The analysis is confined to one space dimension with Laplacian diffusion, where the Stefan condition links $y_{i,d}$5 directly to $y_{i,d}$6; the authors note explicitly that the relationship between $y_{i,d}$7 and the operator changes for other diffusions. Two extensions are posed as open: quasilinear equations $y_{i,d}$8 with $y_{i,d}$9, and degenerate equations $\mathcal{O}_{i,d}$0 with $\mathcal{O}_{i,d}$1, including weak versus strong boundary conditions at $\mathcal{O}_{i,d}$2. For degenerate systems, configuration $\mathcal{O}_{i,d}$3 remains open even on fixed domains. The reversed hierarchy — followers as leaders with $\mathcal{O}_{i,d}$4 restricted to null controls — is also left unexplored, with only a suggested route via existing hierarchical-control techniques. Finally, uniqueness of the full quintuple is not claimed; existence of a solution pair $\mathcal{O}_{i,d}$5 to the Stefan problem itself is assumed per fixed boundary, reflecting the known non-uniqueness of classical Stefan solutions.

Conclusion

The paper extends the Stackelberg–Nash controllability paradigm to a free-boundary parabolic setting, reducing the multi-objective problem to null controllability of a coupled optimality system and closing the loop through Carleman-based observability uniform in the interface position and a Schauder argument fed by Hölder regularity away from the control region. The constructive Fursikov weight adapted to moving boundaries is the principal methodological addition, and its explicit form lends itself to numerical implementation. The open degenerate and quasilinear cases delineate the boundary of the current technique.