Ph. D. Project
Stabilization and controllability of Stefan's problem
2021/09/14 - 2024/09/30
Other supervisor(s):
- General issue, Context

In dimension one, the fusion or solidification of a solid can be represented by a Stefan problem. For a more general presentation of Stefan's problem, we can refer to [4, 9, 10, 7]. More precisely, suppose that the interval (0,1) corresponds to the domain occupied by the liquid and solid phases. We will also assume that the two phases are connected, so we denote by s(t) the point of (0,1) representing the position of the liquid-solid interface. We will also assume that the liquid phase occupies the interval (0,s(t)) and the solid phase, the interval (s(t),1).
Denote by y_l the temperature in the liquid phase and y_s the one in the solid phase and suppose that the melting temperature is null.
The temperatures in the two phases are governed by the heat equation. At the liquid-solid interface, the temperature is equal to the melting temperature, i.e. y_l(t,s(t))=y_s(t,s(t))=0. The displacement velocity of the position of the liquid-solid interface is proportional to the jump of the normal derivative of the temperature in x=s(t).
Finally, in order to avoid the appearance of new phases in the domain, we will impose that the solution satisfies some sign constraints, y_l (respectively y_s) remains nonpositive (respectively nonnegative).
Finally, it will be possible to act on the state of the system using controls. By way of example, we can give ourselves two boundary controls (in x=0 and x=1) of Neumann type.
Of course, all of these equations are subject to initial conditions. Concerning the well-posed character of this system, one can refer to [2, 3].

- Objectives of the Ph.D.

In recent works, [5, 6], stabilization towards trivial stationary states, i.e. associated with null controls, has been proven. This result was obtained using the baskstepping method, cf. [1]. However, this state feedback control requires knowledge of the entire state at all times. This constraint is not feasible in practice. This leads to a first question.
Q.1. Is it possible to stabilize the system towards a trivial stationary state, with a control depend only on a partial measure of the state?

It is also natural to be interested in the stabilization of the system towards non-trivial stationary states.
Q.2. Is it possible to stabilize the system to a non-trivial steady state?

Because of the comparison principle for the heat equation, it is easy to show that any trivial stationary state cannot be reached in a finite time.
Q.3. Is it possible to control the system to a non-trivial steady state?

In [6], the constructed control ensures the sign conditions on the temperatures in the liquid and solid phases. This allows these state constraints to be omitted. However, these state constraints must be taken into account with other control strategies. In particular if the steady state is controllable, these sign constraints will induce a minimum controllability time (cf. [8]).
Q.4. If a steady state is controllable, can we give a characterization of the minimum controllability time, and is there a control in this minimum time?

Finally, to our knowledge, the only stabilization results relate to Stefan's unidimensional system. It is natural to wonder if similar results exist in higher dimension. In this case, a major difficulty is to correctly formalize the control problem. Indeed, in dimension three (respectively two), the solid-liquid interface is a surface (respectively a curve). A simple way to approach this problem is to consider axially symmetrical liquid and solid domains. This allows comming back to a one-dimensional system. The previous questions arise again with the classical Laplace operator replaced by the cylindrical or spherical Laplace operator.

- References

[1] D. M. Bošković, A. Balogh and M. Krstić. Backstepping in infinite dimension for a class of parabolic distributed parameter systems. Math. Control Signals Systems, 16 (1): 44-75, 2003.
[2] J. R. Cannon and M. Primicerio. A two phase Stefan problem with flux boundary conditions. Ann. Mast. Pura Appl. (4), 88: 193-205, 1971.
[3] J. R. Cannon and M. Primicerio. A two phase Stefan problem with temperature boundary conditions. Ann. Mast. Pura Appl. (4), 88: 177-191, 1971.
[4] S. C. Gupta. The classical Stefan problem. Elsevier, Amsterdam, 2018. Basic concepts, modeling and analysis with quasi-analytical solutions and methods.
[5] S. Koga, M. Diagne, S. Tang and M. Krstic. Backstepping control of the one-phase Stefan problem. In 2016 American Control Conference (ACC), pages 2548-2553, July 2016.
[6] S. Koga and M. Krstic. Single-boundary control of the two-phase Stefan system. Systems Control Lett., 135: 104573, 9, 2020.
[7] S. Koga and M. Krstic. Two-Phase Stefan Problem, pages 139-157. Springer International Publishing, Cham, 2020.
[8] J. Lohéac, E. Trélat and E. Zuazua. Minimal controllability time for the heat equation under unilateral state or control constraints. Math. Models Methods Appl. Sci., 27 (9): 1587-1644, 2017.
[9] L. I. Rubenštĕın. The Stefan problem. American Mathematical Society, Providence, R.I., 1971. Translated from the Russian by A. D. Solomon, Translations of Mathematical Monographs, Vol. 27.
[10] L. Rubinstein. The Stefan problem: comments on its present state. J. Inst. Math. Appl., 24 (3): 259-277, 1979.
Control, Stabilization, Heat equation, Stefan's problem, Constrained state
Control Identification Diagnosis