# Input-to-State Stability for the Control of Stefan Problem with Respect to Heat Loss

###### Abstract

This paper develops an input-to-state stability (ISS) analysis of the Stefan problem with respect to an unknown heat loss. The Stefan problem represents a liquid-solid phase change phenomenon which describes the time evolution of a material’s temperature profile and the liquid-solid interface position. First, we introduce the one-phase Stefan problem with a heat loss at the interface by modeling the dynamics of the liquid temperature and the interface position. We focus on the closed-loop system under the control law proposed in [16] that is designed to stabilize the interface position at a desired position for the one-phase Stefan problem without the heat loss. The problem is modeled by a 1-D diffusion Partial Differential Equation (PDE) defined on a time-varying spatial domain described by an ordinary differential equation (ODE) with a time-varying disturbance. The well-posedness and some positivity conditions of the closed-loop system are proved based on an open-loop analysis. The closed-loop system with the designed control law satisfies an estimate of norm in a sense of ISS with respect to the unknown heat loss. The similar manner is employed to the two-phase Stefan problem with the heat loss at the boundary of the solid phase under the control law proposed in [25], from which we deduce an analogous result for ISS analysis.

UCSD]Shumon Koga, iasson]Iasson Karafyllis, UCSD]Miroslav Krstic

Department of Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, CA 92093-0411 USA

Department of Mathematics, National Technical University of Athens, 15780 Athens, Greece

Key words: Input-to-state stability (ISS), Stefan problem, moving boundaries, distributed parameter systems, nonlinear stabilization

## 1 Introduction

Liquid-solid phase transitions are physical phenomena which appear in various kinds of science and engineering processes. Representative applications include sea-ice melting and freezing [20], continuous casting of steel [34], cancer treatment by cryosurgeries [35], additive manufacturing for materials of both polymer [23] and metal [6], crystal growth [7], lithium-ion batteries [21], and thermal energy storage systems [39]. Physically, these processes are described by a temperature profile along a liquid-solid material, where the dynamics of the liquid-solid interface is influenced by the heat flux induced by melting or solidification. A mathematical model of such a physical process is called the Stefan problem[12], which is formulated by a diffusion PDE defined on a time-varying spatial domain. The domain’s length dynamics is described by an ODE dependent on the Neumann boundary value of the PDE state. Apart from the thermodynamical model, the Stefan problem has been employed to model several chemical, electrical, social, and financial dynamics such as tumor growth process [11], domain walls in ferroelectric thin films [32], spreading of invasive species in ecology [9], information diffusion on social networks [30], and optimal exercise boundary of the American put option on a zero dividend asset [5].

While the numerical analysis of the one-phase Stefan problem is broadly covered in the literature, their control related problems have been addressed relatively fewer. In addition to it, most of the proposed control approaches are based on finite dimensional approximations with the assumption of an explicitly given moving boundary dynamics [8, 2]. For control objectives, infinite-dimensional approaches have been used for stabilization of the temperature profile and the moving interface of a 1D Stefan problem, such as enthalpy-based feedback [34] and geometric control [31]. These works designed control laws ensuring the asymptotical stability of the closed-loop system in the norm. However, the results in [31] are established based on the assumptions on the liquid temperature being greater than the melting temperature, which must be ensured by showing the positivity of the boundary heat input.

Recently, boundary feedback controllers for the Stefan problem have been designed via a “backstepping transformation” [27, 28, 37] which has been used for many other classes of infinite-dimensional systems. For instance, [16] designed a state feedback control law by introducing a nonlinear backstepping transformation for moving boundary PDE, which achieved the exponentially stabilization of the closed-loop system in the norm without imposing any a priori assumption. Based on the technique, [17] designed an observer-based output feedback control law for the Stefan problem, [18] extended the results in [16, 17] by studying the robustness with respect to the physical parameters and developed an analogous design with Dirichlet boundary actuation, [19] designed a state feedback control for the Stefan problem under the material’s convection, and [22, 26] developed a delay-compensated control for the Stefan problem with proving the robustness to the mismatch between the actuator’s delay and the compensated delay.

While all the aforementioned results are based on the one-phase Stefan problem which neglects the cooling heat caused by the solid phase, an analysis on the system incorporating the cooling heat at the liquid-solid interface has not been established. The one-phase Stefan problem with a prescribed heat flux at the interface was studied in [36]. The author proved the existence and uniqueness of the solution with a prescribed heat input at the fixed boundary by verifying positivity conditions on the interface position and temperature profile using a similar technique as in [10].

Regarding the added heat flux at the interface as the heat loss induced by the remaining other phase dynamics, it is reasonable to treat the prescribed heat flux as the disturbance of the system. The norm estimate of systems with a disturbance is often analyzed in terms of ISS [38], which serves as a criteria for the robustness of the controller or observer design [1]. Recently, the ISS for infinite dimensional systems with respect to the boundary disturbance was developed in [13, 14, 15] using the spectral decomposition of the solution of linear parabolic PDEs in one dimensional spatial coordinate with Strum-Liouville operators. An analogous result for the diffusion equations with a radial coordinate in -dimensional balls is shown in [3] with proposing an application to robust observer design for battery management systems [33].

The present paper revisits the result of our conference paper [24] which proved ISS of the one-phase Stefan problem under the control law proposed in [16] with respect to the heat loss at the interface, and extends the ISS result to the two-phase Stefan problem with an unknown heat loss at the boundary of the solid phase under the control law developed in [25]. First, we consider a prescribed open-loop control of the one-phase Stefan problem in which the solution of the system is equivalent with the proposed closed-loop control. Using the result of [36], the well-posedness of the solution and the positivity conditions are proved. Next, we apply the closed-loop control through the backstepping transformation as in [16]. The well-posedness of the closed-loop solution and the positivity conditions for the model to be valid are ensured by showing the differential equation of the control law. The associated target system has the disturbance at the interface dynamics due to the heat loss. An estimate of the norm of the closed-loop system is developed in the sense of ISS through the analysis of the target system by Lyapunov method. The similar procedure is performed to extend the ISS result to the two-phase Stefan problem with the unknown heat loss at the boundary of the solid phase.

This paper is organized as follows. The one-phase Stefan problem with heat loss at the interface is presented in Section 2. The well-posedness of the system with an open-loop controller is proved in Section 3. Section 4 introduces our first main result for the ISS of the closed-loop system under the designed feedback control law with providing the proof. The extension of the result to the two-phase Stefan problem is addressed in Section 5. Supportive numerical simulations are provided in Section 6. The paper ends with some final remarks in Section 7.

## 2 Description of the Physical Process

Consider a physical model which describes the melting or solidification mechanism in a pure one-component material of length in one dimension. In order to mathematically describe the position at which phase transition occurs, we divide the domain into two time-varying sub-domains, namely, the interval which contains the liquid phase, and the interval that contains the solid phase. A heat flux enters the material through the boundary at (the fixed boundary of the liquid phase) which affects the liquid-solid interface dynamics through heat propagation in liquid phase. In addition, due to the cooling effect from the solid phase, there is a heat loss at the interface position . As a consequence, the heat equation alone does not provide a complete description of the phase transition and must be coupled with the dynamics that describes the moving boundary. This configuration is shown in Fig. 1.

Assuming that the temperature in the liquid phase is not lower than the melting temperature of the material , the energy conservation and heat conduction laws yield the heat equation of the liquid phase as follows

(1) |

with the boundary conditions

(2) | ||||

(3) |

and the initial values

(4) |

where , , , , and are the distributed temperature of the liquid phase, the manipulated heat flux, the liquid density, the liquid heat capacity, and the liquid heat conductivity, respectively. The first problem we consider in this paper assumes that the heat loss at the interface caused by the solid phase temperature dynamics is assumed to be a prescribed function in time, denoted as . The local energy balance at the liquid-solid interface yields

(5) |

where represents the latent heat of fusion. In (2), the left hand side represents the latent heat, and the first and second term of the right hand side represent the heat flux by the liquid phase and the heat loss caused by the solid phase, respectively. As the governing equations (2)–(2) suffice to determine the dynamics of the states , the temperature in the solid phase is not needed to be modeled.

###### Remark 1

There are two underlying assumptions to validate the model (2)-(2). First, the liquid phase is not frozen to solid from the boundary . This condition is ensured if the liquid temperature is greater than the melting temperature. Second, the material is not completely melt or frozen to single phase through the disappearance of the other phase. This condition is guaranteed if the interface position remains inside the material’s domain. In addition, these conditions are also required for the well-posedness (existence and uniqueness) of the solution in this model. Taking into account of these model validity conditions, we emphasize the following remark.

###### Remark 2

Based on the above conditions, we impose the following assumption on the initial data.

###### Assumption 1

, for all , and is continuously differentiable in .

###### Lemma 2

The proof of Lemma 2 is established by maximum principle and Hopf’s lemma as shown in [12]. For the heat flux at the interface, the following assumptions are imposed.

###### Assumption 3

The heat loss remains non-negative, bounded, and continuous for all , and the total energy is also bounded, i.e.,

(8) | ||||

(9) |

The steady-state solution of the system (2)-(2) with zero manipulating heat flux and zero heat loss at the interface yields a uniform melting temperature and a constant interface position given by the initial data. In [16], the authors developed the exponential stabilization of the interface position at a desired reference setpoint with zero heat loss through the following state feedback control design of :

(10) |

where is the control gain which can be chosen by user. To maintain the positivity of the heat input as stated in Lemma 2, the following restriction on the setpoint is imposed.

###### Assumption 4

The setpoint is chosen to verify

(11) |

where .

Finally, we impose the following condition of the control gain.

###### Assumption 5

The control gain is chosen sufficiently large to satisfy , where .

## 3 Open-Loop System and Analysis

The well-posedness of the one-phase Stefan problem with heat flux at the interface was developed in [36] with a prescribed open-loop heat input for . To apply the result, in this section we focus on an open-loop control which has an equivalent solution as the closed-loop control introduced later, and we prove the well-posedness of the open-loop system.

###### Lemma 6

PROOF. Since Assumption 65 leads to , the open-loop controller (6) remains positive and continuous for all by Assumption 3. Hence, applying the theorem in [36] (page 3) proves Lemma 6.

PROOF. We prove by contradiction. Suppose there exists such that . Let be an internal energy of the system defined by

(14) |

with by the imposed assumption. Taking the time derivative of (3) yields the energy conservation

(15) |

In addition, the time derivative of (6) yields . Combining these two and taking integration on both sides gives

(16) |

By (6) and (3), we get . Substituting this and (6) into (3), we have

(17) |

Let be a function in time defined by

(18) |

Since , we can see that for all if and only if for all . By (3), we have . Taking the time derivative of (3) yields

(19) |

By Assumption 5, (3) leads to for all . Therefore, for all , and we conclude for all which contradicts with the imposed assumption where . Hence, Lemma 6 holds globally.

## 4 ISS for One-Phase Stefan Problem

While the open-loop input (6) ensures the well-posed solution of the system, the analysis does not enable to prove the ISS property of the norm estimate. In addition, the open-loop design requires the heat loss at the interface . However, measuring is not practically doable. In this section, we show that the closed-loop solution with the control law proposed in [16] is equivalent to the open-loop solution introduced in Section 3. The controller is feedback design of liquid temperature profile and the interface position . The heat loss is regarded as a disturbance, and the norm estimate of the reference error is derived in a sense of input-to-state stability.

Our first main result is stated in the following theorem.

###### Theorem 8

Under Assumptions 1-5, the closed-loop system consisting of (2)–(2) with the control law (2) satisfies the model validity conditions (2) and (2), and is ISS with respect to the heat loss at the interface, i.e., there exist a class- function and a class- function such that the following estimate holds:

(20) |

for all , in the norm

. Moreover, there exist positive constants and such that the explicit functions of and are given by , , where , which ensures the exponentially ISS.

The proof of Theorem 8 is established through the remainder of this section.

### 4.1 Reference error system

### 4.2 Backstepping transformation

Introduce the following backstepping transformation

(27) |

which maps into

(28) | ||||

(29) | ||||

(30) | ||||

(31) |

The objective of the transformation (4.2) is to add a stabilizing term in (31) of the target -system which is easier to prove the stability for than -system. By taking the derivative of (4.2) with respect to and respectively along the solution of (4.1)-(4.1), to satisfy (4.2), (30), (31), we derive the conditions on the gain kernel solution which yields the solution as

(32) |

By matching the transformation (4.2) with the boundary condition (29), the control law is derived as

(33) |

### 4.3 Inverse transformation

Consider the following inverse transformation

(34) |

Taking the derivatives of (4.3) in and along (4.2)-(31), to match with (4.1)-(4.1), we obtain the gain kernel solution as

(35) |

where , , , and is to be chosen later. Finally, using the inverse transformation, the boundary condition (29) is rewritten as

(36) |

In other words, the target -system is described by (4.2), (30), (31), and (4.3). Note that the boundary condition (30) and the kernel function (32) are modified from the one in [16], while the control design (33) is equivalent. The target system derived in [16] requires -norm analysis for stabilty proof. However, with the prescribed heat loss at the interface, -norm analysis fails to show the stability due to the non-monotonic moving boundary dynamics. The modification of the boundary condition (30) enables to prove the stability in norm as shown later.

### 4.4 Analysis of closed-loop system

Here, we prove the well-posedness of the closed-loop solution and the positivity conditions of the state variables.

###### Lemma 9

PROOF. Taking the time derivative of the control law (2) along with the energy conservation leads to the following differential equation

(37) |

which has the same explicit solution as the open-loop control (6). Hence, the closed-loop solution is equivalent to to the open-loop solution with (6). Since the open-loop system has a unique solution as shown in Lemma 7, the closed-loop solution has a unique solution as well.

###### Lemma 10

The closed-loop solution verifies the following properties:

(38) | ||||

(39) | ||||

(40) |

###### Lemma 11

The closed-loop system has the following property:

(41) |

### 4.5 Stability analysis

To conclude the ISS of the original system, first we show the ISS of the target system (4.2), (30), (31), and (4.3) with respect to the disturbance . We consider the following functional

(42) |

where denotes norm defined by . Taking the time derivative of (42) along with the solution of (4.2)–(31), (4.3), we have

(43) |

Applying Young’s inequality to the second and third lines of (4.5) twice, we get

(44) |

Again applying Young’s inequality to the forth line of (4.5),

(45) |

where for . Applying (4.5)-(4.5) and Cauchy Schwarz, Poincare, and Agmon’s inequalities to (4.5) with choosing , , and , we have

(46) |

where , and . Since and for all , there exists such that for all and . Thus, setting , the inequality (4.5) leads to

(47) |

where . Since and , we have

(48) |

Let be defined by

(49) |

The time derivative of (49) becomes . In addition, applying (3) and (41) to (49) leads to

(50) |

Applying (48) and Young’s inequality to (4.5), we have

(51) |

where . Consider the following functional

(52) |

Taking the time derivative of (52) with the help of (51) and applying (50), we deduce

(53) |

Since (53) leads to the statement that either or is true, following the procedure in [38] (proof of Theorem 5 in Section 3.3), one can derive

(54) |

By (52) and (54) and applying (50), finally we obtain the following estimate on the norm of the target system

(55) |

By the invertibility of the transformation, the norm equivalence of the target -system to the -system holds, and hence the norm estimate on the original -system is derived, which completes the proof of Theorem 8.

## 5 ISS for Two-Phase Stefan Problem

In this section, we extend the results we have established up to the last section to the ”two-phase” Stefan problem, where the heat loss at the interface is precisely modeled by the temperature dynamics in the solid phase, following the work in [25]. An unknown heat loss is then accounted not at the interface, but at the boundary of the solid phase. This configuration is depicted in Fig. 2.

### 5.1 Problem statement

The governing equations are descried by the following coupled PDE-ODE-PDE system:

(56) | ||||

(57) | ||||

(58) | ||||

(59) | ||||

(60) |

where , and all the variables denote the same physical value with the subscript ”l” for the liquid phase and ”s” for the solid phase, respectively. The solid phase temperature must be lower than the melting temperature, which serves as one of the conditions for the model validity, as stated in the following.

###### Remark 3

To keep the physical state of each phase meaningful, the following conditions must be maintained:

(61) | ||||

(62) | ||||

(63) |

The following assumptions on the initial data

are imposed.

###### Assumption 12

, for all , for all , and and are continuously differentiable in and , respectively.

###### Assumption 13

The initial values satisfy

(64) |

The following lemma is proven in [4] (page 4, Theorem 1).

###### Lemma 14

Furthermore, we impose Assumptions 3 and 5, and the restriction for the setpoint is given as follows.

###### Assumption 15

The setpoint is chosen to satisfy

(65) |

where and for .

### 5.2 Control design and an equivalent open-loop analysis

We apply the boundary feedback control design developed in [25]

(66) |

to the two-phase Stefan problem with the unknown heat loss at the boundary of the solid phase governed by (56)–(60). As in the previous procedure, the equivalence of the closed-loop system under the control law (5.2) with the system under an open-loop input is presented in the following lemma.

###### Lemma 16

PROOF. Taking the time derivative of the control law (5.2) together with (56)–(60) leads to . The solution to the differential equation is equivalent with the formulation of (67) with the initial condition . The setpoint restriction given in Assumption 15 provides , and hence for all . Applying Lemma 14, we ensure the existence of stated in Lemma 14. Moreover, we prove by contradiction, similarly to the proof of Lemma 7. Assume the existence of such that the conditions (61)–(63) are satisfied for . Taking into account the specific heat in the solid phase, the internal energy of the total system is defined by

(69) |

Taking the time derivative of (5.2) leads to the energy conservation law given by the same formulation as (3) for the case of one-phase Stefan problem. In the similar manner as the proof of Lemma 7, we deduce

(70) |

If , then substituting it into (5.2) with the help of (62) leads to , which contradicts with (70). Furthermore, applying and the condition (61) for to the control law (5.2), we get

(71) |

for all , which ensures at least . Hence, we conclude that , which guarantees the well-posedness of the closed-loop solution and the conditions for the model validity (61)–(63) to hold globally, i.e., for all , which completes the proof of Lemma 16.

### 5.3 Main result and proof

We present ISS result for the closed-loop system in the following theorem.

###### Theorem 17

Under Assumptions 3, 5, and 12–15, the closed-loop system consisting of (56)–(60) with the control law (5.2) satisfies the model validity conditions (61)–(63), and is ISS with respect to the heat loss at the boundary of the solid phase. Moreover, there exist positive constants and such that the following estimate holds:

(72) |

for all , where , in the norm