Control and Controllability of PDEs with Hysteresis

Abstract

A controllability problem for a diffusion equation with a complex hysteresis operator is proposed here and consists in finding a control which guarantees that the solution reaches a desired value at a given time. We propose a constructive method based on a two-parameter penalty argument. One small parameter penalizes the distance of the solution at final time from the expected value, the second one is used to approximate the underlying rate independent variational inequalities in the hysteresis term by smooth viscous constitutive relations. We prove that a solution to the controllability problem can be obtained by passing to the limit in a doubly degenerate control system, and the convergence is strong in space and uniform in time.

Appendix: A Physical Model

Assume that a domain $\varOmega \subset {\mathbb {R}}^{N}$ (in practice, we choose $N=3$) is filled with a material substance in which two phases may coexist: solid and liquid. The state variables are the following functions of the space variable $x\in \varOmega$ and time $t\in [0,T]$:

$$\begin{aligned} \begin{array}{ll} s(x,t)\in [-1,1]&{}\ \hbox {phase fraction}: s=-1\ \hbox {solid}, s=1\ \hbox {liquid}, s\in (-1,1)\ \hbox {mixture};\\ \theta (x,t)>0 &{}\ \hbox {absolute temperature}. \end{array} \end{aligned}$$

The process of transition between the two phases is governed by the first and the second principle of thermodynamics involving two state functions: The internal energy $U=U(\theta ,s)$ and the entropy $S=S(\theta ,s)$, which have to satisfy the energy balance equation

$$\begin{aligned} U_t+\mathrm{div}\varPhi =h \end{aligned}$$

(4.26)

and the Clausius–Duhem inequality

$$\begin{aligned} S_t+\mathrm{div}\left( \frac{\varPhi }{\theta }\right) \ge \frac{h}{\theta } \end{aligned}$$

(4.27)

for all processes. Here we denote by $\varPhi$ the heat flux vector and by h the heat source density. We further introduce the free energy $F=F(\theta ,s)$ by the formula

$$\begin{aligned} F(\theta ,s)=U(\theta ,s)-\theta S(\theta ,s), \end{aligned}$$

(4.28)

so that in terms of F the second principle (4.27) can be equivalently stated as the condition that the inequality

$$\begin{aligned} F_t+\theta _t S+\frac{1}{\theta }\left\langle \varPhi ,\nabla \theta \right\rangle \le 0 \end{aligned}$$

(4.29)

holds for all processes. Since the term $\left\langle \varPhi ,\nabla \theta \right\rangle$ does not contain time derivatives and $F_t+\theta _t S$ does not contain space derivatives, they must both be non-positive for all processes. Assuming for the heat flux the Fourier law

$$\begin{aligned} \varPhi =-\kappa \nabla \theta \end{aligned}$$

(4.30)

with a constant heat conductivity $\kappa >0$, the non-positivity of $\left\langle \varPhi ,\nabla \theta \right\rangle$ is obvious. We now compare the remaining inequality $F_t+\theta _t S \le 0$ for all processes with the chain rule identity $F_t=F_\theta \theta _t+F_s s_t$ and obtain another formally equivalent reformulation of the second principle, namely

$$\begin{aligned} S= & {} -F_\theta , \end{aligned}$$

(4.31)

$$\begin{aligned} F_s s_t\le & {} 0. \end{aligned}$$

(4.32)

From (4.28) and (4.31) we deduce the following differential equation for F

$$\begin{aligned} F-\theta F_\theta = U, \end{aligned}$$

(4.33)

which can be solved if we know the internal energy U. For simplicity, we assume the internal energy in the form

$$\begin{aligned} U=c\theta +L(s), \end{aligned}$$

(4.34)

where $c>0$ is the specific heat capacity which we assume constant, and L is an increasing $C^1$-function representing the latent heat. Solutions to (4.33) can be explicitly found and they all differ only by an additive “integration” constant which may depend on s. A “minimal” choice in the sense that no unphysical constants are involved and all values of the phase fraction s outside the admissible interval $[-1,1]$ are excluded is given by the formula

$$\begin{aligned} F(\theta ,s)=-c\theta \log (\theta /\theta _c)+\frac{L(s)}{\theta _c} (\theta _c-\theta )+I(s), \end{aligned}$$

where I(s) is as before the indicator function of the interval $[-1,1]$, and $\theta _c>0$ is a fixed reference temperature (the melting temperature). Condition (4.32) then reads for $s\in (-1,1)$ (note that L is an increasing function)

$$\begin{aligned} s_t(\theta -\theta _c)\ge 0, \end{aligned}$$

(4.35)

which has a clear meaning: The substance has the tendency to melt for low temperatures, and the tendency to solidify for high temperatures. A natural choice for the phase dynamics equation is then

$$\begin{aligned} {\rho {s}}_t+\partial I(s)\ni \frac{1}{\theta _c}(\theta -\theta _c) \end{aligned}$$

(4.36)

with phase relaxation time $\rho >0$. Together with the energy balance equation

$$\begin{aligned} (c\theta + L(s))_t - \kappa \varDelta \theta = h \end{aligned}$$

(4.37)

resulting from (4.26), (4.30), and (4.34), we obtain a system called the relaxed Stefan problem, see [20]. Note also the role of the parameter $\rho$: The smaller it is, the faster the phase transition takes place. When $\rho \rightarrow 0$, the phase transition becomes instantaneous, which corresponds to the classical Stefan problem.

We now show that the energy balance equation (4.26) can be transformed into the form (1.1). Indeed, we define a new unknown u by the formula

$$\begin{aligned} u_t=\frac{1}{\rho \theta _c}(\theta -\theta _c). \end{aligned}$$

(4.38)

Then the phase dynamics equation in (4.36) reads

$$\begin{aligned} s_t+\partial I(s)\ni u_t, \end{aligned}$$

which is nothing but the definition of the stop operator with threshold 1

$$\begin{aligned} s={\mathfrak {s}}_1[u], \end{aligned}$$

see Sect. 2. This enables us to rewrite (4.37) in the form

$$\begin{aligned} cu_{tt}+\frac{1}{\rho \theta _c}L({\mathfrak {s}}_1[u])_t-\kappa \varDelta u_t=\frac{1}{\rho \theta _c} h. \end{aligned}$$

Integrating the above equation in time leads to

$$\begin{aligned} cu_t+\frac{1}{\rho \theta _c}L({\mathfrak {s}}_1[u])-\kappa \varDelta u=v, \end{aligned}$$

with v containing the time integral of h and additional terms coming from the initial conditions. Up to the physical constants, this is precisely (1.1) with ${\mathscr {F}}[u] = L({\mathfrak {s}}_1[u])$. The homogeneous Neumann boundary condition for $\theta$ (and therefore for u in (3.1)) has the physical meaning of a thermally insulated body.