Elliptic control problems with Neumann boundary conditions

The following elliptic control problem with control and state constraints constitutes a generalization of elliptic problems considered in Casas [9], Casas et al. [10,11], Ito, Kunisch [16], Kunisch, Volkwein [18]. The problem is to determine a control $\,u \in L^{\infty}(\Gamma)\,$ that minimizes the functional

$$F(y,u) = \, \int \limits_{\Omega} f(x,y(x))\,dx \, + \, \int \limits_{\Gamma}\,g(x,y(x),u(x))\,dx$$ (2.1)

subject to the state equation

$$\begin{array}{rllll} -\Delta y(x) + d(x,y(x)) & = & 0 \,, & ... ...u(x)) \,, & \quad \mbox{for} \quad x \in \Gamma \,, \end{array}$$ (2.2)

and the inequality constraints on control and state

$$C(x,y(x),u(x)) \leq 0 \,, \qquad \mbox{for } \quad x \in \Gamma \,,$$ (2.3)

$$S(x,y(x)) \leq 0 \,,\hspace*{18.5mm} \mbox{for } \quad x \in \bar{\Omega}\,.$$ (2.4)

In this setting, $\Omega \subset I\!\! R^2\,$ is a bounded domain with piecewise smooth boundary $\, \Gamma = \partial\,\bar{\Omega}\,$. The derivative in the direction of the outward unit normal $\,\nu\,$ of $\,\Gamma\,$ is denoted by $\,\partial_{\nu}\,$ in (2.2). Note that the state inequality constraints (2.4) are supposed to hold on the closure of $\,\Omega\,$.

The Laplacian $\,-\Delta\,$ in (2.2) can be replaced by an elliptic operator

$$Ay(x) = - \sum_{k,j=1}^2\, \partial_ {x_k} (a_{kj}(\cdot)\, \partial_{x_j}\,y)(x) \,,$$

where the coefficients $\,a_{kj} \in C^2(\bar{\Omega})\,$ satisfy the following coercivity condition with some $\, c > 0 \,$:

$$\sum_{k,j=1}^2\, a_{kj}(x) v_k v_j \,\geq \, c(v_1^2 + v_2^2) \quad \forall x \in \bar{\Omega}, \; v \in I\!\! R^2 \,.$$

However, in the sequel we restrict the discussion to the operator $\,A=-\Delta\,$ which simplifies the form of the necessary conditions and the presentation of the numerical approach in section 4. The functions $\,f: \Omega \times I\!\! R\rightarrow I\!\! R, \, g: \Gamma \times I\!\! R^2 \... ...\! R^2 \rightarrow I\!\! R, \, C: \Gamma \times I\!\! R^2 \rightarrow I\!\! R\,$ and $\,S:\bar{\Omega} \times I\!\! R\rightarrow I\!\! R\,$ are assumed to be $\,C^2$-functions. It is straightforward to include more than one inequality constraint into (2.3) or (2.4). However, since both the state and control variable are scalar variables, the active sets for different inequality constraints are disjoint and hence can be treated separately.

Then under appropriate assumptions on the function $\,d\,$ it can be shown that the state equation (2.2) admits for each $\,u \in L^{\infty}(\Gamma)\,$ a weak solution $\,y\in Y=C(\bar{\Omega}) \cap H^1(\Omega)\,$ (cf. Casas et al. [10]), i.e., it holds

$$\int \limits_{\Omega} \, [\,\Delta y(x)\,\Delta v(x) + d(x,y... ...v(x)\,]\,dx = \int \limits_{\Gamma}\,b(x,y(x),u(x))\,v(x)\,dx$$

for all $\,\, v \in H^1(\Omega) \,.$ An optimal solution of problem (2.1)-(2.4) will be denoted by $\,\bar{u}\,$ and $\,\bar{y}\,$. From [10] we infer the further assumption that the function $\,b\,$ in the Neumann condition (2.2) is sufficiently smooth and satisfies the following inequality with suitable $\,\epsilon > 0 \,$,

$$b_y(x,y,u) \leq 0 \qquad \mbox{for all} \quad x \in \Gamma, ... ... \epsilon , \; \vert\,u-\bar{u}(x)\,\vert < \epsilon \,. \quad$$ (2.5)

Questions of existence of optimal solutions will not be discussed here. The active sets for the inequality constraints (2.3) and (2.4) are defined by

$$J(C):= \{\,x\in \Gamma\, \vert \; C(x,\bar{y}(x),\bar{u}(x))... ...\{\,x\in \bar{\Omega}\, \vert \; S(x,\bar{y}(x)) = 0 \, \} \,.$$

It is required that the following regularity conditions hold:

$$\hspace*{-5mm} C_u(x,\bar{y}(x),\bar{u}(x)) \not= 0 \quad \f... ...\;\; S_y(x,\bar{y}(x)) \not= 0 \quad \forall \; x \in J(S) \,.$$ (2.6)

Here and in the following, partial derivatives are denoted by subscripts.

First order optimality conditions for a local optimal solution $\,\bar{u}\,$ and $\,\bar{y}\,$ can be derived by generalizing the line of proof in Casas [9], Casas et al. [10,11]. Problem (2.1)-(2.4) is considered as a mathematical programming problem in Banach spaces to which the first order Kuhn-Tucker conditions are applicable. In particular, this approach requires that the regularity condition given in Zowe, Kurcyusz [24] is satisfied; cf. Casas et al. [11]. We do not discuss this regularity condition in detail although condition (2.6) forms part of it. The first order necessary conditions imply that there exist an adjoint state $\, \bar{q} \in W^{1,1}(\Omega)\,$, a multiplier $\, \bar{\lambda} \in L^{\infty}(\Gamma)\,$, and a bounded Borel measure $\,\bar{\mu}\,$ on $\,\bar{\Omega}\,$ such that the following three conditions hold,

1. adjoint equation:

$$\hspace*{-10mm} -\Delta\bar{q}(x) + \bar{q}(x)\,d_y(x,\bar{y}(x)) + f_y(x,\bar{y}(x)) + S_y(x,\bar{y}(x))\,\bar{\mu} \hspace*{-4mm} = 0 \quad \mbox{on} \;\; \Omega \,, \quad$$ (2.7)

$$\hspace*{-8mm} \partial_{\nu} \,\bar{q}(x) - \bar{q}(x)\,b_y(x,\bar{y}(x),\bar{u}(x)) + g_y(x,\bar{y}(x),\bar{u}(x)) + \hspace*{4mm}$$

$$+ \bar{\lambda}(x)\,C_y(x,\bar{y}(x),\bar{u}(x)) + S_y(x,\bar{y}(x))\,\bar{\mu} \hspace*{-4mm} = 0 \quad \mbox{on} \;\; \Gamma \,,$$ (2.8)

2. minimum condition on $\,\Gamma\,$:

$$\hspace*{-8mm} g_u(x,\bar{y}(x),\bar{u}(x)) - \bar{q}(x)\, b... ...)) + \bar{\lambda}(x)\,C_u(x,\bar{y}(x),\bar{u}(x)) = 0 \,,\;$$ (2.9)

3. complementarity condition:

$$\begin{array}{rlllll} \bar{\lambda}(x) \geq 0 & \; \mbox{on}... ... 0 & \; \mbox{on} & \bar{\Omega} \setminus J(S) \,. \end{array}$$ (2.10)

The adjoint equations (2.7), (2.8) are understood in the weak sense, cf. Casas et al. [11]. According to Bourbaki [6], Chapter 9, the bounded Borel measure $\,\bar{\mu}\,$ appearing in the adjoint equation (2.7), (2.8) has the decomposition

$$\hspace*{15mm} \bar{\mu} = \bar{\nu} \cdot dx\,+\,\bar{\nu}_s \cdot \bar{\mu}_s\,,$$ (2.11)

where $\,dx\,$ represents the Lebesgue measure and $\,\bar{\mu}_s\,$ is singular with respect to $\,dx\,$; the functions $\,\bar{\nu},\,\bar{\nu}_s\,$ are measurable on $\,\bar{\Omega}\,$. The problem of obtaining the decomposition (2.11) explicitly is related to the difficulty of determining the structure of the active set $\,J(S)\,$. To our knowledge, the literature does not contain any numerical examples where the decomposition (2.11) has actually been computed. In section 4, we shall make an attempt to approximate the measure by the multipliers of the discretized problem.

In later applications, we shall mostly deal with cost functionals of tracking type (cf. Ito, Kunisch [16]),

$$F(y,u) = \frac{1}{2}\,\int \limits_{\Omega}\,(y(x) - y_d(x))... ...ac{\alpha}{2}\, \int \limits_{\Gamma}\, (u(x) - u_d(x))^2\,dx$$ (2.12)

with given functions $\,y_d \in C(\bar{\Omega}), \, u_d \in L^{\infty}(\Gamma)\,$, and a nonnegative weight $\,\alpha \geq 0 \,.$ The control and state constraints (2.3) and (2.4) are supposed to be box constraints of the simple type

$$y(x) \leq \psi(x) \quad \mbox{on} \; \Omega \,, \qquad u_1(x) \leq u(x) \leq u_2(x) \quad \mbox{on} \; \Gamma \,,$$ (2.13)

with functions $\,\psi \in C(\bar{\Omega})\,$ and $\, u_1, u_2 \in L^{\infty}(\Gamma)\,$. For these data the adjoint equation (2.7), (2.8) become

$$\hspace*{-10mm} \begin{array}{ll} -\Delta \bar{q}(x) + \bar{... ...),\bar{u}(x)) \,= 0 & \;\; \mbox{on} \; \Gamma \,, \end{array}$$ (2.14)

If the function $\,b\,$ in (2.2) has the form $\, b(x,y,u) = u + b_0(x,y) \,$, i.e., if $\,b_u \equiv 1 \,$ holds, then the minimum condition (2.9) reduces to

$$\hspace*{-10mm} [\alpha (\bar{u}(x)-u_d(x)) - \bar{q}(x)] \,... ...\quad \forall \, x \in \Gamma, \; u \in [u_1(x),u_2(x)] \,. \;$$ (2.15)

Case $\,\alpha > 0\,$: condition (2.15) shows that the control $\,u(x)\,$ is given by the projection of $\, u_d(x) + \bar{q}(x)/ \alpha \,$ on the interval $\,[u_1(x),u_2(x)]\,$ which can be stated more explicitly as

$$\hspace*{-8mm} \bar{u}(x) = \left \{ \begin{array}{llllll} u... ...) + \bar{q}(x)/\alpha \, \geq u_2(x) \,. \end{array}\right \}$$ (2.16)

Case $\,\alpha=0\,$: we obtain an optimal control of bang-bang or singular type:

$$\hspace*{-8mm} \bar{u}(x) = \left \{ \begin{array}{llllll} u... ...et \Gamma \,, \; meas(\Gamma_s) > 0 \,. \end{array}\right \}$$ (2.17)

Thus, for $\,\alpha=0\,$ the adjoint function $\,\bar{q}(x)\,$ on the boundary plays the role of a switching function. The isolated zeros of $\,\bar{q}(x)\vert _{\Gamma}\,$ are the switching points of a bang-bang control.