Boundary control problem

We consider the problem of determining a boundary control function $\,u \in L^{\infty}(\Gamma)\,$ which minimizes the functional

$$\hspace*{-10mm} F(y,u) = \, \int \limits_{\Omega} f(x,y(x))\... ...),u(x))\,dx \,+\, \int \limits_{\Gamma_2}\,k(x,u(x))\,dx \quad$$ (1)

subject to the elliptic state equation,

$$-\Delta y(x) + d(x,y(x)) = 0 \,, \quad \mbox{for} \quad x \in \Omega \,,$$ (2)

boundary conditions of Neumann or Dirichlet type,

$$\partial_{\nu} y(x) = b(x,y(x),u(x)) \,, \quad \mbox{for} \quad x \in \Gamma_1\,,$$ (3)

$$y(x) = a(x,u(x)) \,, \hspace*{10mm} \quad \mbox{for} \quad x \in \Gamma_2\,,$$ (4)

and control and state inequality constraints,

$$C(x,u(x)) \leq 0 \,, \qquad \mbox{for } \quad x \in \Gamma \,,$$ (5)

$$S(x,y(x)) \leq 0 \,, \hspace*{9mm} \mbox{for } \quad x \in \Omega \,.$$ (6)

The functions $\,f: \Omega \times \rm I\! R\rightarrow \rm I\! R, \; g: \Gamma_1 \times \rm I... ...R\rightarrow \rm I\! R, \; C: \Gamma \times \rm I\! R\rightarrow \rm I\! R\,,$ and
$\,S:\Omega \times \rm I\! R\rightarrow \rm I\! R\,$ are assumed to be $\,C^2$-functions. It is straightforward to include more than one inequality constraint in (5) or (6). 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.

The Laplacian $\,\Delta\,$ in (2) can be replaced by any 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 $\, c > 0 \,$:

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

However, in the sequel we restrict the discussion to the operator $\,A=\Delta\,$ which simplifies the form of the necessary conditions and the numerical analysis.

An optimal solution of problem (1)-(6) will be denoted by $\,\bar{u}\,$ and $\,\bar{y}\,$. The active sets for the inequality constraints (5), (6) are defined by

$$\hspace*{-8mm} J(C):= \{\,x\in \Gamma\, \vert \; C(x,\bar{u}... ...= \{\,x\in \Omega \, \vert \; S(x,\bar{y}(x)) = 0 \, \} \,. \;$$ (7)

The following regularity conditions are supposed to hold,

$$\begin{array}{rl} C_u(x,\bar{u}(x)) \not= 0 & \quad \forall ... ...ar{y}(x)) \not= 0 & \quad \forall \; x \in J(S) \,. \end{array}$$ (8)

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

First order necessary conditions for the rather general problem (1)-(6) are not yet available in the literature. The main difficulty results from the Dirichlet condition (4) which prevents solution from being sufficiently regular. First order necessary conditions for problems with linear elliptic equations $\, -\Delta y(x) + y(x) = 0 \,$ and pure Neumann conditions may be found in Casas [14], Casas et al. [15,16]. A weak formulation for linear elliptic equations and Dirichlet conditions is due to Bergounioux, Kunisch [4].

We shall present first order conditions in a form that can be derived at least in a purely formal way. This form will turn out to be consistent with the first order conditions of Kuhn-Tucker for the discretized elliptic control problem in section 3.1. We assume that there exists an adjoint state $\, \bar{q} \in W^{1,1}(\bar{\Omega})\,$, a multiplier $\, \bar{\lambda} \in L^{\infty}(\Gamma)\,$, and a regular Borel measure $\,\bar{\mu}\,$ on $\,\Omega \,$ such that the following conditions hold:

adjoint equation and boundary conditions:

$$- \Delta\bar{q}(x) + \bar{q}(x)\,d_y(x,\bar{y}(x)) + f_y(x,\bar{y}(x)) + \hspace*{13mm}$$

$$+ S_y(x,\bar{y}(x))\,\bar{\mu} = 0 \quad \mbox{on} \;\; \Omega \,, \quad$$ (9)

$$\partial_{\nu} \bar{q}(x) - \bar{q}(x) b_y(x,\bar{y}(x),\bar{u}(x)) + g_y(x,\bar{y}(x),\bar{u}(x)) = 0 \quad \mbox{on} \;\; \Gamma_1 \,, \quad$$ (10)

$$\bar{q}(x) = 0 \quad \mbox{on} \;\; \Gamma_2 \,, \quad$$ (11)

minimum condition for $\,x \in \Gamma_1\,$:

$$\hspace*{-4mm} g_u(x,\bar{y}(x),\bar{u}(x)) - \bar{q}(x)\, b... ...),\bar{u}(x)) + \bar{\lambda}(x)\,C_u(x,\bar{u}(x)) = 0 \,,\;$$ (12)

minimum condition for $\,x \in \Gamma_2\,$:

$$k_u(x,\bar{u}(x)) + \partial_{\nu} \bar{q}(x)\, a_u(x,\bar{u}(x)) + \bar{\lambda}(x)\,C_u(x,\bar{u}(x)) = 0 \,,\;$$ (13)

complementarity conditions:

$$\hspace*{-4mm} \begin{array}{rllrll} \bar{\lambda}(x) \geq 0... ...& \quad \mbox{in} & \;\; \Omega \setminus J(S) \,. \end{array}$$ (14)

The adjoint equations (9)-(11) are understood in the weak sense, cf. Casas et al. [16]. According to Bourbaki [10], Chapter 9, the regular Borel measure $\,\bar{\mu}\,$ appearing in the adjoint equation (9) has the decomposition

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

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 $\,\Omega \,$. The problem of obtaining the decomposition (15) explicitly is related to the difficulty of determining the structure of the active set $\,J(S)\,$. In section 3, we shall make an attempt to approximate the measure by the multipliers of the discretized control problem.

In many applications, the cost functional (1) is of tracking type, cf. [2,4,22,24],

$$F(y,u) = \frac{1}{2}\,\int \limits_{\Omega}\,(y(x) - y_d(x))... ...\alpha}{2}\, \int \limits_{\Gamma}\, (u(x) - u_d(x))^2\,dx \,,$$ (16)

with given functions $\,y_d \in C(\bar{\Omega}), \, u_d \in L^{\infty}(\Gamma)\,$, and nonnegative weight $\,\alpha \geq 0 \,.$ The control and state constraints (5) and (6) are taken to be box constraints of the simple type

$$y(x) \leq \psi(x) \quad \mbox{in} \;\; \Omega \,, \qquad u_1(x) \leq u(x) \leq u_2(x) \quad \mbox{on} \;\; \Gamma \,,$$ (17)

with functions $\,\psi \in C(\bar{\Omega})\,$ and $\, u_1, u_2 \in L^{\infty}(\Gamma)\,$. Here, in particular we assume that the functions $\,g\,$ and $\,k\,$ in (1) coincide. For these data the adjoint equations (9)-(11) become

$$\hspace*{-10mm} \begin{array}{rll} -\Delta \bar{q}(x) + \bar... ...x) & = 0 & \quad \mbox{on} \quad \Gamma_2 \,. \quad \end{array}$$ (18)

If the function $\,b\,$ in the Neumann condition (3) has the special form $\, b(x,y,u) = b_0(x,y) + u \,$, then the minimum condition (12) reduces to

$$\hspace*{-8mm} [\alpha (\bar{u}(x)-u_d(x)) - \bar{q}(x)] \, ... ...uad \forall \, x \in \Gamma_1, \; u \in [u_1(x),u_2(x)] \,. \;$$ (19)

Likewise, if the function $\,a\,$ in (4) is given by $\, a(x,u) = a_0(x) + u \,$, the minimum condition (13) yields

$$\hspace*{-7mm} [\alpha (\bar{u}(x)-u_d(x)) + \partial_{\nu} ... ...uad \forall \, x \in \Gamma_2, \; u \in [u_1(x),u_2(x)] \,. \;$$ (20)

Case $\,\alpha > 0 \,$: The previous conditions determine the following control laws:
for $\,x \in \Gamma_1\,$,

$$\hspace*{-8mm} \bar{u}(x) = \left \{ \begin{array}{llllll} u... ...d(x) + \bar{q}(x)/\alpha \geq u_2(x) \,, \end{array}\right \}$$ (21)

for $\,x \in \Gamma_2\,$,

$$\hspace*{-9mm} \bar{u}(x) = \left \{ \begin{array}{llllll} u... ...u}\bar{q}(x)/\alpha \,\geq u_2(x) \,. \end{array}\right \} \;$$ (22)

Case $\,\alpha=0\,$: We obtain an optimal control of bang-bang or singular type:
for $\,x \in \Gamma_1\,$,

$$\hspace*{-8mm} \bar{u}(x) = \left \{ \begin{array}{llllll} u... ...amma_1 \,, \; meas(\Gamma_{s1}) > 0 \,, \end{array}\right \}$$ (23)

for $\,x \in \Gamma_2\,$,

$$\hspace*{-8mm} \bar{u}(x) = \left \{ \begin{array}{llllll} u... ...a_2 \,, \; meas(\Gamma_{s2}) > 0 \,. \end{array}\right \} \;$$ (24)

Hence in case $\,\alpha=0\,$, the so-called switching function is given by the adjoint function $\,\bar{q}(x)\,$ on the boundary $\,\Gamma_1\,$ resp. by the outward normal derivative $\,\partial_{\nu}\bar{q}(x)\,$ on the boundary $\,\Gamma _2\,$. The isolated zeros of the switching function are the switching points of a bang-bang control; cf. the example in section 4.1.