Elliptic control problems with Dirichlet boundary conditions

The following elliptic control problem with control and state constraints generalizes the elliptic problems considered in Bergounioux, Kunisch [3], Ito, Kunisch [16]. When admitting Dirichlet boundary conditions, the functions $\,g\,$ in the cost functional (2.1), $\,b\,$ in the boundary conditions (2.2), and $\,C\,$ in the constraints (2.3) do not depend on the state variable $\,y\,$. This leads to the problem of finding a control $\, u \in L^{\infty}\,$ that minimizes the functional

$$F(y,u) = \, \int \limits_{\Omega} f(x,y(x))\,dx \, + \, \int \limits_{\Gamma} g(x,u(x))\,dx$$ (3.1)

subject to the state equation

$$\begin{array}{rlll} -\Delta y(x) + d(x,y(x)) & = 0 \,, & \; ... ... b(x,u(x)) \,, & \; \mbox{for} \; x \in \Gamma \,, \end{array}$$ (3.2)

and the inequality constraints on control and state

$$C(x,u(x)) \leq 0 \,, \qquad \mbox{for } \quad x \in \Gamma \,,$$ (3.3)

$$S(x,y(x)) \leq 0 \,, \hspace*{9mm} \mbox{for } \quad x \in \Omega \,.$$ (3.4)

When treating Dirichlet boundary conditions, an intrinsic difficulty arises from the fact that the solution operator $\,u(\cdot)\vert _{\Gamma} \rightarrow y(\cdot)\,$ for (3.2) is not sufficiently smooth as to give an appropriate form of first order necessary conditions: cf., Lions [19], Lions, Magenes [20], Chapter 2. A weak formulation of first order necessary conditions for linear elliptic equations may be found in Bergounioux, Kunisch [3]. Instead of trying to prove first order necessary conditions under strong assumptions we content ourselves with deriving first order necessary conditions in a purely formal way. This form of the necessary conditions will be justified by its analogy in the first order necessary conditions for the discretized version of the elliptic problem; cf. Section 4.2.

Denote an optimal solution of problem (3.1)-(3.4) by $\,\bar{u}\,$ and $\,\bar{y}\,$. The active sets of inequality constraints (3.3) and (3.4) are

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

We require the following regularity conditions:

$$C_u(x,\bar{u}(x)) \not= 0 \quad \mbox{on} \; J(C) \,, \quad S_y(x,\bar{y}(x)) \not= 0 \quad \mbox{on} \; J(S) \,.$$ (3.5)

Let $\,q\,$ and $\,p\,$ be multipliers associated with the elliptic equation and the Dirichlet boundary condition in (3.2), and let $\,\lambda\,$ resp. $\,\mu\,$ be the multiplier resp. the Borel measure associated with the control and state inequality constraints (3.3) and (3.4). Then the Lagrangian for problem (3.1)-(3.4) becomes

$$\begin{eqnarray*} \hspace*{-9mm} {\cal L}(y,u,q,p,\lambda,\mu): && \hspace*{-5mm... ...p(x) dx + \, \int \limits_{\Gamma} \lambda(x)\,C(x,u(x))\,dx . \end{eqnarray*}$$

The first order necessary condition with respect to the state variable gives

$${\cal L}_y(\bar{y},\bar{u},\bar{q},\bar{p},\bar{\lambda},\bar{\mu}) y = 0 \quad \mbox{for all } \; y \, .$$

Function evaluation along a stationary solution $\, (\bar{y},\bar{u},\bar{q},\bar{p},\bar{\lambda},\bar{\mu}) \,$ will be denoted henceforth by a bar, e.g., $\, \bar{f}=f(x,\bar{y}),$ etc. Using partial integration (Green's theorem),

$$\int \limits_{\Omega} (-\Delta y \,q + y\,\Delta q)\,dx \,= ... ...its_{\Gamma} (-q\,\partial_{\nu}y + y\,\partial_{\nu}q)\,dx\,,$$

we rewrite $\,\bar{\cal L}_y y = 0 \,$ as

$$\begin{eqnarray*} \hspace*{-10mm} \bar{\cal L}_y y =\int \limits_{\Omega} [\,-\... ..., - \, \int \limits_{\Gamma} \bar{q}\,\partial_{\nu}y\,dx \; = 0 \end{eqnarray*}$$

which holds for all functions $\,y\,$. From this we obtain the adjoint equations:

$$\hspace*{-12mm} -\Delta\bar{q}(x) + \bar{q}(x)\,d_y(x,\bar{y}(x)) + f_y(x,\bar{y}(x)) + S_y(x,\bar{y}(x))\,\bar{\mu} = 0 \hspace*{-3mm} \mbox{on} \;\;\Omega \,, \quad$$ (3.6)

$$\hspace*{41mm} \bar{q}(x) = 0 \hspace*{-3mm} \mbox{on} \;\;\Gamma \,. \quad$$ (3.7)

Moreover, we find the multiplier $\, p = - \partial_{\nu}q\,$ on $\,\Gamma\,$. Then the optimality condition for the control variable is evaluated as

$$\bar{\cal L}_u\,u = [\,\bar{g}_u + (\partial_{\nu}\bar{q})\... ...{\lambda} \bar{C}_u\,]u \, = 0 \quad \mbox{for all} \;\; u \,,$$

which yields the minimum condition:

$$g_u(x,\bar{u}(x)) + \partial_{\nu}\bar{q}(x)\, b_u(x,\bar{u}... ...}(x)\,C_u(x,\bar{u}(x)) = 0 \qquad \mbox{on} \; \Gamma \,. \;$$ (3.8)

The complementarity condition is similar to (2.10):

$$\begin{array}{rllrll} \bar{\lambda}(x) \geq 0 & \; \mbox{on}... ...} \,= 0 & \; \mbox{on} & \Omega \setminus J(S) \,. \end{array}$$ (3.9)

The Borel measure $\,\bar{\mu}\,$ has a decomposition similar to the one in (2.11).

Note that the minimum condition (3.8) agrees with the condition (2.9) if we replace $\,\bar{q}\,$ formally by $\,-\partial_{\nu}\bar{q}\,$. Hence, assuming functionals of tracking type (2.12), box constraints (2.13) for control and state and the property $\,b_u \equiv 1 \,$, we obtain the counterparts of the control laws (2.16), (2.17).

Case $\,\alpha > 0\,$: the control is determined by

$$\hspace*{-9mm} \bar{u}(x) = \left \{ \begin{array}{llllll} u... ...u}\bar{q}(x)/\alpha \,\geq u_2(x) \,. \end{array}\right \} \;$$ (3.10)

Case $\,\alpha=0\,$: the optimal control is of bang-bang or singular type

$$\hspace*{-8mm} \bar{u}(x) = \left \{ \begin{array}{llllll} u... ...\Gamma \,, \; meas(\Gamma_s) > 0 \,. \end{array}\right \} \;$$ (3.11)

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