Necessary conditions for elliptic control problems with control and state constraints

Let $\Omega \subset I\!\! R^2\,$ be a bounded domain with piecewise smooth boundary $\, \Gamma = \partial\,\bar{\Omega}\,$. The derivative in the direction of the outward unit normal $\,\nu\,$ of $\,\Gamma\,$ will be denoted by $\,\partial_{\nu}\,$. Suppose that the boundary is partioned as $\, \Gamma = \Gamma_1 \cup \Gamma_2\,$ with disjoints sets $\,\Gamma_1, \Gamma_2 \subset \Gamma\,$ that are composed of finitely many smooth and connected components.

We consider the optimal control problem of determining a distributed control function $\, u \in L^{\infty}(\Omega)\,$ that minimizes the functional

$$F(y,u) = \, \int \limits_{\Omega} f(x,y(x),u(x))\,dx \, + \, \int \limits_{\Gamma_1}\,g(x,y(x))\,dx \quad$$ (2.1)

subject to the elliptic state equation,

$$-\Delta y(x) + d(x,y(x),u(x)) = 0 \,, \quad \mbox{for} \quad x \in \Omega \,,$$ (2.2)

Neumann and Dirichlet boundary conditions,

$$\partial_{\nu} y(x) = b(x,y(x)) \,, \quad \mbox{for} \quad x \in \Gamma_1\,,$$ (2.3)

$$y(x) = y_2(x) \,, \hspace*{7mm} \quad \mbox{for} \quad x \in \Gamma_2\,,$$ (2.4)

and mixed control-state inequality constraints, resp. pure state inequality constraints,

$$C(x,y(x),u(x)) \leq 0 \,, \qquad \mbox{for } \quad x \in \Omega \,,$$ (2.5)

$$S(x,y(x)) \leq 0 \,, \hspace*{18.2mm} \mbox{for } \quad x \in \Omega\cup\Gamma_1 \,.$$ (2.6)

The split boundary formulation permits simultaneous treatment of various boundary conditions while in [23] this was done in separate sections. The functions $\,f: \Omega \times I\!\! R^2 \rightarrow I\!\! R, \; g: \Gamma_1 \times I\!\! ... ...!\! R\rightarrow I\!\! R, \; C: \Omega \times I\!\! R^2 \rightarrow I\!\! R\,,$ and $\,S:\Omega\cup\Gamma_1 \times I\!\! R\rightarrow I\!\! R\,$ are supposed to be $\,C^1$-functions and $\,y_2 \in \,C^1(\Gamma_2)\,$ is assumed in the Dirichlet condition (2.4). We have to admit that no numerical example with a spliting of the boundary $\Gamma$ into $\Gamma_1, \Gamma_2$ will be considered in this paper. However, the spltting has been introduced to allow for a general discussion of necessary conditions. A practical example with splitted boundary may be found in [25].

The Laplacian $\, - \Delta \,$ in the elliptic equation (2.2) can be replaced by an elliptic operator $\,A\,$ in divergence form. We refer to section 2 of [23] for a precise definition. The above distributed control problem is slightly more general than the one considered in Bonnans and Casas [6] where first order conditions have been given in terms of a weak and strong Pontryagin principle. For linear elliptic equations, first order conditions may also be found in Bergounioux et al. [3], Bonnans and Casas [5]. Nonlinear elliptic equations of Lotka-Volterra type have been considered in Cañada et al. [8] and Leung, Stojanovic [21,28].

Throughout this paper, it wil be assumed that an optimal solution $\,\bar{u}\,$ and $\,\bar{y}\,$ of problem (2.1)-(2.6) exists. To ensure well-posedness of the elliptic problem (2.1)-(2.3) we require as in Bonnans, Casas [6], condition (2.3), that

$$\hspace*{15mm} d_y(x,y,u) \,\geq 0 \qquad \forall \; (x,y,u) \in V \,,$$ (2.7)

holds where $\,V\,$ is a suitable bounded set containing the graph of the optimal solution. In special cases, such as Examples 1 and 2 below, one can dispense with this condition since well-posedness follows from special results cf. Gunzburger et al. [15]. However, we should note that condition (2.7) is not satisfied for all numerical examples in section 4.

The active sets for the inequality constraints (2.5), (2.6) are given by

$$\begin{array}{l} J(C):= \{\,x\in \Omega\, \vert \; C(x,\bar... ...amma_1 \, \vert \; S(x,\bar{y}(x)) = 0 \, \} \,. \; \end{array}$$ (2.8)

We do not study regularity conditions in detail and require that the following ones hold:

$$\begin{array}{rl} C_u(x,\bar{y}(x),\bar{u}(x)) \not= 0 & \qu... ...ar{y}(x)) \not= 0 & \quad \forall \; x \in J(S) \,. \end{array}$$ (2.9)

Extending in a purely formal way the first order necessary conditions in Bonnans and Casas [6], we arrive at the following. There exists an adjoint state $\, \bar{q} \in W^{1,1}({\Omega})\,$, a multiplier $\, \bar{\lambda} \in L^{\infty}(\Omega)\,$, and a regular Borel measure $\,\bar{\mu}\,$ in $\,\Omega\,$ such that the following conditions hold:
adjoint equation and boundary conditions:

$$\hspace*{-12mm} -\Delta\bar{q}(x) + \bar{q}(x)\,d_y(x,\bar{y}(x),\bar{u}(x)) + f_y(x,\bar{y}(x),\bar{u}(x)) +$$

$$+ \bar{\lambda}(x) \,C_y(x,\bar{y}(x),\bar{u}(x)) +S_y(x,\bar{y}(x))\,\bar{\mu} \hspace*{-3mm} = 0 \quad \mbox{in} \quad \Omega \,, \quad$$ (2.10)

$$\hspace*{-10mm} \partial_{\nu} \bar{q}(x) - \bar{q}(x) b_y(x,\bar{y}(x)) + g_y(x,\bar{y}(x))$$

$$+ S_y(x,\bar{y}(x))\,\bar{\mu} \hspace*{-3mm} = 0 \quad \mbox{on} \quad \Gamma_1 \,, \quad$$ (2.11)

$$\hspace*{-10mm} \bar{q}(x) \hspace*{-3mm} = 0 \quad \mbox{on} \quad \Gamma_2 \,. \quad$$ (2.12)

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

$$\hspace*{-8mm} f_u(x,\bar{y}(x),\bar{u}(x)) + \bar{q}(x)\,d... ...+ \bar{\lambda}(x)\,C_u(x,\bar{y}(x),\bar{u}(x)) \,= 0 \, . \;$$ (2.13)

complementarity conditions:

$$\hspace*{-7mm} \begin{array}{rllrll} \bar{\lambda}(x) \geq 0... ...} & \quad (\Omega\cup\Gamma_1)\,\setminus J(S) \,. \end{array}$$ (2.14)

The adjoint equations (2.10)-(2.12) are understood in the weak sense. According to Bourbaki [7], Chapter 9, the regular Borel measure in the adjoint equations (2.10) and (2.11) possesses a decomposition

$$\hspace*{15mm} \bar{\mu} = \bar{\nu} \cdot dx\,+\,\bar{\nu}_s \cdot \bar{\mu}_s\,,$$ (2.15)

where $\,dx\,$ represents the Lebesgue measure, the measure $\,\bar{\mu}_s\,$ is singular with respect to $\,dx\,$ and $\,\bar{\nu},\,\bar{\nu}_s \,$ are measurable.

In many applications, the control and state constraints (2.5) and (2.6) are simple box constraints of the type

$$u_1(x) \leq u(x) \leq u_2(x) \, , \quad y(x) \leq \psi(x) \qquad \mbox{a.e.} \;\; x \in \Omega \,,$$ (2.16)

with functions $\,\psi \in C(\bar{\Omega})\,$ and $\, u_1, u_2 \in L^{\infty}(\Omega)\,$. In this case, the adjoint equation (2.10) reduces to

$$\hspace*{-7mm} -\Delta\bar{q}(x) + \bar{q}(x)\,d_y(x,\bar{y... ...x)) + \bar{\mu} \, = 0 \quad \mbox{in} \quad \Omega \,, \quad$$ (2.17)

while the the minimum condition (2.13) yields the control law

$$\begin{array}{l} [\,f_u(x,\bar{y}(x),\bar{u}(x)) + \bar{q}(x... ...ll \; u \in [u_1(x),u_2(x)] \,, \; x \in \Omega \,. \end{array}$$ (2.18)

A further specialization refers to a cost functional (2.1) of tracking type which has been considered frequently, cf. [1,2,19,20],

$$F(y,u) = \frac{1}{2}\,\int \limits_{\Omega}\,(y(x) - y_d(x))... ...\alpha}{2}\, \int \limits_{\Omega}\, (u(x) - u_d(x))^2\,dx \,,$$ (2.19)

with given functions $\,y_d \in C(\bar{\Omega}), \, u_d \in L^{\infty}(\Omega)\,$, and a nonnegative weight $\,\alpha \geq 0 \,.$ Furthermore, let the function $\,d(x,y,u)\,$ in the state equation (2.2) be linear in the control variable with $\,d(x,y,u) = d_0(x,y) - u\,$. Then we deduce from (2.18) the minimum condition

$$\hspace*{-8mm} [\alpha (\bar{u}(x)-u_d(x)) - \bar{q}(x)] \, ... ... \; u \in [u_1(x),u_2(x)]\,, \; \mbox{a.e. in } \Omega \, . \;$$ (2.20)

Case $\,\alpha > 0 \,$: The previous conditions show that the optimal control $\,\bar{u}(x)\,$ is the projection of $\,u_d(x)+\bar{q}(x)\,$ onto the interval $\,[u_1(x),u_2(x)]\,$. More precisely, we have for $\,x \in \Omega\,$:

$$\hspace*{-8mm} \bar{u}(x) = \left \{ \begin{array}{llllll} u... ... \bar{q}(x)/\alpha \, \geq u_2(x) \,. \end{array}\right \} \;$$ (2.21)

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

$$\hspace*{-8mm} \bar{u}(x) = \left \{ \begin{array}{llllll} u... ...mega \,, \; meas(\Omega_{s}) > 0 \,. \end{array}\right \} \;$$ (2.22)