Distributed Control Problem

Here the problem is to determine a distributed control function $\, u \in L^{\infty}(\Omega)\,$ that minimizes the functional

$$F(y,u) = \, \int \limits_{\Omega} f(x,y(x),u(x))\,dx \, + \, \int \limits_{\Gamma_1}\,g(x,y(x))\,dx \quad$$ (25)

subject to the elliptic state equation,

$$-\Delta y(x) + d(x,y(x),u(x)) = 0 \,, \quad \mbox{for} \quad x \in \Omega \,,$$ (26)

Neumann and Dirichlet boundary conditions,

$$\partial_{\nu} y(x) = b(x,y(x)) \,, \quad \mbox{for} \quad x \in \Gamma_1\,,$$ (27)

$$y(x) = y_2(x) \,, \hspace*{7mm} \quad \mbox{for} \quad x \in \Gamma_2\,,$$ (28)

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

$$C(x,y(x),u(x)) \leq 0 \,, \qquad \mbox{for } \quad x \in \Omega \,,$$ (29)

$$S(x,y(x)) \leq 0 \,, \hspace*{18.2mm} \mbox{for } \quad x \in \Omega \,.$$ (30)

The functions $\,f: \Omega \times \rm I\! R^2 \rightarrow \rm I\! R, \; g: \Gamma_1 \times \r... ...ightarrow \rm I\! R, \; C: \Omega \times \rm I\! R^2 \rightarrow \rm I\! R\,,$ and $\,S:\Omega \times \rm I\! R\rightarrow \rm I\! R\,$ are assumed to be $\,C^2$-functions, while the Dirichlet condition (28) holds with $\,y_2 \in \,C^1(\Gamma_2)\,$.

The above distributed control problem is slightly more general than the one considered in Bonnans and Casas [9] 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 [8]. Nonlinear elliptic equations of Lotka-Volterra type have been treated in Canada et al. [12] and Leung, Stojanovic [25,34].

Denote an optimal solution of problem (25)-(30) by $\,\bar{u}\,$ and $\,\bar{y}\,$. The active sets corresponding to the inequality constraints (29), (30) are given by

$$\begin{array}{l} J(C):= \{\,x\in \Omega\, \vert \; C(x,\bar... ...\Omega \, \vert \; S(x,\bar{y}(x)) = 0 \, \} \,. \; \end{array}$$ (31)

It is required that the following regularity conditions analogous to (8) hold:

$$\begin{array}{rl} C_u(x,\bar{y}(x),\bar{u}(x)) \not= 0 & \qu... ...ar{y}(x)) \not= 0 & \quad \forall \; x \in J(S) \,. \end{array}$$ (32)

Then first order necessary conditions can be stated in the following form. There exist an adjoint state $\, \bar{q} \in W^{1,1}(\bar{\Omega})\,$, 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:

$$\hspace*{-6mm} -\Delta\bar{q}(x) + \bar{q}(x)\,d_y(x,\bar{y}(x),\bar{u}(x)) + f_y(x,\bar{y}(x),\bar{u}(x)) +$$

$$+ \bar{\lambda}(x) C_y(x,\bar{y}(x),\bar{u}(x)) + S_y(x,\bar{y}(x))\,\bar{\mu} = 0 \quad \mbox{in} \quad \Omega \,, \;$$ (33)

$$\hspace*{-10mm} \partial_{\nu} \bar{q}(x) - \bar{q}(x) b_y(x,\bar{y}(x)) + g_y(x,\bar{y}(x)) = 0 \quad \mbox{on} \quad \Gamma_1 \,, \;$$ (34)

$$\hspace*{-10mm} \bar{q}(x) = 0 \quad \mbox{on} \quad \Gamma_2 \,, \;$$ (35)

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

$$\hspace*{-7mm} f_u(x,\bar{y}(x),\bar{u}(x)) + \bar{q}(x)\,d... ...) + \bar{\lambda}(x)\,C_u(x,\bar{y}(x),\bar{u}(x)) \,= 0 \, ,$$ (36)

complementarity conditions:

$$\begin{array}{rllrll} \bar{\lambda}(x) \geq 0 & \quad \mbox{... ...= 0 & \quad \mbox{in} & \; \Omega\setminus S(S) \,. \end{array}$$ (37)

The adjoint equations (33)-(35) are understood in the weak sense. The regular Borel measure in the adjoint equation (33) has a decomposition similar to that in (15),

$$\hspace*{15mm} \bar{\mu} = \bar{\nu} \cdot dx\,+\,\bar{\nu}_s \cdot \bar{\mu}_s\,,$$ (38)

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

With regard to Example 4.2 in section 4 we shall discuss the minimum condition (36) in case that the control and state constraints (29) and (30) are box constraints

$$y(x) \leq \psi(x) \;, \quad u_1(x) \leq u(x) \leq u_2(x) \qquad \forall \;\; x \in \Omega \,,$$ (39)

with functions $\,\psi \in C(\bar{\Omega})\,$ and $\, u_1, u_2 \in L^{\infty}(\Omega)\,$. We immediately derive from (36) the control law

$$\begin{array}{l} [\,f_u(x,\bar{y}(x),\bar{u}(x)) + \bar{q}(x... ...ll \; u \in [u_1(x),u_2(x)] \,, \; x \in \Omega \,. \end{array}$$ (40)

It is straightforward to obtain analogous control laws for tracking functionals similar to (16).