Parabolic and Elliptic Control Problems

In this section we describe the continuous control problems exemplarily considered in the following. The first is a one-dimensional parabolic boundary control problem formulated such that it includes problems from [1,9]

$$\begin{eqnarray*} &&\mbox{minimize}\\ f(y,u)&=&\frac12\int^l_0(y(x,T)-y_T(x))^2... ...u(t)^2dt\\ &&+\int^T_0(a_y(t)y(l,t)+a_u(t)u(t))dt,\quad\alpha>0 \end{eqnarray*}$$

    subject to(P)

$$\begin{eqnarray*} &&y_t-y_{xx}=0\quad\mbox{in }(0,l)\times(0,T)\\ &&y(x,0)=a(x)... ...{in }(0,T)\\ &&\alpha_1\le u(t)\le\alpha_2\quad\mbox{in }(0,T). \end{eqnarray*}$$

The notation above is that of [1] and the corresponding data will be defined in section 5 below. The problem does, however, include also one considered in [9], namely by choosing $b(t)=0$, $\varphi(y(l,t))=y^2(l,t),a_y(t)=0,a_u(t)=0$.

We define the following discretization of problem (P).

$$\begin{eqnarray*} &&\mbox{minimize}\quad f_h(y_h,u_h)=\frac{dx}4((y_{0,m}-y_T(x_... ...y(t_j)y_{n,j}+a_u(t_j)u_j) +\frac12a_y(T)y_{n,m}+a_u(T)u_m\bigg) \end{eqnarray*}$$

         subject to(P$_h$)

$$\begin{eqnarray*} &&\quad\frac{y_{j,i+1}-y_{j,i}}{dt}=\frac12(y_{j-1,i}-2y_{j,i}... ...ldots,m\\ &&\quad\alpha_1\le u_i\le\alpha_2,\quad i=1,\ldots,m. \end{eqnarray*}$$

Here $x_j=jdx$, $dx=1/n$, $t_j=jdt$, $dt=T/m$.

For the problem (P) above and specific data an analytic solution is given in [1] and this also permits the authors to verify the necessary and sufficient optimality conditions they had stated and proved.

For the elliptic control problems we consider the class defined in [19]. It includes boundary and distributed controls which are addressed separately in [16], respectively [17] as well as Dirichlet, Neumann, and mixed boundary conditions. In the case of boundary control the underlying continuous problem is

$$\begin{eqnarray*} \mbox{minimize}\\ F(y,u)&=&\int_{\Omega_0}f(x,y(x))dx+\int_{\Gamma_1}g(x,y(x),u(x))dx\\ &&+\int_{\Gamma_2}k(x,u(x))dx \end{eqnarray*}$$

    subject to(EB)

$$\begin{eqnarray*} &&-\Delta y(x)+d(x,y(x))=0,\quad\mbox{for }x\in\Omega,\\ &&\p... ...}x\in\Gamma_1,\\ &&y(x)=a(x,u(x)),\quad\mbox{for }x\in\Gamma_2, \end{eqnarray*}$$

and

$$\begin{array}{l} \qquad\qquad\qquad\qquad C(x,u(x))\le0,\qua... ...uad\qquad S(x,y(x))\le0,\quad\mbox{for }x\in\Omega. \end{array}$$

Here $\Omega$ is a bounded, plane domain with piecewise smooth boundary $\Gamma$. $\Omega_0\subset\Omega$ is equal to $\Omega$ unless noted otherwise. $\partial_\nu$ denotes the derivative in the direction of the outward unit normal $\nu$ on $\Gamma$ and the boundary is partitioned as $\Gamma=\Gamma_1\cup\Gamma_2$ with disjoint sets $\Gamma_1,\Gamma_2$ consisting of finitely many connected components. For the general formulation given above in [17,19] necessary optimality conditions are stated, a discretization is described in full detail and the corresponding optimality conditions are related carefully to those for the continuous problem. With concrete applications in mind then, however, the following data of the problem are specialized, the objective function

$$F(y,u)=\frac12\int_{\Omega_0}(y(x)-y_d(x))^2dx+\frac\alpha2\int_{\Gamma}(u(x)-u_d(x))^2dx$$ (2.1)

with $\alpha\ge0$ and the inequality constraints

$$y(x)\le\varphi(x),\quad x\mbox{ in }\Omega,\quad u_1(x)\le u(x)\le u_2(x)\quad\mbox{on }\Gamma.$$ (2.2)

This last set of conditions is also considered as a special case for the distributed control problem in [17,19], the data of which are

$$\begin{eqnarray*} &&\mbox{minimize}\\ &&\qquad F(y,u)=\int_{\Omega}f(x,y(x)),u(x))dx+\int_{\Gamma_1}g(x,y(x))dx \end{eqnarray*}$$

            subject to(ED)

$$\begin{eqnarray*} &&\qquad\quad-\Delta y(x)+d(x,y(x),u(x))=0,\quad x\in\Omega,\\... ...d x\in\Gamma_1,\\ &&\qquad\quad y(x)=y_2(x),\quad x\in\Gamma_2, \end{eqnarray*}$$

and the bounds as in (2.2).

As a special objective function

$$F(y,u)=\frac12\int_{\Omega}(y(x)-y_d(x))^2dx+\frac\alpha2\int_{\Omega}(u(x)-u_d(x))^2dx$$ (2.3)

is considered.