Known SSC Results for the Continuous Problems

No attempt will be made to quote all relevant results on optimality conditions. In the case of problem (P) the following set of SSC is stated in [1]. First, the Lagrange function

$$\begin{eqnarray*} L(y,u,p)&=&F(y,u)-\int^T_0\int^l_0(y_t-y_{xx})p(x,t)dxdt\\ &&... ...(l,t))p(l,t)dt\\ &&+\int^T_0(b(t)+u(t)-\varphi(y(l,t)))p(l,t)dt \end{eqnarray*}$$

is defined with the Lagrange multiplier function $p$. Then the second derivative of $L$ with respect to $(y,u)$ is called $L''$ and is evaluated at a point $(\bar y,\bar u,\bar p)$ satisfying the first order optimality conditions. The key requirement is the inequality

$$L''(\bar y,\bar u,\bar p)(y,u)^2\ge\Vert u\Vert^2_{L^2(0,T)}$$ (3.1)

which has to hold for all $(y,u)$ which satisfy the linearized (at $(\bar y,\bar u)$) constraints. In the example considered in [1] and below, the inequality even holds for all $(y,u)$. Locally, then, $(\bar y,\bar u)$ is a minimizer of problem (P).

SSC have not been stated exactly for the two elliptic control problems (EB), (ED) in the previous section, but a series of papers address special cases. In [4], for example, a good overview of the literature is given and the boundary control problem covered in much technical detail is nearly identical to (EB) except for

$$d(x,y(x))=y(x),\quad \Gamma_1=\Gamma.$$

On the other hand, a problem of type (ED) but with

$$\begin{eqnarray*} &&d(x,y,u)=u-\varphi(y),\quad\Gamma_2=\Gamma,\quad y_2\equiv0 \end{eqnarray*}$$

and the tracking type objective function (2.3) is extensively analyzed in [2]. Finally, the state-constrained case is addressed in [5] for (EB) and in [20] for (P).