First order necessary conditions

Let the control $\bar{u}$ be locally optimal for (P) with associated state $\bar{y}$, i.e.

$$J(y,u) \ge J(\bar{y},\bar{u})$$ (2.1)

holds for all $(y,u)$ satisfying the constraints (1.2-1.4), where $u$ belongs to a sufficiently small $L^\infty$-neighborhood of $\bar{u}$. Suppose further that $(\bar{y},\bar{u})$ is regular. Then there exist Lagrange multipliers $\bar{p}\in W(0,T) \cap C(\bar{Q})$ (the adjoint state) and $\bar{\lambda} \ge 0$ such that the adjoint equation

$$\begin{array}{rclll} -\bar{p}_t - \bar{p}_{xx}&= &\alpha(\ba... ...l,t)\, \bar{p}(l,t)&=&a_y(t)&\mbox{ in }&(0, T),\\ \end{array}$$ (2.2)

the variational inequality

$$\int \limits_0^T (\nu \bar{u}(t) + \bar{p}(l,t) + a_u(t))(u(t) - \bar{u}(t))\, dt \ge 0 \quad \forall u \in U_{ad},$$ (2.3)

and the complementary slackness condition

$$\bar{\lambda} \int \int_Q \bar{y}(x,t)\, dxdt = 0$$ (2.4)

are fulfilled, see [3] or [11]. We mention that (2.3) is equivalent to the well-known projection property

$$\bar{u}(t) = \Pi_{[u_a,u_b]}\{ -\frac{1}{\nu}(\bar{p}(l,t) + a_u(t))\},$$ (2.5)

where $\Pi_{[u_a,u_b]}: \mbox{\piz R}\rightarrow [u_a,u_b]$ denotes projection onto $[u_a,u_b]$. Moreover, we recall that these conditions can be derived by variational principles applied to the Lagrange function ${\cal L}$,

$$\begin{array}{rcl} {\cal L}(y,u,p,\lambda) &=&J(y,u) - \int ... ... + y^2(l,t)- u(t) - e_{\Sigma }(t) ) p(l,t) \, dt . \end{array}$$

Defining ${\cal L}$ in this way, we tacitly assume that the homogeneous initial and boundary conditions of $y$ are formally included in the state space. The conditions (2.2-2.3) follow from ${\cal L}_y(\bar{y},\bar{u},\bar{p},\bar{\lambda}) y = 0$ for all admissible increments $y$ and ${\cal L}_u(\bar{y},\bar{u},\bar{p},\bar{\lambda}) (u -\bar{u}) \ge 0 \quad \forall u \in U_{ad}$. Let $\tau > 0$ be given. We define

$$\begin{array}{rcl} A^+(\tau) &=& \{t \in (0,T) \, \vert \, \... ...\nu \bar{u}(t) + \bar{p}(l,t) + a_u(t) \ge \tau \}. \end{array}$$

It holds $\bar{u}= u_b$ on $A^+$ and $\bar{u}= u_a$ on $A^-$. These sets indicate strongly active control constraints.