Second order sufficient optimality conditions

Let $(\bar{y},\bar{u},\bar{p},\bar{\lambda})$ be given such that the system of first order necessary conditions is satisfied, i.e. the relations (1.2-1.4), (2.2-2.4) and $\bar{\lambda} \ge 0$ are fulfilled. Now we state second order conditions, which imply local optimality of $\bar{u}$. For this purpose, we need the second order derivative of ${\cal L}$ with respect to $(y,u)$,

$${\cal L}''(\bar{y},\bar{u},\bar{p},\bar{\lambda}) [y,u]^2= \... ..._0^T u^2\, dt + 2 \int \limits_0^T \bar{p}(l,t) y^2(l,t)\, dt.$$ (2.6)

Let us assume as in the example below that the state-constraint (1.4) is active at $\bar{y}$ and $\bar \lambda = 1$. Then we require the following second-order sufficient optimality condition:

(SSC)         There exist positive $\delta$ and $\tau$ such that

$${\cal L}''(\bar{y},\bar{u},\bar{p},\bar{\lambda}) [y,u]^2 \ge \delta\, \int \limits_0^T u^2\, dt$$ (2.7)

holds for all $y \in W(0,T), \, u \in L^2(0,T)$ such that

$$\begin{array}{rcl} y_t - y_{xx}&= &0\\ y(x,0)&=&0\\ y_x(0,t)&=&0\\ y_x(l,t)+2\bar{y}(l,t)y(l,t)&=&u(t) \end{array}$$ (2.8)

and

$$u(t) = 0 \mbox{ on } A^+(\tau) \cup A^-(\tau)$$ (2.9)

$$u(t) \ge 0 \mbox{ if } \bar{u}(t) = u_a \mbox{ but } t \notin A^-(\tau)$$ (2.10)

$$u(t) \le 0 \mbox{ if } \bar{u}(t) = u_b \mbox{ but } t \notin A^+(\tau)$$ (2.11)

$$\qquad \int \int_Q y(x,t)\, dxdt = 0.$$ (2.12)

It is known that (SSC) implies local optimality of $\bar{u}$ in a neighborhood of $L^\infty(0,T)$, see [4]. In our example, we shall verify a slightly stronger condition. We require (2.7) for all $(y,u)$, which satisfy only (2.8-2.9).