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The theory of second-order sufficient optimality conditions (SSC) for the optimal control of semilinear elliptic and parabolic equations is a field of active research. Conditions of this type play an important role in the associated numerical analysis. Their verification is a basic and important issue. Although a numerical confirmation of SSC cannot yet give a definite answer whether they really hold in the infinite-dimensional problem, it provides some evidence about their validity. We refer to recent papers by Mittelmann [7,8], who confirmed that second order sufficient conditions can be checked effectively by numerical techniques. Here, we consider the numerical verification of second order sufficient optimality conditions for the following class of nonlinear optimal control problems of parabolic equations with constraints on the control and the state.

(P)         Minimize

J(y,u) &= &\frac{1}{2} \int \int_Q \alpha... \limits_0^T [ a_y(t) y(l,t) + a_u(t) u(t)]\, dt
\end{array}\end{displaymath} (1.1)

subject to
y_t - y_{xx}&= &e_Q&\mbox{ in }&Q\\
...2(l,t)&=&e_{\Sigma}(t) + u(t)&\mbox{ in }&(0, T)\\
\end{array}\end{displaymath} (1.2)

and to
u_a \le u(t) \le u_b, \, \mbox{ a.e. in } (0,T),
\end{displaymath} (1.3)

\int \int_Q y(x,t) \, dxdt \le 0.
\end{displaymath} (1.4)

In this setting, $T, \, \nu, \, l >0, \, u_a < u_b$ are fixed real numbers, $Q = (0,l) \times (0,T)$. Functions $\alpha, \, y_d,$ and $ e_Q$ are given in $L^{\infty}(Q)$, and $a_y, \, a_u,\, e_{\Sigma}$ are fixed in $L^\infty(0,T)$. We shall denote the set of admissible controls by $U_{ad}= \{ u in L^\infty (0,T), \vert \, u_a \le u \le u_b, \quad \mbox
a.e. in (0,T) \} $. Problem (P) is nonconvex, since the state equation is semilinear. Its nonlinearity $y^2$ is not of monotone type, hence standard results on existence and uniqueness of solutions to (1.2) do not apply. However, in our test example we shall construct a pair $(\bar{y},\bar{u})$ solving (1.2). The linearization of (1.2) at $\bar{y}$ is uniquely solvable for all $u \in U_{ad}$ with continuity of the solution mapping $u \mapsto y(u)$ from $L^p(0,T)$ to $W(0,T) \cap C(\bar{Q})$, $p > 2$, where

W(0,T) = \{ y \in L^2(0,T;H^1(0,l)) \, \vert\, y_t \in L^2(0,T;H^1(0,l)^*) \},

see Raymond and Zidani [11]. Thus the implicit function theorem guarantees existence and uniqueness of the solution $y = y(u)$ of (1.2) in $W(0,T) \cap C(\bar{Q})$ for all $u$ in a sufficiently small $L^p$-neighborhood of $\bar{u}$.

We shall consider a particular example of (P), where SSC are fulfilled, although the second order derivative ${\cal L}''$ of the Lagrange function is not positive definite on the whole space. This is possible, since we consider strongly active control constraints. Therefore, the construction of this example is more involved than the analysis of a similar one presented by Arada, Raymond and Tröltzsch in [1], where ${\cal L}''$ was coercive on the whole space. As a natural consequence, the numerical verification is more difficult. In fact, the example from [1] was verified numerically in [7] for coarser and in [8] for finer discretizations establishing the definiteness of a projected Hessian matrix while even the full matrix has this property. This gave rise to our search for the example presented below. The analysis of SSC for semilinear elliptic and parabolic control problems with pure control constraints is already quite well elaborated. We refer to the referenes in [5], [10]. The more difficult case of pointwise state-constraints is investigated, by Casas, Tröltzsch, and Unger [5], or Raymond and Tröltzsch [10], and in further papers cited therein. However, the discussion of SSC for state constraints is still rather incomplete. Problems with finitely many inequality and equality constraints of functional type are discussed quite completely in a recent paper by Casas and Tröltzsch [4].

next up previous
Next: First and second-order optimality Up: paper94 Previous: paper94
Hans D. Mittelmann