Second-order analysis for control constrained optimal control problems of semilinear elliptic systems. (English) Zbl 0917.49020

Bonnans, J. F.

Appl. Math. Optimization 38, No. 3, 303-325 (1998).

The system is \[ -\Delta y(x) + \varphi(y(x)) = u(x) \quad (x \in \Omega),\qquad y(x) = 0 \quad (x \in \Gamma), \tag{1} \] where \(\Omega\) is a bounded domain in the \(n\)-dimensional Euclidean space with \(C^{(2)}\) boundary \(\Gamma\) and \(\varphi\) is a \(C^{(2)}\) nondecreasing function. The control \(u(\cdot)\) belongs to \(L^s(\Omega)\) for some \(s \geq 2\). The optimal control problem is that of minimizing a quadratic cost functional \[ J(y, u) = {1 \over 2} \int_{\Omega} (y(x) - \overline y(x))^2 dx + {N \over 2} \int_{\Omega} u(x)^2 dx \] among the solutions \(y_u(x)\) of (1) under a control constraint \(u \in K\). The object is to obtain necessary and sufficient second order optimality conditions, which is done requiring that the feasible set defined by the control constraint be polyhedral. Assuming that this second order condition holds, the author gives a formula for the second order expansion of the value, as well as for the directional derivative of the optimal control when the cost function is perturbed. Finally, the second order theory is extended to vector valued controls.

Reviewer: H.O.Fattorini (Los Angeles)

MSC:

49K20	Optimality conditions for problems involving partial differential equations
49K40	Sensitivity, stability, well-posedness
35J60	Nonlinear elliptic equations
90C31	Sensitivity, stability, parametric optimization

Keywords:

second-order necessary and sufficient optimality conditions; elliptic systems; optimal control