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

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.

### MSC:

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