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Primal-dual active set strategy for a general class of constrained optimal control problems. (English) Zbl 1028.49027

The authors investigate an optimal control problem \[ \text{min} F(y,u)=\frac 12|y-y_d|^2_Y+\frac \mu 2\|u\|^2_U \] subject to \(\;y=Su+f\) and \(u\in U_{ad}=\{u\in U=L^2(\Sigma):\;a\leq u(x)\leq b\;\text{a.e. in }\Sigma\},\) where \(Y\) is a Hilbert space, \(S:U\to Y\) is a linear and compact operator, \(f,\;y_d \in Y,\;\mu>0,\;-\infty\leq a<b\leq \infty\) and at least \(a\) or \(b\) is finite. Using the generalized Moreau-Yosida approximation the above problem is regularized. The optimality conditions have the form \(y^*=Su^*+f,\;p^*=S^*(y_d-y^*),\;p^*=\mu^*+\lambda^*,\;\lambda^*=c\left(u^*+\frac 1c \lambda^*-P\left(u^*+\frac 1c \lambda^*\right)\right)\) for each \(c>0\), where \(P:\mathbb R\to \mathbb R\) is defined by \[ \;P(r)=\begin{cases} r\;\text{if} a\leq r \leq b,\\ a\;\text{if} r<a,\\ b\;\text{if} r>b.\end{cases} \] A primal-dual active set algorithm is presented and its convergence analysis is performed. The efficiency of the algorithm is demonstrated for a constrained parabolic control problem of the type \[ \text{min} \tfrac 12\|y(T,.)-y_d\|^2_{L^2(\Omega)}+\tfrac \mu 2\|u\|_{L^2(\Sigma)} \] subject to \(\;y_t=\Delta_x y,\;\text{in} Q, \;\frac{\partial y}{\partial n}=u\;\text{on} \Sigma,\;y(0,.)=0\in \Omega,\;a\leq u\leq b,\;Q=(0,T)\times \Omega,\;\Sigma =(0,T)\times \partial \Omega.\)

MSC:

49M29 Numerical methods involving duality
49J20 Existence theories for optimal control problems involving partial differential equations
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