Farag, M. H. The gradient projection method for solving an optimal control problem. (English) Zbl 0871.49006 Appl. Math. 24, No. 2, 141-147 (1996). The author deals with the optimal control problem \[ \min\limits_{v\in V} f(v),\;f(v)=\int _0^l|u(x,T;v)-g(x)|^2dx+\beta\int_0^T|v_1(t)|^2dt, \]\[ u_t=a^2u_{xx}+B(x,t)u+v_2(x,t),\;(x,t)\in\varOmega=(0,l)\times (0,T], \]\[ u(x,0)=\phi(x),\;u_x(0,t)=0,\;u_x(l,t)=\nu[v_1(t)-u(l,t)], \]\[ \begin{split} V=\{v\mid v=(v_1(t),v_2(x,t));v_1(t)\in L_2(0,T),\;v_{1\text{min}}\leq v_1(t)\leq v_{1\text{max}};\\ v_2(x,t)\in L_2(\varOmega),\int_0^l\int_0^T|\;v_2(x,t)|^2dxdt\leq R^2\}.\end{split} \] He derives the conjugate problem and the gradient of the cost functional \(f\) at \(v\in V\) using the solution of the conjugate problem. Further, the gradient projection method for the constrained problem is derived. The achieved results seem to be out of date. More complicated nonlinear parabolic control problems have been solved in monographs of Barbu, Tiba, Neittaanmäki and Tiba etc. Reviewer: I.Bock (Bratislava) Cited in 1 Document MSC: 49J20 Existence theories for optimal control problems involving partial differential equations 65K10 Numerical optimization and variational techniques 90C52 Methods of reduced gradient type Keywords:optimal control problem; parabolic equation; gradient projection method; distributed parameter system PDFBibTeX XMLCite \textit{M. H. Farag}, Appl. Math. 24, No. 2, 141--147 (1996; Zbl 0871.49006) Full Text: DOI EuDML