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The gradient projection method for solving an optimal control problem. (English) Zbl 0871.49006

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 The achieved results seem to be out of date.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.

MSC:

49J20 Existence theories for optimal control problems involving partial differential equations
65K10 Numerical optimization and variational techniques
90C52 Methods of reduced gradient type
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