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Numerical approximation of the optimal inputs for an identification problem. (English) Zbl 0915.65069

An optimal control problem for a parabolic equation of the following form is considered: \[ J_\alpha(u)= {1\over 2}\cdot\| \lambda(u)- \overline\lambda\|^2_{L^2(0,T)}+{\alpha\over 2}\cdot\| u\|^2_{L^2(0,T)}\to \min, \] where \(\lambda(u)\) is an evolutive parameter defined by \[ \lambda(u)(t)= u\cdot(B^*Y(u))(t)\quad\text{on }[0,T] \] with \(Y: u\to y= Y(u)\) determined by the parabolic equation: \[ y_t= d\cdot\Delta y+ (Bu)(1- y)\quad\text{on}\quad Q_T:= \Omega\times (0,T), \]
\[ y(.,0)= y_0\quad\text{on\;}\Omega,\quad y= 0\quad\text{on }\Sigma:= \Gamma\times (0,T). \] This problem is related to an identification problem. The author gives an approximation scheme and an algorithm. Some numerical results are presented.

MSC:

65K10 Numerical optimization and variational techniques
49M25 Discrete approximations in optimal control
49J20 Existence theories for optimal control problems involving partial differential equations
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References:

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