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.


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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