Coefficient control problems for high-order elliptic equations (Q1281229)

The author considers the problem of minimizing the functional \[ F(u)=\int_{\Omega} f ( x, u, \Delta u) dx, \] taking into account solutions of the family of boundary value problems \[ \Delta [a (x, k(x)),u(x), \Delta u(x)]=g(x),\quad x\in \Omega,\qquad u|_{\partial \Omega}=\Delta u|_{\partial \Omega}=0, \] where \(\Omega \) is a bounded region of \(\mathbb{R}^{m}\) , \( m\geq 1\), \(\partial \Omega \) is a smooth boundary of \(\Omega\), \( a , g , f\) are given functions and \( k(x)\) is a control coefficient. A method of optimal sampling of the control coefficient of some class of nonlinear partial differential equations of higher order is presented. The main goal is to set a connection between a state equation and the functional to be minimized so that the optimal problem has a unique solution on the given set. The existence of the optimal control coefficient in the class of differential equations is proved. The algorithm of solving the optimal problem is proposed. It is noted that in particular cases it is possible to find an exact solution.