Maximum principle for optimal boundary control of vibrating structures with applications to beams
The paper considers the optimal boundary control of a transversely vibrating beam. The state equation is of the form \[ w_{tt}+ L_1\Biggl(x, {\partial\over\partial x}\Biggr)[w_t]+ L_2\Biggl(x, {\partial\over\partial x}\Biggr)[w]= 0,\quad (x,t)\in(0, T)\times (0,1), \] with fourth-order differential operators \(L_1\) and \(L_2\), and with a general linear boundary condition. The necessary optimality conditions in the form of the maximum principle are obtained for a quadratic cost functional. A method of solution of the problem, based on the expansions in series of the eigenfunctions of the operators \(L_2\) (for the case \(L_1\equiv 0\)), is proposed and some numerical results are presented.
scientific article; zbMATH DE number 1236229