On approximate solution of mobile (scanning) control problems

We describe an approximate technique for solving the so– called mobile (scanning) control problems. The method is based on the Bubnov–Galerkin procedure and allows us to reduce the control problem, in which the unknown function is included in nonlinear manner, to a finite–dimensional system of integral constraints of equality type. An efficient numerical scheme is described reducing the solution of the nonlinear system to a problem of nonlinear programming. The proposed method is described thoroughly for nonlinear equations with linear boundary conditions. Two particular problems of heating by a moving source and vibration damping by a moving absorber are considered. The system of necessary and sufficient conditions for controllability are obtained in both cases. Main points of numerical implementations are discussed.