Vibrations control: singular solutions for mean square optimization
We consider a control problem for longitudinal vibrations of a nonhomogeneous bar with clamped ends. The vibrations of the bar are controlled by an external force which is distributed along the bar. For the minimization problem of mean square deviation of the bar we construct optimal solutions in the form of the Fourier series. To find Fourier coefficients we consider an optimal control problem in the space l^2. For the control problem in l^2 we show that
in a certain neighborhood of the origin the structure of the optimal solutions is the following one: for the finite time the optimal nonsingular trajectory enters the singular surface with infinite numbers of control switchings, after that the optimal trajectory remains on the singular surface and attains the origin for the infinite time.