Optimal control for a vibrating string with variable axial load and damping gain

We consider an optimal control problem for a vibrating string with multiplicative distributive control. Multiplicative control is the coefficient u(x, t) of velocity term yt. Using multiplicative control, we bring the state solution to the desired profile under a quadratic cost of control.We prove the existence of optimal control. Criterion for characterization of the unique optimal control is provided in the form of optimality system.