ИСТИНА

We consider a vibrating string with Dirichlet control action at the booth ends. We investigate the optimal control problem of steering this system from given initial data to rest, in time T, by minimizing an objective functional that is the convex sum of the L2-norm of the control with weight coefficient λ. We provide an explicit solution of this optimal control problem and showing that if the weight of the tracking term is positive, then the optimal control action is concentrated at the beginning and at the end of the time interval, and in-between it decays exponentially. Will be found the explicit form of boundary control which transfers the system from a given initial state to a given final state for a predetermined period of time T. Boundary control provide minimum the following energy functional ∫(1−λ)∥μ′(t)∥^2+λ∥ν′(t)∥^2dt