Boundary control by an elastic force at one endpoint of the string in the presence of a model nonlocal boundary condition of one of four types,and its optimization

In the present paper, in terms of a generalized solution of the wave equation, we perform an exhaustive study of the problem on the boundary control by an elastic force u _x (0, t) = µ(t) at one endpoint x = 0 of a string in the presence of a model nonlocal boundary condition of one of four types relating (with the sign “+” or “?”) the values of the displacement u(x, t) or its derivative u _x (x, t) at the boundary point x = l of the string to their values at some interior point (mathop xlimits^ circ ) of the string (0 < (mathop xlimits^ circ ) < l). We prove necessary and sufficient conditions for the existence of such boundary controls. Under these conditions, we optimize the controls by minimizing the boundary energy integral and then write out the optimal boundary controls in closed analytic form.