Optimization of a boundary control of an elastic force at one end of a string based on minimization of the integral of the modulus of the elastic force raised to an arbitrary power p≥1

An analytical form of an optimal boundary control at the end of a string by an elastic force that transfers the string oscillation process from an arbitrary preset initial state into an arbitrary preset final state in time T is presented. The method is based on minimizing the integral of the modulus of the elastic force raised to an arbitrary fixed power. To solve this optimization problem, lemma is used, which reduces minimizing the sum of integrals to pointwise minimization of the sums under the integral signs. The pointwise minimization is performed by the method of Lagrange multipliers. It is important to be able to prevent the boundary control from being resonant by taking a sufficiently large time interval T.