Mathematical Analysis of Monotonic Stability of the Amplitude of Forced Oscillations of a String

The stability of forced oscillations of a finite-length string is considered. The driving force is specified as a known expression containing one harmonic of the time of the string's motion. Monotonic stability of string oscillations is understood as a monotonic decrease in the oscillation amplitude of the modulus of the difference of the solutions describing forced and free oscillations observed at an arbitrary point of the string. In this case, the solutions of the equation of string oscillations in partial derivatives of the second order for free and forced oscillations are assumed to be known. The work aims to analyze three conditions for a monotonic change in the modulus of the difference in the amplitude of forced and free oscillations of a string on a semi-infinite time interval: monotonicity condition, nonlinearity condition, and convergence condition. The analysis of the conditions for monotonic stability of string oscillations is also carried out in the example given in the article.