Optimal singular and chattering modes in the problem of controlling the vibrations of a string with clamped ends

✍ Scribed by L.A. Manita

• Publisher: Elsevier Science
• Year: 2010
• Tongue: English
• Weight: 231 KB
• Volume: 74
• Category: Article
• ISSN: 0021-8928
• DOI: 10.1016/j.jappmathmech.2010.11.011

✦ Synopsis

The problem of minimizing the root mean square deviation of a uniform string with clamped ends from an equilibrium position is investigated. It is assumed that the initial conditions are specified and the ends of the string are clamped. The Fourier method is used, which enables the control problem with a partial differential equation to be reduced to a control problem with a denumerable system of ordinary differential equations. For the optimal control problem in the l 2 space obtained, it is proved that the optimal synthesis contains singular trajectories and chattering trajectories. For the initial problem of the optimal control of the vibrations of a string it is also proved that there is a unique solution for which the optimal control has a denumerable number of switchings in a finite time interval.