Numerical solution of minimum norm problems of distributed control of vibrations (Q917386)

The distributed control of a vibration process, described by an abstract wave equation for a Hilbert-space-valued function is considered. It is searched for a control function guaranteeing minimal possible time \(T=T(M)\), \(\| f(t)\|_ H\leq M\), for weak solution of the equation y(t), \(y(T)=\dot y(T)=0\), whose norm \(\| f\|_{\infty,T}= \sup_{t\in [0,T]}\| f(t)\|_ H\) is minimal. The problem of determining \(f_ T: \| f_ T\|_{\infty,T}=\min \| f\|_{\infty,T}\) is called the problem of minimum norm control and the full problem is called time-minimal null-controllability. The dual problem is formulated and solved approximately. The results are applied to the vibrating strings and beams.