How violent are fast controls? II
[For Part I by the first author see ibid. 1, No. 1, 89-95 (1988; Zbl 0663.49018).] The authors consider a linear time-dependent control system \[ \dot x=Ax+Bu \] and the problem of minimizing the \(L^p\)-norm \(\| u\|_p\) subject to \[ x(0)=0, \qquad x(T)=\xi \] when \(T\rightarrow 0\). The main result is that the asymptotic part of the minimizing norm is \[ (1/T)^{k+1-1/p},\quad p\in [1,+\infty] \] where \(k=\min\{j: \xi\in \text{range} (B+AB+\cdots +A^jB\}\). Finally a representation formula is given for the minimizing control.
scientific article; zbMATH DE number 1018714