How violent are fast controls?
As is clearly stated in the title, this paper investigates the question of the asymptotic growth of the minimum norm control to reach a fixed target as the horizon is decreased. Consider the linear control system: \[ \dot x=Ax+Bu,\quad x(0)=0,\quad x\in {\mathbb{R}}^ n,\quad u\in {\mathbb{R}}^ m \] where A and B are fixed matrices; the target \(\xi\) is to be reached at time T: \(x(T;u)=\xi\). One seeks the control of minimal \(L^ 2\) norm: \[ \| u\|^ 2=\int^{T}_{0}\| u(t)\|^ 2dt,\quad \| u_{opt}\| =\min \{\| u\| /x(T;u)=\xi \}. \] Let \(C_ T\) be defined by: \[ C_ T:\xi \to u_{opt}(.;T,\xi). \] The system is supposed to be controllable. Let K be the smallest integer such that \(rank[B,AB,...,A^ kB]=n\). Then the main result of the paper is that \(\| C_ T\| \sim \gamma T^{-(k+)}\) as \(T\to 0\).
scientific article; zbMATH DE number 4084442