Exponential stabilization of non-stationary systems (Q1902747)

The stabilization of systems of the form \(\dot x= A(t) x+ b(t) u(t)\) is studied by applying a time-varying similarity transformation \(x(t)= L(t) y(t)\). It is shown that if the ratio of the maximum and minimum singular values of \(L(t)\) is bounded, then the stability of \(\dot x= A(t) x\) is equivalent to that of the system \(\dot y= L^{-1} (t) A(t) L(t) y(t)\) (i.e. in a sense, we can neglect the derivative term \(L^{-1} (t) dL/ dt)\). From this, it is shown that by appropriate choice of \(L(t)\), a stabilizing control of the form \(u(t)= s^T (t) L(t) x\) can be derived, where \(s^T (t)y\) stabilizes the `quasi-similar' system in \(y\).