How Violent Are Fast Controls?

Abstract

Consider acontrollable system: x = Ax + Bu, with x(0) = 0. Given any time T > 0 there is then a control operator C_T : EPSILON --> U(DOT) giving the (unique) minimum norm control such that x(T) = EPSILON. We show that ||C_T|| ~ LAMBDA T ^(-(K+1/2)) where LAMBDA is computable from A, B and K is the minimal exponent giving the rank condition.