Differential calculus for some p-norms of the fundamental matrix with applications

For the fundamental matrix Φ(t)=eA t of a complex n×n matrix A, the differential properties of the mapping t↦∥Φ(t)∥p at every point t=t0∈R0+ ≔ {t∈R | t⩾0} are investigated, where ∥·∥p is the matrix operator norm associated with the vector norm ∥·∥p in Cn or Rn as the case may be, for p∈{1,2,∞}. Moreover, formulae for the first two right derivatives D+k∥Φ(t)∥p, k=1,2, are calculated and applied to determine the best upper bounds on ∥Φ(t)∥p in certain classes of bounds. These results cannot be obtained by the methods used so far. The systematic use of the differential calculus for norms, as done here for the first time, could lead to major advances also in other branches of mathematics and of other sciences, notably in engineering, for example in the simulation of dynamic problems with excitation.