Differential calculus for some \(p\)-norms of the fundamental matrix with applications (Q5949502)

Let \(\Phi(t)=e^{At}\) be the fundamental matrix of a complex \(n \times n\)-matrix \(A\). The author considers the differential properties of the mapping \(t\mapsto\|\Phi(t)\|_p\) on \([0,\infty)\), \(p=1,2,\infty\). Here, \(\|\cdot \|_p\) is the matrix operator norm associated with the vector norm \(\|\cdot \|_p\) in \(\mathbb{C}^n\) or \(\mathbb{R}^n\). In addition, the formulae for the first two right derivatives \(D^k_+\|\Phi (t)\|_p\) on \([0,\infty)\), \(k=1,2\), are calculated. The results are applied to a vibration problem to obtain the best upper bounds on \(\|\Phi(t) \|_p\), \(p=2,\infty\), in certain classes of bounds.