Constant upper bounds on the matrix exponential norm

Abstract This work is devoted to the constant (time-independent) upper bounds on the function ∥ exp(tA)∥2 where t ⩾ 0 and A is a square matrix whose eigenvalues have negative real parts. Along with some constant upper bounds obtained from known time-dependent exponential upper bounds based on the solutions of Lyapunov equations, a new constant upper bound is proposed that has significant advantages. A detailed comparison of all these constant upper bounds is carried out using 2 × 2 matrices and matrices of medium size from the well-known NEP collection.