Let \(g_1(t),\dots,g_n(t)\) be continuous functions in the interval \([a,b]\). Given \(s=(s_1,\dots,s_n)\in\mathbb{R}^n\), the moment problem is that of finding a measurable function \(f(t)\) in \([a,a+\theta]\subseteq[a,b]\) such that \[ \int^{a+\theta}_a g_k(t)f(t)dt= s_k,\quad k=1,\dots,n,\quad |f(t)|\leq L. \] The min-moment problem does the above and also requires \([a,a+\theta]\) to be minimal. It is obviously connected with the time optimal control problem for a linear constant coefficient system \(y'(t)= Ay(t)+ Bu(t)\), where \([a,b]= [0,T]\) and \(g_k(t)= e^{\lambda_kt}\); via a change of variables, we may take \(g_k(t)=t^{m_k}\) and, under special conditions on the \(\lambda_k\), we may assume that the \(m_k\) are integers. This is the problem under study, together with its infinite companion, where the sequence \(m_k\) is infinite. The method used involves generating functions and is related to previous work of the authors.