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In this paper we study first order hyperbolic systems with multiplecharacteristics (weakly hyperbolic) and time-dependent analytic coefficients.The main question is when the Cauchy problem for such systems is well-posed in$C^{\infty}$ and in $\mathcal{D}'$. We prove that the analyticity of thecoefficients combined with suitable hypotheses on the eigenvalues guarantee the$C^\infty$ well-posedness of the corresponding Cauchy problem. This result isan extension to systems of the analogous results for scalar equations recentlyobtained by Jannelli and Taglialatela in \cite{JT} and by the authors in\cite{GR:13}.

On $C^\infty$ well-posedness of hyperbolic systems with multiplicities