The paper studies a linear continuous-time descriptor system over an infinite time horizon under disturbances of bounded energy, i.e., bounded \(L_2\) norm. The concept of a generalized \(\mathcal {H}_2\) norm is introduced as the norm of the linear operator generated by this system. It is shown that the generalized \(\mathcal {H}_2\) norm can be computed via solutions to projected generalized continuous-time algebraic Lyapunov equation, and the synthesis of optimal controls, which include multi-objective cases, minimizing the generalized \(\mathcal {H}_2\) norm of multiple outputs is formulated in terms of linear matrix inequalities. As applications, multi-objective problems of vibration isolation and oscillation damping are considered.