This paper considers a linear continuous-discrete time-varying system described by a set of differential and difference equations on a finite horizon. For such a hybrid system, the concept of the generalized \({{\mathcal{H}}}_{2}\) norm is introduced, representing the induced norm of a linear operator generated by the system under consideration. This norm is characterized in terms of Lyapunov difference equations and also in terms of recursive linear matrix inequalities. Discrete time-varying optimal controllers, including multiobjective ones, that minimize the generalized \({{\mathcal{H}}}_{2}\) norm of the closed loop system are designed.

本文考虑了由有限域上的一组微分和差分方程描述的线性连续离散时变系统。对于这种混合系统，引入了广义\（{{{mathcal {H}}} _ {2} \）范数的概念，表示考虑中的系统生成的线性算子的诱导范数。该范数以Lyapunov差分方程为特征，还以递归线性矩阵不等式为特征。设计了离散时变最优控制器，包括多目标控制器，该控制器最小化了闭环系统的广义\（{{\\ mathcal {H}}} _ {2} \）范数。