A necessary and sufficient condition for ℋ- index of linear time-varying systems

The increasing interests in H_- index leads to some recent development in time domain investigation. This paper presents a necessary and sufficient condition of H_- index in a finite time horizon for linear time-varying systems. It shows that the existence of a lower bound of H_- index is equivalent to the existence of a certain backward differential Riccati equation. When applied to a linear time-invariant system with infinite time horizon, the result leads to an identical one given in frequency domain study in terms of algebraic Riccati equation. Detailed discussions of the property and special cases are presented. Finally, an example is given to illustrate the main results.