# Finite-Time \(L_1\) Control for Positive Markovian Jump Systems with Partly Known Transition Rates

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## Abstract

The paper deals with the problem of finite-time \(L_1\) control for positive Markovian jump systems with partly known transition rates. Firstly, by constructing a linear co-positive Lyapunov function, sufficient conditions for finite-time boundedness and \(L_1\) finite-time boundedness of the open-loop system are developed. Then, an effective method is proposed for the construction of the state feedback controller. These sufficient criteria are derived in the form of linear programming. A key point of this paper is to extend the special requirement of completely known transition rates to more general form that covers completely known and completely unknown transition rates as two special cases. Finally, two examples are given, which include a mathematical model of virus mutation treatment to illustrate the validity of the obtained results.

## Keywords

Positive Markovian jump systems Partly known transition rates Finite-time boundedness Linear programming## Notes

### Acknowledgments

This work is supported by the Key Program of the National Natural Science Foundation of China under Grant No. 61433004.

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