Now we seek the switching controller for jump system (4.1) so that the closed-loop system could be JISpS in probability. Perform a new transformation as

For simplicity, we just denote , by , , , , where , and the new coordinate is .

According to stochastic differential equation (2.11), one has:

Here we define

From assumption (A2), one gets that there exist nonnegative smooth functions , satisfying

The inequality (5.4) could easily be deduced by using Lemma 4.1.

Now we turn to the martingale process ; according to Lemma 4.2, there exist a function and an -dimensional standard Wiener noise satisfying , where , and is a positive bounded constant. Therefore we have

Remark 5.1.

Differential equation of new coordinate is deduced as above. The martingale process resulting from Markov process is transformed into Wiener noise by using Martingale representation theorem, and it will affect the Lyapunov function construction and affect the remainder of the control design process; for nonjump systems with uncertainty, a quadratic Lyapunov is chosen to meet control performance in most cases [32, 35, 36]. However, for jump systems, this choice will fail because of the existence of martingale process (or Wiener noise). To solve this problem, we suggest using quartic Lyapunov function instead of quadratic one, and this will increase largely the difficulty of design.

Choose the quartic Lyapunov function as

In the view of (5.5) and (5.6), the infinitesimal generator of satisfies

The following inequalities could be deduced by using Young's inequality and norm inequalities with the help of changing the order of summations or exchanging the indices of the summations:

where , and are design parameters.

Based on assumption (A2) and (5.4), we obtain the following inequality by applying Lemma 4.1:

Here , are design parameters.

Submit (5.9) into (5.7), there is

Choose the virtual control signal as

Thus the real control signal is such that

where , , and function is chosen to satisfy .

Theorem 5.2.

If Assumptions (A1) and (A2) hold and a switching control law (5.11) is adopted, the interconnected Markovian jump system (4.1) is JISpS in probability, and all solutions of closed-loop system are ultimately bounded. Furthermore, the system output could be regulated to a small neighborhood of the equilibrium point with preset precision in probability within finite time.

Proof.

From Assumption (A1), the subsystem is JISpS in probability. There exist such that

Considering (5.12), for any given , there is

Notice the fact that stands up as long as and vice versa. Thus we have

In (5.15), appropriate parameter can be chosen to satisfy .

According to Theorem 3.6 and (5.12), with switching controller adopted, the subsystem of jump system (4.1) is JISpS in probability with as system state and as input, which means for any given , there exists function , function , and such that

On the other hand, according to Assumption (A1), there is

Similarly, by choosing parameter , for any given , there exist function , function , and such that

Here parameter can be chosen to satisfy .

By combining (5.16) and (5.18) we choose parameters guaranteeing that

According to stochastic small-gain theorem, for any given , there exists function such that

where , is given as in [32]. From (5.20) it can be seen that all solutions of closed-loop system are ultimately bounded in probability.

According to (5.20) and the property of function, for any given , there exists . If , there is . At the same time by choosing approximate parameter, it can be guaranteed that .

Let , thus we have that for any given , there exists and such that if , the output of jump system satisfies

meanwhile can be made as small as possible by choosing approximate parameters . The proof is completed.

Remark 5.3.

Theorem 5.2 shows that if both subsystem and subsystem are JISpS in probability, the jump system (4.1) is JISpS in probability with appropriate control parameters chosen. Meanwhile the system output can be regulated to a small region in probability with preset precision within finite time. In the following simulation, we will show how different parameters affect the system states and output.