Feedback Stabilization for a Class of Nonlinear Stochastic Systems with State-and Control-Dependent Noise

Stability and stabilization are two important topics inmodern control theory, which are first of considered issues in the systems analysis and synthesis. It is well known that stochastic control has become a very popular research area, which has been applied to mathematical finance [1], quantum systems [2], and so forth; stochastic stability and stabilization have been studied by many researchers; we refer the reader to the celebrated book [1] for the discussions of various stabilities. A series of works on robustly exponential stability can be found in [3–6]. While the pth moment stability were discussed in [7, 8], in particular, the asymptotic mean square stability has been studied for a long time; see [9–13]. The stabilizability of linear stochastic control systems has been investigated by [9, 10, 12–17]. In recent years, the study for stabilization of nonlinear stochastic systems has attracted great attention; the methods appearring in studying this topic can be summarized as follows: GARE-based method [9, 12, 18, 19]; control Lyapunov function method [1, 3–6, 20]; passive system method [21], and spectral analysis method based on generalized Lyapunov operators [13, 16, 17].We refer the reader to [19] for the stabilization of general nonlinear stochastic systems, where a class of new Hamilton-Jacobi inequalities were presented. It can be seen that most of the previous works were on the systems with only the state-dependent noise. In the present paper, we deal with a class of linearized systems with both the stateand control-dependent noise. Some sufficient conditions on local state feedback stabilization are given via LMIs and GAREs, respectively, which not only generalize but also improve the results of [18]. We also investigate the global state feedback stabilization and a sufficient condition is also given in terms of LMIs. A numerical example verifies the effectiveness of our results.


Introduction
Stability and stabilization are two important topics in modern control theory, which are first of considered issues in the systems analysis and synthesis.It is well known that stochastic control has become a very popular research area, which has been applied to mathematical finance [1], quantum systems [2], and so forth; stochastic stability and stabilization have been studied by many researchers; we refer the reader to the celebrated book [1] for the discussions of various stabilities.A series of works on robustly exponential stability can be found in [3][4][5][6].While the th moment stability were discussed in [7,8], in particular, the asymptotic mean square stability has been studied for a long time; see [9][10][11][12][13].The stabilizability of linear stochastic control systems has been investigated by [9,10,[12][13][14][15][16][17].In recent years, the study for stabilization of nonlinear stochastic systems has attracted great attention; the methods appearring in studying this topic can be summarized as follows: GARE-based method [9,12,18,19]; control Lyapunov function method [1,[3][4][5][6]20]; passive system method [21], and spectral analysis method based on generalized Lyapunov operators [13,16,17].We refer the reader to [19] for the stabilization of general nonlinear stochastic systems, where a class of new Hamilton-Jacobi inequalities were presented.
It can be seen that most of the previous works were on the systems with only the state-dependent noise.
In the present paper, we deal with a class of linearized systems with both the state-and control-dependent noise.Some sufficient conditions on local state feedback stabilization are given via LMIs and GAREs, respectively, which not only generalize but also improve the results of [18].We also investigate the global state feedback stabilization and a sufficient condition is also given in terms of LMIs.A numerical example verifies the effectiveness of our results.

Main Results.
In this section, we obtain two theorems on locally asymptotic stabilization of (10) as follows.
Theorem 2. Suppose and the following LMI, has a solution  > 0,  ∈ R × ; then the equilibrium point  ≡ 0 of system (10) is locally asymptotically stabilizable in probability with control law The following theorem is another description for locally asymptotic stabilization in probability via GARE.Theorem 3.Under the condition of (11), if for any  > 0,  > 0, GARE, has a positive solution  > 0, then system (10) is locally asymptotically stabilizable in probability with control law To prove our main results, we first consider the linear constant coefficient stochastic control system System ( 16) is said to be asymptotically mean square stable if, for any () = , lim  → ∞ ‖ , ()‖ 2 = 0.
Lemma 4 (see [23]).System (16) is asymptotically mean square stable if and only if the following Lyapunov-type inequality, has at least one solution  > 0.
Lemma 5 (see [13]).System ( 16) is asymptotically mean square stable if and only if its dual system, is asymptotically mean square stable.
for  ̸ = 0. Therefore, the system (10) is locally asymptotically stabilizable in probability with control law The proof of Theorem 2 is completed.Remark 6.If there is a constant matrix  of suitable dimension such that system ( 22) is asymptotically mean square stable, then the following control system, is called stabilizable in mean square sense [9,12,13].
Proof of Theorem 3. Note that if we let then GARE ( 14) can be written as By repeating the proof of Theorem 2, Theorem 3 is easy to be proved.
Remark 8.Although Theorem 2 is equivalent to Theorem 3, it seems that Theorem 2 is more convenient in actual use than Theorem 3, because we can easily test whether or not LMI ( 12) is feasible by existing convex optimization tools; see [10,24].However, we would like to point out that if GARE ( 14) has a positive solution  > 0, by applying Theorem 10 of [9],  must solve the following semidefinite programming problem: subject to The semidefinite programming problem (36)-( 37), as LMI (12), can also be verified via some convex optimization tools [10,24].
Remark 10.It is not convenient to use Theorem 9 in practice, because the condition (39) is difficult to verify for all real nonnegative symmetric matrices.
Remark 11.Checking the proof of Theorem 9 in [18], we can find that Theorem 9 of [18] required that the smallest eigenvalue of  should be larger than zero; that is,  > 0; so ( 1/2 , ) is certainly observable.GARE (41) is a special case of (14).We should point out that (39) and the controllability of (, ) are only sufficient but not necessary conditions for the existence of positive solutions of GARE (41) with  > 0,  > 0; see [25] and the following counterexample.
Example 12.In GARE (41), we set  = 1,  = 1,  = , and In this case, GARE (41) reduces to It is easy to test that is stabilizable in mean square sense.By [9,13], (43) must have a unique positive definite solution  > 0. However, (39) is not satisfied; this can be seen by setting Considering Proposition 7, Theorem 2 not only has computational advantage but also generalizes and improves Theorem 9 given in [18].

Globally Asymptotic Stabilization
has solutions  > 0,  ∈ R × ; then the equilibrium point  ≡ 0 of system (10) is globally asymptotically stabilizable with the control law Repeating the same procedure as in Theorem 2, we can prove L() < 0 for all  ∈ R  .The theorem is shown.
Remark 15.Obviously, (11) and (46) do not imply each other, which motivates us to search for other less conservative conditions in the future.

Numerical Example
In this section, we present the following numerical example to illustrate the effectiveness of our main results.
Example 1.Consider the following two-dimensional nonlinear stochastic system: with 0 (, ) = 2,  1 (, ) = −.Obviously,  0 (, ) and  1 (, ) satisfy condition (11).According to Theorem 2, a feasible solution is derived by solving LMI ( 12 The state responses of the unforced system ( = 0) and the controlled system ( = ) are shown in Figures 1  and 2, respectively.From Figure 2, it can be found that the controlled system can achieve stability by using the proposed controller.

Conclusion
In this paper, we have studied the feedback stabilizability of nonlinear stochastic systems with state-and controldependent noise.Some sufficient conditions on stabilization have been derived in terms of LMIs and GAREs.A numerical example is presented to show the validity of the obtained results.T r a c eo fas q u a r em a t r i x  0 2 ({ > 0} × ): Class of functions (, ) twice continuously differential with respect to  ∈  and once continuously differential with respect to  > 0 except possibly at the point  = 0.

Figure 2 :
Figure 2: The state responses of the controlled system ( = ).