Input-to-state stability in the meaning of switching for delayed feedback switched stochastic financial system

Financial system is essentially chaotic and unstable if there is not any external inputs. By means of Lyapunov function method, design of switching law, novel fuzzy assumption, Lp estimation technique and Laplace semigroup theory, the author presents the boundedness and LMI-based (globally) asymptotical input-to-state stability criteria of financial systems. Particularly, the globally asymptotical stability in the meaning of switching implies that when the time t is big enough, the dynamic of any subsystem must approach its unique equilibrium point. Besides, the global financial crisis often erupts periodically, which illuminates that the global stability in the classical sense is actually meaningless. So the stability in the meaning of switching proposed in this paper is suitable and appropriate. Numerical examples illuminate the effectiveness of the obtained results.

the symmetric matrix (B − A) is a positive definite matrix.

System description
The following financial system has been investigated in many existing literature( [1][2][3][4][5][6][7][8][9]), (2.1) where x represents the interest rate, y represents the investment demand, z represents the price index, a represents savings, b represents the unit investment cost, and c represents the elasticity of commodity demand. It is well known that if c − b − abc 0, the financial system (2.1) has the unique equilibrium point P 2 (0, 1 b , 0); if c − b − abc > 0, the financial system (2.1) owns three equilibrium point P 1 Chaos appears in the financial system (2.1) if c−b−abc > 0 and c+a− 1 b < 0. For example, let a = 0.9, b = 0.2, c = 1.2, then there is a chaos phenomenon in the financial system (2.1) (see, e.g. [2,3]). Throughout this paper, we assume 1 X 3 = − X 1 − cX 3 , (2.11) orẊ (t) = H 3 X(t) + f (X(t)), (2.12) where In the real financial market, the economic system is a dynamic market with randomness. Economic activities do not necessarily pursue a fixed interest rate. The null solution of the following switched systeṁ X(t) = H σ X(t) + f (X(t)), σ ∈ T, where T = {1}, or {1, 2}, or {1, 2, 3} (2.14) corresponds to a series of equilibrium points P σ with respective interest rates. The switched system (2.14) means that the financial system is adjusted macroscopically, or the market adjusts itself when the economic situation changes to a critical value. For example, when the economic crisis breaks out, in order to promote economic development, the interest rate is adjusted to zero or even negative. Thereby, the global asymptotic stability of the switched system (2.14) reflects the activity of real financial market and the stability.
In financial management, the Uncertainty of parameters sampling conforms to the fuzzy model to a great extent.
So we consider the following T-S fuzzy rule for the financial system (2.14).

Remark 1.
Since the important economic parameters, such as savings, the unit investment cost, and the elasticity of commodity demand, usually come from some small sample statistics, we hope that the financial system can be stabilized in a certain range of parameter ambiguity.
Since some stochastic disturbance factors in the real financial market, such as the input and outflow of foreign capital, these additional funds will produce immediate and delayed stochastic disturbance to some important financial parameters, such as savings, the unit investment cost, and the elasticity of commodity demand. So, in this paper, we have to consider the following delayed feedback stochastic financial system: where v is the external input, time delay τ(t) satisfies τ(t) ∈ [0, τ], and X(t −τ(t)) is the delayed feedback state variable.
The noise perturbation φ : )), and w(t) = (w 1 (t), w 2 (t), w 3 (t)) T ∈ R 3 is a 3-dimensional Brownian motion defined on a complete probability space (Υ, F , P) with a natural filtration {F t } t 0 generated by {w(s) : 0 s t}, where we associate Υ with the canonical space generated by w(t), and denoted by F the associated σ-algebra generated by w(t) with the probability measure P, and E{dw(t)} = 0. Besides, If the fuzzy factors are ignored, the system (2.17) is degenerated into the following system: Remark 2. The globally asymptotical stability in the meaning of switching means that when the time t > 0 is big enough, and the system is switched into any ith subsystem, the dynamic of the subsystem must approach its equilibrium point P i . For example, the financial crisis breaks out, the financial system is switched into a subsystem, say, the 2th subsystem whose equilibrium point is P 2 , and the dynamic of the subsystem must approach P 2 if the time t > 0 is big enough. Moreover, if the stability is local, then the dynamic of the subsystem may not approach P 2 due to various different initial values. Besides, the global financial crisis often erupts periodically, which illuminates that the global stability in the classical sense is actually meaningless. So the stability in the meaning of switching proposed in this paper is suitable and appropriate.
In this paper, we assume (A1) There are constant symmetric matrices Λ σ1 and Λ σ2 with Λ σ2 < α σ I such that where I represents the identity matrix.

Main result in the case of non-fuzzy factors
Define the switching law for the switched system (2.18) as follows, switching law F a : At each switching we determine the next mode according to the following minimum law : On the other hand, if σ(t − ) = i but ξ Υ i . i.e., hitting a switching surface, choose the next mode by applying (3.1) and begin to switch.
(A3) Since X 1 represents the interest rate which is actually bounded in real world, there exists the positive number (A4) There are constant matrices D σ and D σ such that the external input satisfies α, and there is a positive definite diagonal matrix P ρI with P = diag(p 1 , p 2 , p 3 ) with p 1 = p 2 , a positive definite matrix Q, and a sequence of then the zero solution of the system (2.18) is the global asymptotical input-to-state stability. That is, the switched system (2.18) is said to be the global asymptotical input-to-state stability in the meaning of switching.
Proof. Consider the following Lyapunov-Krasovskii function The condition (A3) and Let R + be the set of positive numbers. Denote by C 1,2 (R + × R n → R + ) the family of nonnegative functions V(t, X) on R + × R n which are continuous once differentiable in t and twice differentiable in X. For each such V, one can define an operator LV associated with equations (2.18) as .
In fact, it follows by (3.2) and the proof by contradiction that ∪ Besides, (3.6) So it follows by (3.6) that which implies that Moreover, it is obvious that Combining (3.8) and (3.9) results in that the zero solution of the system (2.18) is the global asymptotical input-tostate stability. That is, the switched system (2.18) is said to be the global asymptotical input-to-state stability in the meaning of switching.
Remark 4. The global stability in the meaning of switching has practical significance. In fact, the global financial crisis often erupts periodically, which illuminates that the global stability in the classical sense is actually meaningless.

Main result in the case of fuzzy system
Define the switching law as follows, switching law F: At each switching we determine the next mode according to the following minimum law : , hitting a switching surface, choose the next mode by applying (4.1) and begin to switch.
Here, Ψ is a positive definite symmetric matrix with λ min Ψ = λ > 0, and Υ σ is defined as follows, where λ min Ψ represents the minimum of all the eigenvalues of the symmetric matrix Ψ > 0.
then the fuzzy switched system (2.17) is exponential input-to-state stability.
Proof. Consider the following Lyapunov function: Let R + be the set of positive numbers. Denote by C 1,2 (R + × R n → R + ) the family of nonnegative functions V(t, X) on R + × R n which are continuous once differentiable in t and twice differentiable in X. For each such V, we define an operator LV associated with equations (2.17) as .
Let γ be a positive number with 0 < γ < min{λ, ς, 1 2 λ 2 } and where λ 2 > 0 is the first positive eigenvalue of the following Neumann boundary problem and Ω is a bounded domain in R 3 with smooth boundary ∂Ω.
It is obvious that U is continuous for t −τ. For t 0 and γ > 0, ) .

Ito formula yields
and hence ) .
Integrating the above inequality from t to t + ε, and taking the mathematical expectation, one may derive that, for all t 0 and any ε > 0, ) ds and then Now we claim that there is a positive constants C 0 > 1 and K ∈ R with K > 1 such that Indeed, suppose this assertion is not true, then it is not difficult to prove (see, e.g. [7]) that there exists positive constant q with K q > 1 such that and It follows from (4.6), (4.7) and the definition of U(t, X(t)) that for s ∈ [−τ, 0] and t ∈ [t * * , t * ] which yields that for any s ∈ [−τ, 0], Combining (4.9) and (A1) results in which derives For any given t t 0 , according to the switching law F, when σ(t − ) = i and X(t) ∈ Υ i , then keep σ(t) = i, and we can conclude that −λE[X T X] = −λEV(t, X(t)).

(4.12)
When σ(t − ) = i and X(t) Υ i , which means that the trajectory hits a switching surface. On the other hand, it is not difficult to deduce from (4.2) that , which together with the minimum law (4.1) yields (4.12), too.
Remark 5. It is the first paper to design and investigate the stability of switched fuzzy delayed feedback financial system with stochastic perturbation, which is in line with the fact that a prosperous and stable financial market may not necessarily pursue a equilibrium point with a fixed interest rate. In most cases, financial market is more stable through switching law under macro control and market self-regulation.
Remark 6. Although our ordinary differential equations model successfully explains the financial market stability criterion under switching law, state variables are still related to regions, such as certain countries and regions. So we consider the partial differential equations model for the financial system (2.17).

Remark 7.
Considering that the interest rate, the investment demand and the price index are usually invariable in a country, we see, the state variable u(t, X) = (u 1 (t, X), u 2 (t, X), u 3 (t, X)) T is actually uniform with respect to the region, which implies that the system (4.14) is exactly the system (2.17). That is, u i (t, X) of the system (4.14) is exactly equivalent to X i (t) in the system (2.17). Lemma 2.3] in the case of p = 2, we know that the fuzzy system (4.14) is exponentially stable under the same assumptions of Theorem 4.1.
To obtain the boundedness criterion of the system (4.14) or (2.17), we need the following assumption on the Laplace operator.
Below, we assume that the condition (A3) holds. Besides, due to the meaning of Remark 7, we propose the boundedness definition on the system (2.17) or (4.14).
Definition 2. The fuzzy system (2.17) is said to be bounded under the meaning of L ∞ if for any given T > τ 1 > 0 such that for all t ∈ [τ 1 , T ], the each state variable u i (t, X) of the system (4.14) satisfies In order to verify the correctness of Theorem 4.2, we may firstly present the following technical lemma with regard to the system (4.14). Proof. For the convenience of the proof, we may rewrite the system (4.14) as follows, where H σr = (h σri j ) 3×3 .
It follows from the system (4.15) that for any given τ 1 > 0, which implies that for any given τ 1 > 0, (4.16) We can see it from the proof of Theorem 4.1 that then we know from Remark 8 that where 2γ < λ 2 , and C > 0 is a constant which is big enough.
Since C is big enough, we can make a series of estimates with Lemma 1.1, where δ ∈ ( 1 3 , 2 3 ), and λ 2 is defined in (4.3).
Similarly, we can get by (4.17) Since the constant C is big enough, the condition (A1) yields Moreover, since the constant C is big enough, we can estimate  And then the proof is completed.
Now we can know from Remark 7 that Theorem 4.2 has been proved. In the case σ = 1,

Conclusions
Although the non-Lipschitz functions make the free weight matrix technique unsuitable for financial systems, in this paper, LMI-based criterion of globally asymptotical stability in the meaning of switching for chaos delayed feedback switched stochastic financial system was obtained, and the globally asymptotical stability in the meaning of switching is more suitable for the financial system, which implies that when the time t is big enough, the dynamic of any subsystem must approach its unique equilibrium point. Besides, the global financial crisis often erupts periodically, which illuminates that the global stability in the classical sense is actually meaningless. In addition, boundedness result and the exponential stability criterion was also derived for fuzzy financial system in the meaning of switching.
Numerical examples shows the effectiveness of the proposed methods.

Funding Statement
The work is supported by the Application basic research project of science and Technology Department of Sichuan Province (No. 2020YJ0434) and the Major scientific research projects of Chengdu Normal University in 2019 (No.