On Robust Stability and Stabilization of Networked Evolutionary Games with Time Delays
Abstract
:1. Introduction
- Using STP of matrices and dimension augmenting technique, an auxiliary system is constructed to formulate the dynamics of NEGs with time delays and disturbances. The auxiliary system is a linear-like system. It reduces the difficulty of analyzing NEG dynamics with time-varying delays.
- Based on the auxiliary system, an explicit criterion is derived for robust stability. It is presented as a matrix and is easily verified by mathematical software such as Matlab.
- In order to stabilize NEG to the target equilibrium, the robust stability problem is transformed into the robust stabilization problem. Based on the auxiliary system, the necessary and sufficient condition is derived for set stabilization. Moreover, an algorithm is developed to design the set stabilization controller.
2. Preliminaries
- (1)
- is the set of all real matrices
- (2)
- ,
- (3)
- (4)
- () denotes the i-th column (row) of matrix A
- (5)
- (6)
- (7)
- is a logical matrix, which is abbreviated as
- (8)
- represents the set of -dimensional logical matrices
- (9)
- ∘ denotes the Hadamard product of matrices
- (1)
- Define . Then, .
- (2)
- Define and . Then, and
3. Formulation and Robust Stability Analysis of NEGs with Time Delays
3.1. Model Description
- (1)
- The set of players ;
- (2)
- Each player has a strategy set . The strategies of all players constitute a profile, and the set of a profile is denoted by ;
- (3)
- Each player has a payoff function, .
- (1)
- is a network graph with node set and edge set ;
- (2)
- is a fundamental game set, where is an edge-related fundamental game played by players i and j;
- (3)
- is an SUR set, where is the SUR of player ;
- (4)
- is the time-varying delay that occurs when players receive information from others;
- (5)
- is a disturbance set.
3.2. Algebraic Formulation
3.3. Robust Stability Analysis
4. Stabilization Analysis of NEGs with Time Delays and Disturbances
4.1. Model Description
4.2. Stabilization Analysis
Algorithm 1: Find the largest control-invariant subset . |
Step 1: Assume and . Set and Step 2: Let , . Step 3: If , set , stop. Step 4: Compute and set . Step 5: If , let , stop; otherwise, let , return to Step 3 and repeat the calculation. |
- (1)
- is stabilized at the robust-Nash equilibrium under control.
- (2)
- System (19) is set stabilized at .
- (3)
- There exists a positive integer such that
Algorithm 2: Design control matrix such that and . |
Step 1: Let . Assume . Step 2: For , if , set Step 3: Calculate . Set . Step 4: If , there is no . Stop the calculation. Step 5: If , set ; otherwise, set and go back to Step 2. Step 6: Set . is designed as where Stop. |
5. Example
5.1. Model Description
- (1)
- The network graph is shown in Figure 1.
- (2)
- (3)
- Imitating the strategy of the neighbor who has the optimal payoff is the SUR of each player, namely,
- (4)
- .
- (5)
- The external disturbance system is
5.2. Robust Stability Analysis
5.3. Robust Stabilization Analysis
6. Problems
7. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
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An, Q.; Guo, H.; Zheng, Y. On Robust Stability and Stabilization of Networked Evolutionary Games with Time Delays. Mathematics 2022, 10, 2695. https://doi.org/10.3390/math10152695
An Q, Guo H, Zheng Y. On Robust Stability and Stabilization of Networked Evolutionary Games with Time Delays. Mathematics. 2022; 10(15):2695. https://doi.org/10.3390/math10152695
Chicago/Turabian StyleAn, Qiguang, Hongfeng Guo, and Yating Zheng. 2022. "On Robust Stability and Stabilization of Networked Evolutionary Games with Time Delays" Mathematics 10, no. 15: 2695. https://doi.org/10.3390/math10152695
APA StyleAn, Q., Guo, H., & Zheng, Y. (2022). On Robust Stability and Stabilization of Networked Evolutionary Games with Time Delays. Mathematics, 10(15), 2695. https://doi.org/10.3390/math10152695