Spatial Structure Analysis Of Finite Games And Its Applications In Optimization And Control Of Facility-based Systems

Finite potential games,as a special kind of finite games,have nice properties,among which the existence of Nash equilibrium and reachability of Nash equilibrium under dynamic evolution make them popular.They occupy an important position in game theory,and especially in recent years,as an interface of the utility design and learning design in game theoretic control,play a crucial role in the optimal control of distributed systems.Using the static and dynamic properties of finite potential games and by resorting to the semi-tensor product of matrices,two aspects are studied in this thesis.On the one hand,facility-based systems are equivalently converted into finite potential games,therefore,the optimal control can be realized by only optimizing the potential function.On the other hand,the space of finite games is regarded as the Euclidean space with the same topological structure and the vector space structure,hence,the analysis of finite games can be put into the linear algebra frameworkThis dissertation investigates the space structure of symmetric games,skew-symmetric games and finite games with symmetric potential,and the optimal control of single-objective facility-based systems and multi-criteria facility-based system,as well as numerical solution for continuous potential functions.The main contents of this thesis are listed as follows:1.The distributed optimal control of single-objective facility-based systems is considered.First,The congestion game is converted into a matrix form.Then,a necessary and sufficient condition is given to assure that the system is convertible into a congestion game with the given system performance criterion as its potential function by designing proper facility-cost functions.Meanwhile,the dynamic be-havior of the system is analyzed.Finally,the approach is extended to those systems which are partly or nearly convertible,enhancing the system robustness2.The optimization of multi-criteria facility-based systems(MFBSs)is investi-gated through vector-potential function approach.To begin with,the algebraic con-ditions for the verification of Pareto equilibrium is given according to set relations,and the relationship between Nash equilibrium and Pareto equilibrium is discussed.A MFBS is transformed into a multi-criteria potential game with given system per-formance criteria as its vector-potential function by designing proper facility-cost functions(FCFs).A linear system is presented,which turns the problem of designing suitable FCFs into checking whether a solution of the linear system exists.Then,the approach is extended to constrained MFBSs.Finally,with help of this technology,a dynamic MFBS is considered.The existence and convergence of Nash equilibrium are proved.3.A symmetry-based decomposition of finite games is firstly proposed.First,for skew-symmetric games,by analyzing the characteristics of payoff functions,nec-essary and sufficient conditions are given to verify whether a finite game is skew-symmetric.In addition,it linear representation is presented to reveal the mathemat-ical nature of skew-symmetry.Then,bases of subspaces of symmetric games and skew-symmetric games are constructed respectively,and proved to be orthogonal.Consequently,the space of finite games is decomposed into three mutually orthogo-nal subspaces.4.The structure of finite games with symmetric potential is investigated.First,necessary and sufficient conditions are given to verify the symmetry of the potential function.Then,a basis of symmetric functions is constructed.Based on the basis,a set of linear equations is presented to verify whether a finite game is a potential game with symmetric potential or not.This verification can be solved by checking whether the linear system has a solution.Third,the dimension and a basis of the subspace of finite games with symmetric potential are both provided.Finally,the subspace of symmetric potential games are explored.The dimension and a basis are also obtained.5.Finite element approach to continuous potential games is proposed.First,a linear interpolation function based on the finite potential function of a sub-game is constructed to approximate the continuous potential function.As the number of sample points increases,its convergence is proved.Further,the merge interpolation is proposed to reduce its computational complexity and improve its accuracy when combining more finite potential functions together.Finally,?-potential functions are proposed and constructed for near potential games,as well as the error estimation of?.