Research On Several Problems Of Hybrid Dynamic Game Based On The Viability Theory

Hybrid dynamic systems is a new interdisciplinarity originated in the computer aided control,systems engineering,the autopilot design,manufacturing,ommunication network,traffic management system,industrial process control and hierarchical organization of complex control system.It combines the control engineering,mathematics,systems science and computer science.As an important type of hybrid dynamic systems,the research of dynamic game system has a high value no matter in theory or in practical applications.At present,the dynamic game theory has been widely used in biology,economy,management,and even language research and so on all aspects of life.The research of viability is an important direction in system control,and discriminating domain is an extension of viability in the game theory.This dissertation is devoted to the discriminating domain of the hybrid dynamic game.Based on the viability theory,nonsmooth analysis and reachability theory,the problem of determining the discriminating domain of the hybrid differential game in continuous dynamic systems,discrete dynamic system and hybrid systems,and the algorithm of the discriminating kernel of continuous dynamic systems,discrete dynamic system are studied for some important hybrid dynamic game systems which are the linear systems,the affine systems,and so on.Based on nonsmooth analysis,determining the viability of control systems is discussed.Firstly,determining the viability of hybrid differential inclusion systems on a region which is expressed by piecewise smooth functions is studied.Determining the viability can be transformed into the solving of linear inequality group.Secondly,the viability conditions for hybrid affine control systems in a region defined by inequality constraints is studied.The viability condition that is similar to KKT condition for the continuous affine dynamic systems is discussed.Finally,the results are generalized to the viability of hybrid systems.Aimed at the phenomenon that the discriminating theorem of discriminating domain of the general nonlinear system is difficult to used,determining the bounded discriminating domain for continuous linear dynamic game is studied based on viability theory and nonsmooth analysis.There have two conclusions.The first conclusion is: if a closed set is a discriminating domain of the dynamic game,the convex hull of the set is also a discriminating domain of the dynamic game.The convex hull is a polyhedral discriminating domain.Then,an efficient algorithm isgiven.Finally,using alternative theorem,we can get the victory domain of the dynamic game.The second conclusion is: researching a bounded polyhedron who is a convex hull of finite points for the discriminating domain of linear dynamic game,it just need to test whether the extreme points of the polyhedron meet the viability conditions.Then,using the relationship between viability and discriminating domain,it can determine whether the polyhedron is the discriminating domain of the dynamic game.A bounded discriminating domain for hybrid linear dynamic game with two players and two targets is studied using viability theory.Firstly,if a closed set is a discriminating domain of hybrid linear dynamic game,the convex hull of the set is also a discriminating domain.Secondly,researching whether a bounded polyhedron is a discriminating domain of hybrid linear dynamic game,we just need to test whether the extreme points of the polyhedron meet the viability conditions.Then,using the relationship between viability and discriminating domain,whether the polyhedron is a discriminating domain is determined.The discriminating domain for pursuit-evasion dynamic games in a region defined by inequality constraints which is smooth or non-smooth is studied.Using the converse-negative proposition of Gordan Lemma,we obtain a determining condition of the discriminating domain who is similar to the KKT condition in the optimization.Through the example,we illustrate the effectiveness of the method.In the end,we generalize the result to the nonlinear control system.The determining of discriminating domain for hybrid dynamic game with two targets and two players on a region where the functions are piecewise smooth is discussed.Determining the discriminating domain can be transformed into the solving of inequality group.Then,the result is generalized to hybrid dynamic game.Finally,an algorithm for hybrid discriminating kernel is introduced.The calculation of discriminating kernel for the discrete and continuous dynamic game is discussed using viability kernel and reachable set.For the discrete dynamic game,we give an approximation of the viability kernel by the maximal reachable set.For the continuous dynamic game,we compute an under-approximation of the viability kernel by the backward reachable set from a closed target.Then,based on the relationship between viability kernel and discriminating kernel,we propose an algorithm of discriminating kernel for discrete and continuous system,respectively.Through the examples,we illustrate the correctness and effectiveness of the method.The novelty is that we give two algorithms of discriminating kernel for dynamic game which contain two control variables,not one control variable as in differential inclusion.