tailieunhanh - Optimal Control with Engineering Applications Episode 12

Tham khảo tài liệu 'optimal control with engineering applications episode 12', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 4 Differential Games A differential game problem is a generalized optimal control problem which involves two players rather than only one. One player chooses the control u t G Q c Rmu and tries to minimize his cost functional while the other player chooses the control v t G Qv c Rmv and tries to maximize her cost functional. A differential game problem is called a zero-sum differential game if the two cost functionals are identical. The most intriguing differential games are pursuit-evasion games such as the homicidal chauffeur game which has been stated as Problem 12 in Chapter 1 on p. 15. For its solution consult 21 and 28 . This introduction into differential games is very short. Its raison d etre here lies in the interesting connections between differential games and the H-x_ theory of robust linear control. In most cases solving a differential game problem is mathematically quite tricky. The notable exception is the LQ differential game which is solved in Chapter . Its connections to the H-x_ control problem are analyzed in Chapter . For more detailed expositions of these connections see 4 and 17 . The reader who is interested in more fascinating differential game problems should consult the seminal works 21 and 9 as well as the very complete treatise 5 . Theory Conceptually extending the optimal control theory to the differential game theory is straightforward and does not offer any surprises initially In Pontryagin s Minimum Principle the Hamiltonian function has to be globally minimized with respect to the control u. In the corresponding Nash-Pontryagin Minimax Principle the Hamiltonian function must simultaneously be globally minimized with respect to u and globally maximized with respect to v. 104 4 Differential Games The difficulty is In a general problem statement the Hamiltonian function will not have such a minimax solution. Pictorially speaking the chance that a differential game problem with a quite general formulation has a solution is .

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