Appl Math Optim 52:1–22 (2005)
2005 Springer Science+Business Media, Inc.
Differential Games of inf-sup Type and Isaacs Equations
and Shuenn-Jyi Sheu
Graduate School of Information Science, Nagoya University,
Furo-cho, Chikusa-ku, Nagoya 464-8601, Japan
Institute of Mathematics, Academia Sinica,
Nankang, Taipei 11529, Taiwan, Republic of China
Abstract. Motivated by the work of Fleming , we provide a general framework
to associate inf-sup type values with the Isaacs equations. We show that upper and
lower bounds for the generators of inf-sup type are upper and lower Hamiltonians,
respectively. In particular, the lower (resp. upper) bound corresponds to the progres-
sive (resp. strictly progressive) strategy. By the Dynamic Programming Principle
and identiﬁcation of the generator, we can prove that the inf-sup type game is char-
acterized as the unique viscosity solution of the Isaacs equation. We also discuss
the Isaacs equation with a Hamiltonian of a convex combination between the lower
and upper Hamiltonians.
Key Words. Differential game, Dynamic Programming Principle, Viscosity solu-
tions, Inﬁnitesimal generators, Strategy classes.
AMS Classiﬁcation. 49L20, 49L25, 49N70.
We consider the state dynamics inﬂuenced by two factors:
(s) = f (s, x(s), a(s), b(s)), t ≤ s ≤ T ,
x(t) = x ∈ R
The second author was supported by Grant NSC 92-2115-M-001-035.