Volume 19, Number 2, April-June 2013
|Page(s)||404 - 437|
|Published online||23 January 2013|
Hamilton–Jacobi equations and two-person zero-sum differential games with unbounded controls∗
Department of Mathematics, Harbin Institute of
264209, Shandong, P.R.
2 Department of Mathematics, University of Central Florida, Orlando, 32816 FL, USA
Revised: 22 September 2011
A two-person zero-sum differential game with unbounded controls is considered. Under proper coercivity conditions, the upper and lower value functions are characterized as the unique viscosity solutions to the corresponding upper and lower Hamilton–Jacobi–Isaacs equations, respectively. Consequently, when the Isaacs’ condition is satisfied, the upper and lower value functions coincide, leading to the existence of the value function of the differential game. Due to the unboundedness of the controls, the corresponding upper and lower Hamiltonians grow super linearly in the gradient of the upper and lower value functions, respectively. A uniqueness theorem of viscosity solution to Hamilton–Jacobi equations involving such kind of Hamiltonian is proved, without relying on the convexity/concavity of the Hamiltonian. Also, it is shown that the assumed coercivity conditions guaranteeing the finiteness of the upper and lower value functions are sharp in some sense.
Mathematics Subject Classification: 49L25 / 49N70 / 91A23
Key words: Two-person zero-sum differential games / unbounded control / Hamilton–Jacobi equation / viscosity solution
© EDP Sciences, SMAI, 2013
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.