TY - JOUR
T1 - On the stability of the best reply map for noncooperative differential games
AU - Bressan, Alberto
AU - Wang, Zipeng
N1 - Funding Information:
This research was partially supported by NSF through grant DMS-0807420, “New problems in nonlinear control”.
PY - 2012/4
Y1 - 2012/4
N2 - Consider a differential game for two players in infinite time horizon, with exponentially discounted costs. A pair of feedback controls (u * 1(x), u * 2(x))is Nash equilibrium solution if u * 1 is the best strategy for Player 1 in reply to u * 2, and u * 2 is the best strategy for Player 2, in reply to u * 1 . The aim of the present note is to investigate the stability of the best reply map: (u 1, 2→R 1(u 2)R 2(u 1)). For linear-quadratic games, we derive a condition which yields asymptotic stability, within the class of feedbacks which are affine functions of the state x ∈ ℝ n. An example shows that stability is lost, as soon as nonlinear perturbations are considered.
AB - Consider a differential game for two players in infinite time horizon, with exponentially discounted costs. A pair of feedback controls (u * 1(x), u * 2(x))is Nash equilibrium solution if u * 1 is the best strategy for Player 1 in reply to u * 2, and u * 2 is the best strategy for Player 2, in reply to u * 1 . The aim of the present note is to investigate the stability of the best reply map: (u 1, 2→R 1(u 2)R 2(u 1)). For linear-quadratic games, we derive a condition which yields asymptotic stability, within the class of feedbacks which are affine functions of the state x ∈ ℝ n. An example shows that stability is lost, as soon as nonlinear perturbations are considered.
UR - http://www.scopus.com/inward/record.url?scp=84859119527&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84859119527&partnerID=8YFLogxK
U2 - 10.1142/S0219530512500066
DO - 10.1142/S0219530512500066
M3 - Article
AN - SCOPUS:84859119527
SN - 0219-5305
VL - 10
SP - 113
EP - 132
JO - Analysis and Applications
JF - Analysis and Applications
IS - 2
ER -