Nonzero-sum differential games of continuous-Time nonlinear systems with uniformly ultimately ε-bounded by adaptive dynamic programming

Highlights：

• In this paper, due to the introduction of Osgood condition, a more general condition than Lipschitz condition, the uniqueness of the solution of the system is obtained. So the requirement of the nonlinear term f (x) is greatly relaxed by using Osgood condition, which makes the nonlinear term f (x) is not limited to a function of the first degree, but can be quadratic, cubic, N-degree.

• Compared with the works in [23, 30] and [31], the assumptions that g(x) and k(x) are bounded functions are removed, which can also be reflected in the subsequent simulation.

• In [23], the system state and the weight estimation errors of critic NNs are uniformly ultimately bounded (UUB), where the boundary contains constants and the errors generated by the NNs. However, in our paper, the boundary is only related to the errors generated by the NNs themselves. Further, when the error is small enough or even tend to zero, the system is asymptotically stable.

摘要

•The proposed control scheme can solve the coupled HJ equations online and forward in time without the requirement of initial stabilizing control policies through adding a novel term of only relying on value function V(x) on the weight tuning law of critic NN for each player in this paper.•In this paper, due to the introduction of Osgood condition, a more general condition than Lipschitz condition, the uniqueness of the solution of the system is obtained. So the requirement of the nonlinear term f (x) is greatly relaxed by using Osgood condition, which makes the nonlinear term f (x) is not limited to a function of the first degree, but can be quadratic, cubic, N-degree.•Compared with the works in [23, 30] and [31], the assumptions that g(x) and k(x) are bounded functions are removed, which can also be reflected in the subsequent simulation.•In [23], the system state and the weight estimation errors of critic NNs are uniformly ultimately bounded (UUB), where the boundary contains constants and the errors generated by the NNs. However, in our paper, the boundary is only related to the errors generated by the NNs themselves. Further, when the error is small enough or even tend to zero, the system is asymptotically stable.