Nash equilibrium seeking with prescribed performance

In this work, we study a Nash equilibrium (NE) seeking problem for strongly monotone non-cooperative games with prescribed performance. Unlike general NE seeking algorithms, the proposed prescribed-performance NE seeking laws ensure that the convergence error evolves within a predefined region. Thus, the settling time, convergence rate, and maximum overshoot of the algorithm can be guaranteed. First, we develop a second-order Newton-like algorithm that can guarantee prescribed performance and asymptotically converge to the NE of the game. Then, we develop a first-order gradient-based algorithm. To remove some restrictions on this first-order algorithm, we propose two discontinuous dynamical system-based algorithms using tools from non-smooth analysis and adaptive control. We study the special case in optimization problems. Then, we investigate the robustness of the algorithms. It can be proven that the proposed algorithms can guarantee asymptotic convergence to the Nash equilibrium with prescribed performance in the presence of bounded disturbances. Furthermore, we consider a second-order dynamical system solution. The simulation results verify the effectiveness and efficiency of the algorithms, in terms of their convergence rate and disturbance rejection ability.