e-Equilibrium in LQ differential games with bounded uncertain disturbances: robustness of standard strategies and new strategies with adaptation

A finite time multi-persons linear-quadratic differential game (LQDG) with bounded disturbances and uncertainties is considered. When players cannot measure these disturbances and uncertainties, the standard feedback Nash strategies are shown to yield to an "-(or quasi) Nash equilibrium depending on an uncertainty upper bound that confirms the robustness property of such standard strategies. In the case of periodic disturbances, another concept, namely adaptive concept, is suggested. It is defined an ‘‘adaptation period’’ where all participants apply the standard feedback Nash strategies with the, so-called, ‘‘shifting signal’’ generated only by a known external exciting signal. Then, during the adaptation, the readjustment (or correction) of the control strategies is realized to estimate the effect of unknown periodic disturbances by the corresponding correction of the shifting vector. After that adaptation period, the complete standard strategies including the recalculated shifting signal are activated allowing the achievement of pure ("1⁄4 0) Nash equilibrium for the rest of the game. A numerical example dealing with a two participants game shows that the cost functional for each player achieves better values when the adaptive approach is applied.