Existence and uniqueness of open-loop Stackelberg equilibria in linear-quadratic differential games

We present existence and uniqueness results for a hierarchical or Stackelbe rg equilibrium in a two-player differential game with open-loop information structure. There is a known convexity condition ensuring the existence of a Stackelberg equilibrium, which was derived by Simaan and Cruz (Ref. 1). T his condition applies to games with a rather nonconflicting structure of th eir cost criteria. By another approach, we obtain here new sufficient exist ence conditions for an open-loop equilibrium in terms of the solvability of a terminal-value problem of two symmetric Riccati differential equations a nd a coupled system of Riccati matrix differential equations, The latter co upled system appears also in the necessary conditions, but contrary to the above as a boundary-value problem. In case that the convexity condition hol ds, both symmetric equations are of standard type and admit globally a posi tive-semidefinite solution. But the conditions apply also to more conflicti ng situations. Then, the corresponding Riccati differential equations may b e of H-infinity -type. We obtain also different uniqueness conditions using a Lyapunov-type approach. The case of time-invariant parameters is discuss ed in more detail and we present a numerical example.