Robust Stackelberg Differential Game With Model Uncertainty

Abstract

This article formalizes two types of modeling uncertainties in a stochastic Stackelberg linear-quadratic (LQ) differential game and then discusses the associated robust Stackelberg strategy design for either the leader or follower. Both uncertainties are primarily motivated by practical applications in engineering and management. The first uncertainty is connected to a disturbance unknown to the follower but known to the leader. A <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">soft-constraint</i> min-max control is applied by the follower to determine the optimal response, and then an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">augmented</i> LQ forward-backward stochastic differential equation control is solved by the leader to ensure a robust strategy design. The second uncertainty involves a disturbance, the realization of which can be completely observed by the follower, but only its distribution can be accessed by the leader. Thus, a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">hard-constraint</i> min-max control on an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">affine-equality-constraint</i> is studied by the leader to address the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">exact-optimal</i> robust design. Moreover, based on a weak convergence technique, a minimizing sequence of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">near-optimal</i> robust designs is constructed, which is more tractable in computation. Some numerical results of the abovementioned robust strategies are also presented.