Linear-quadratic differential games: from finite to infinite dimension
The object of this paper is the generalization of the pioneering work of P. Bernhard [J. Optim. Theory Appl. 27 (1979)] on two-person zero-sum games with a quadratic utility function and linear dynamics. It relaxes the semidefinite positivity assumption on the matrices in front of the state in the utility function and introduces affine feedback strategies that are not necessarily $L^2$-integrable in time. It provides a broad conceptual review of recent results in the finite-dimensional case for which a fairly complete theory is now available under most general assumptions. At the same time, we single out finite-dimensional concepts that do not carry over to evolution equations in infinite-dimensional spaces. We give equivalent notions and concepts. One of them is the invariant embedding for almost all initial times. Another one is the structural closed loop saddle point. We give complete classifications in terms of open loop values of the game and compare results.