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 L²-integrable in time. It provides a broad conceptual review of recent results in the finite-dimensional case for which...