This paper deals with the infinite horizon linear-quadratic problem with indefinite cost. Given a linear system, a quadratic cost functional and a subspace of the state space, we consider the problem of minimizing the cost functional over all inputs for which the state trajectory converges to that subspace. Our results generalize classical results on the zero-endpoint version of the linear-quadratic problem and more recent results on the free-endpoint version of this problem.
Soethoudt, J. M., & Trentelman, H. L. (1989). The regular indefinite linear-quadratic problem with linear endpoint constraints. Systems and Control Letters, 12(1), 23-31. https://doi.org/10.1016/0167-6911(89)90091-1