Indefinite Linear Quadratic Optimal Control: Strict Dissipativity and Turnpike Properties

We investigate strict dissipativity and turnpike properties for indefinite discrete-time linear quadratic optimal control problems in the presence of constraints on state and input. Previous results provide a geometric characterization of these properties in case that the stage cost is convex. We generalize these results to indefinite cost functions using two approaches: First, we show that the existing framework can be extended to indefinite state weighting if the stage cost accumulated over multiple consecutive time steps is convex. As a second contribution, we study the strict dissipation inequality by taking the particular shape of the constraints into account. This allows us to state sufficient conditions for strict dissipativity and turnpike properties where the occurrence of turnpikes on the boundary of the constraints is directly related to negative eigenvalues of the cost.