OL-NE for LQ differential games: a Port-Controlled Hamiltonian system perspective and some computational strategies

Linear Quadratic differential games and their Open-Loop Nash Equilibrium (OL-NE) strategies are studied with a threefold objective. First, it is shown that the state/costate lifted system (arising from the application of Pontryagin's Minimum Principle) is such that its behaviour restricted to the equilibrium subspace can be interpreted as the (non-power-preserving) interconnection of two cyclo-passive Port-Controlled Hamiltonian systems. Such PCH systems constitute the best response generators for each player, thus mimicking and extending the corresponding interpretation of (single-player) optimal control problems. Second, by realizing that the behaviour of the lifted dynamics off the equilibrium subspace is “irrelevant” for generating the equilibrium strategies, it is shown that such an invariant subspace can be rendered, via a suitably constructed virtual input, externally asymptotically stable while preserving the OL-NE. Finally, based on these premises we provide a closed-form gradient-descent method to solve the asymmetric coupled Riccati equations characterizing the OL-NE strategies.