Strategic control of a tracking system

We consider stochastic dynamic game problems where a trajectory controller takes an action to construct an information bearing signal, namely the control input, and subsequently a tracking system takes an action, i.e., constructs a tracking output, based on the control input. The trajectory controller has access to two Gaussian processes evolving according to first-order autoregressive models, e.g., desired and private states. Different from the design of a measurement or sensing scheme for a tracking system, here the trajectory controller and the tracker have different objectives. Particularly, the trajectory controller aims to drive the tracking system to a desired path, different from the tracker's actual intent, by constructing the measurement signal. For finite horizon problems involving two different quadratic cost functions, we show that the optimal control input policies are linear functions of the current states when the states evolve in parallel. We then extend this result for the general case when the trajectory controller has a myopic objective and show that the optimal control input policies are also linear functions of the current states. Finally, we restrict the policy space for the control input to the set of all linear mappings of the current states and convert the finite horizon stochastic game problem into a discrete time deterministic optimal control problem. We also include some illustrative numerical examples for different strategic control scenarios.