Linear quadratic optimal control for systems with continuous and impulsive inputs

Linear systems described by hybrid dynamics are considered; that is systems described by linear time-varying models with continuous control inputs as well as “jump” dynamics at discrete time instants where impulsive control inputs are applied. A quadratic performance index is formulated and the necessary conditions for optimality are determined using variational calculus. The solution to the control problem is obtained using two types of Riccati equations: the usual continuous-time Riccati differential equation and a discrete Riccati equation which yields discontinuous jumps in the continuous-time solution. Necessary and sufficient conditions for optimal timing of the impulsive inputs are also presented. Numerical examples are provided, which demonstrate the correctness of the solutions.