LQ-based optimization for linear impulsive control systems mixed with continuous-time controls and fixed-time impulses

Despite significant progress in the optimal theory of impulsive control systems, finding the optimal solution for them still remains a challenging task because of the computational complexity. In this paper, we focus our interests on the LQ-based optimization for a specific class of linear impulsive control systems mixed with continuous-time controls and fixed-time impulses so that the problem-solving ideas can be borrowed from the intensively studied and highly mature linear quadratic optimization theory, and the difficulty encountered in the conventional hybrid optimal control theory is bypassed because the impulsive instants are prescribed a priori. Using the classical Bellman Dynamic Programming, a matrix Riccati hybrid equation for the LQ-based optimization problem is derived and its steady-state solution is analyzed. The hybrid-type Riccati equation is formed by concatenating the matrix Riccati differential equation and the difference counterpart. Furthermore, the time-invariant system only with uniformly timing impulses is considered. In this case, the matrix Riccati hybrid equation is degenerated into a difference one, which is related to the discretized continuous-time dynamics. Finally, a simple regulator problem with impulsive control is given to validate the feasibility of the designed optimal feedback impulse control law.