Abstract: The linear-quadratic optimal control by dynamic compensation for rectangular descriptor system is considered in this paper. First, a dynamic compensator with a proper dynamic order is given such that the closed-loop system is regular, impulse-free, and stable (it is called admissible), and its associated matrix inequality and Lyapunov equation have a solution. Also, the quadratic performance index is expressed in a simple form related to the solution and the initial value of the closed-loop system. In order to solve the optimal control problem for the system, the proposed Lyapunov equation is transformed into a bilinear matrix inequality (BMI), and a corresponding path-following algorithm to minimize the quadratic performance index is proposed in which an optimal dynamic compensator can be obtained. Finally, a numerical example is provided to demonstrate the effectiveness and feasibility of the proposed approach.