This work presents new ideas related to the synthesis of nonlinear stabilizing control laws in the framework of optimization approach. The focus is done on the optimal damping concept, proposed by V.I. Zubov in the early 60s of the last century. This theory is used to reduce significant computational costs in solving optimal stabilization problems. The essential features of the aforementioned concept are taken into account, allowing the construction of new methods for practical synthesis of control systems with desired dynamic properties. Various modern aspects of the optimal damping theory's practical implementation are discussed. Special attention is paid to the specific choice of the functional to be damped to provide the desirable stability and performance features of the closed-loop connection.