Over the past few years, convex optimization, and semide?nite programming inparticular, have come to be recognized as a valuable tool for control system analysisand design. A number of important problems in system and control theory can bereformulated as semide?nite programming problems, i.e., minimization of a linear ob-jective subject to linear matrix inequality constraints. From semide?nite programmingduality theory, conditions for infeasibility of the linear matrix inequalitys as well asdual optimization problems can be formulated. These dual problems can in turn bere-interpreted in control or system theoretic terms, often yielding new results or newproofs for existing results from control theory. Moreover, the most e?cient algorithmsfor convex optimization solve the primal and dual problems simultaneously. Insightinto the system-theoretic meaning of the dual problem can therefore be very helpful indeveloping e?cient algorithms. In this paper, we propose the use of complex semide?nite programming, i.e., theextension of semide?nite programming in which one replaces the real symmetric ma-trices by complex Hermitian matrices. In the ?rst chapter, we summarily introducethe development of linear matrix inequality and semide?nite programming as well asthe main achievements of this thesis are summarized. In the second chapter, we ex-tend the results of semide?nite programming to the Hermitian complex form. Boththe weak and strong duality theories corresponding to Lagrange problems and opti-mality conditions are estableshed. Moreover, we propose a primal-dual central pathalgorithm for the solution of large-scale complex semide?nite programming problemsarising in control. Since complex semide?nite programming is reducible to semide?niteprogramming. The polynomial-time solvability of semide?nite programmings impliesthat complex semide?nite programmings are also solvable in polynomial time. In thethird chapter, we present two new proofs for existing results from system and controltheory by means of establishing severl alternative theorems on complex linear matrixinequalitys. Some conclusions and prospects are presented in chapter four.