The descriptor systems have been attracting the attention of many researchers over recent decades due to their capacity to preserve the structure of physical systems and to describe static constraints and impulsive behaviors. Within the descriptor framework, the contributions of this dissertation are threefold: i) review of existing results for state-space systems, ii) generalization of classical results to descriptor systems, iii) exact and analytical solutions to non standard control problems. A realization independent Kalman-Yakubovich-Popov (KYP) lemma and dilated LMI characterizations are deduced for descriptor systems. The solvability and corresponding numerical algorithms of generalized Sylvester equations and generalized algebraic Riccati equations (GARE) associated with descriptor systems are provided. In addition, the simultaneous H∞ control problem is considered through extending recently reported results. A sufficient condition is proposed through a combination of a generalized algebraic Riccati equation and a set of LMIs. Moreover, the nonstandard H2 and H∞ control problems with unstable and/or nonproper weighting functions or subject to regulation constraints are addressed. These contributions allow, without approximation or transformation, dealing with many practical problems defined within H2 or H∞ control methodologies, where the control signals are penalized at high frequency or unstable internal models specified by external signals is involved.
