Solubility of Matrix Inequalities

An important part is played in the theory of control by special matrix equations [i], and also by the matrix inequalities associated with them. The solutions of these equations or inequalities allow us to construct Lyapunov functions for nonlinear systems of automatic regulation, and to solve problems of the synthesis of optimal control with quadratic quality criterion. Convenient and effectively verifiable conditions for the solubility of Lur'e's inequalities are given by the proposition known as the frequency theorem or the Yakubovich-Kalman Lemma, different variants of whose proof may be found in [2-8]. Beginning with Kalman's work [3], the formulation of the frequency theorem in the often-occurring singular case contains requirements of controllability and observability of the linear part of the system, Although for systems with one nonlinearity these requirements are quite natural, and their verification does not present any special difficulties, for systems with several nonlinearities this verification can be extremely complicated (see Sec. 1.2 of [6] and Sec. 5.6 of [9]). Therefore other mathematicians have tried to completely remove, alter or weaken the requirements of controllability and observability in the formulation of the frequency theorem in the degenerate case. The first and most important result is that of Meyer [i0, ii] (which refers to systems with stable linear part and one nonlinearity). A generalization is introduced (without proof) in [12] of a result of Meyer to the case of several nonlinearities. However an extra restriction on the coefficients of the system is introduced. Another close result connected with the removal of the observ~bility condition was obtained in [13]. In this article we show that the conditions of controllability and observability may in practice be omitted for systems with several nonlinearities and a linear part which is not necessarily stable, i.e. for systems of a substantially wider class than that considered in [10-13].