Instability conditions for linear time delay systems: a Lyapunov matrix function approach

The construction of Lyapunov–Krasovskii functionals with prescribed time derivative in the framework of the stability analysis of time delay systems has been addressed in several works, starting with the contributions of Repin (1965) and Datko (1971). In the past decade, this topic has been revisited by Kharitonov and Zhabko (2003), with the so-called functionals of complete type whose time derivative includes the whole state of the system. In contrast with the complete-type functional presented by Huang (1989) and Repin (1965), that were only shown to admit cubic lower bounds (Hale and Verduyn Lunel 1993), the complete-type functionals admit a useful quadratic lower bound (Kharitonov and Zhabko 2003). Using the converse idea of the fact that if a system is exponentially stable, the existence of the functional is guaranteed and it admits a quadratic lower bound, it is possible to provide sufficient instability conditions. Contrasting with the number of papers addressing the stability of functional differential equations, the number of contributions on their instability is, up to the authors knowledge, scarce. The topic has been studied by Hale (1965) and Barnea (1969) who combine the proposal of Lyapunov–Krasovskii functionals tailored to the problem under consideration together with the determination of a particular initial condition that starts a trajectory that results in the violation of the negativity condition of the derivative of the functional. A frequency domain approach based on the analysis of the quasipolynomial of the system in the light of the Theorem of Pontryagin (Pontryagin 1942) is proposed in Buslowicz (1983). The aim of our contribution is to provide instability conditions for retarded, distributed and neutral-type time delay systems based on the violation of the necessary condition given by the lower bound on the functional. For the distributed delay case, we use the functional introduced in Kharitonov (2006). For the delayed and neutral cases, we use the completetype functionals with derivative including special cross terms (Mondié, Kharitonov, and Santos 2005; Ochoa and Mondié 2008) that reduce conservatism. An essential element of our proofs is the connection of the lower bound with the Lyapunov matrix function that determines the functional. This matrix function is the solution of a set of equations called dynamic, symmetric and algebraic equations (Huang 1989, Kharitonov 2006, Ochoa, Velázquez-Velázquez, Kharitonov, and Mondié 2007). It has also been shown to be the solution of a delay-free matrix linear system of differential equations. This key system was introduced in the context of oscillation spectra for single delay systems in Louisell (2001), its theoretical properties were investigated for distributed delay systems in Kharitonov (2006), and for neutral delay systems in Ochoa et al. (2007), respectively. This article is organised as follows: In Section 2, the case of one delay systems traces the road map we will follow. The main results concerning the existence of a functional with prescribed derivative with cross terms are recalled and the quadratic lower bound is obtained. The instability result based on the violation of this lower bound is proved, and corollaries that provide more conservative but easy to check conditions are presented. A parallel development is done for systems