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We propose a novel approach for the optimization of the H 2 norm for time-delay systems, grounded in its characterization in terms of the delay Lyapunov matrix. We show how the partial derivatives of the delay Lyapunov matrix with respect to system or controller parameters can be semi-analytically computed, by solving a delay Lyapunov equation with inhomogeneous terms. It allows us to obtain the gradient of the H 2 norm and in turn to use it in a gradient based optimization framework. We demonstrate the potential of the approach on two classes of problems, the design of robust controllers and the computation of approximate models of reduced dimension. Thereby a major advantage is the flexibility: in the former class of applications the order or structure of the controller can be prescribed, including recently proposed delaybased controllers. For the latter class of applications, approximate models described by both ordinary and delay differential equations (e.g., inhering the structure of the original system) can be synthesized.

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Necessary conditions for the exponential stability of timedelay systems of neutral type, and a new stability criterion for the scalar case are presented in this contribution. They are obtained in the framework of Lyapunov-Krasovskii functionals of complete type. The particularity of these conditions is that they depend exclusively on the delay Lyapunov matrix.Index Terms-Neutral type time-delay systems, stability analysis, Lyapunov-Krasovskii functionals, delay Lyapunov matrix. I. INTRODUCTIONNeutral type delay systems are characterized by the fact that the state rate of change depends not only on delayed states, but also on past state rates of change. This class of systems has proven to be suitable for modeling electric transmission lines [1], population dynamics [2], controlled systems with delay in the communication channel [3], drilling operation [4] (see [5]-[7] for more examples). The stability theory of neutral type delay systems via Lyapunov-Krasovskii functionals with prescribed derivative was first approached by Castelan and Infante [8]. Functionals of complete type were later developed for the one delay case by Rodriguez et. al [9]and by Kharitonov [10], and extended to the multiple delays scalar case by Velázquez and Kharitonov [11]. The comprehensive framework of these functionals is exposed in the recent monograph by Kharitonov [12]. This approach has been applied to robust stability analysis [9], exponential estimates [10], determination of critical frequencies and parameters [13], and recently, to the predictor control scheme for neutral type state delay systems with input delay [14].An obvious question is raised by the recent necessary stability conditions for retarded delay systems with pointwise delays and with distributed delays presented by Egorov and Mondié [15] and by Cuvas and Mondié [16], respectively: Is it possible to find stability conditions for neutral time-delay systems that depend only on the delay Lyapunov matrix? An affirmative response to this question is given in this paper. For the multivariable case, necessary stability conditions are provided, while for the scalar neutral type equation a new stability criterion similar to the one presented in [17] is proven.These conditions, provide a new approach for studying the stability of neutral time-delay systems, that complement the frequency domain techniques [13], [18]-[20], as well as the classical LMI sufficient conditions obtained from proposed Lyapunov-Krasovskii functionals (see [21]-[23], among others).The result is achieved by using the methodology introduced in [15] and [16] which relies on an appropriate choice for the class of piecewise initial functions depending on the fundamental matrix, and new properties of the Lyapunov matrix. Because of the jump discontinuities in the fundamental matrix, the result concerning the computation of the functional [12], which was originally proved for differentiable initial functions, had to be relaxed. The new formulation is given in Theorem 2, while the lengthy and technical proof is

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