## Project

# Looking for Stability: Advances on Spectrum-based Stability and Stabilization of Uncertain Linear Time-delay Systems

Time-delay systems model widespread phenomena, ranging from life sciences and economics to natural sciences and engineering. We focus on linear time-delay systems of time-invariant and periodic type, whose stability properties can be inferred from the spectrum of infinite-dimensional operators. Indeed, the asymptotic growth or decay rate of the solutions towards zero can be determined by the spectral radius of the monodromy operator, or the spectral abscissa of the infinitesimal generator associated with the time-invariant time-delay system. The optimization of these stability measures with respect to system parameters permits to stabilize an unstable system and/or to increase the decay rate of the solutions towards zero.

The thesis aims to develop and analyze accurate and reliable numerical methods for the stability assessment and stabilization of linear time-delay systems, validating the efficiency of the novel methodologies on numerical examples and engineering applications. The main contributions follow two research directions. First, we consider linear time-invariant time-delay systems, whose parameters are affected by uncertainties, modeled by a random vector. Second, we present stability assessment and stabilization methods for linear periodic time-delay systems, where the period and delays are commensurable.

The spectral abscissa of a time-invariant time-delay system, whose uncertainties are modeled by a random vector, is a random variable, and admits a polynomial chaos expansion. Other than explaining the parallelism between the polynomial chaos and the polynomial approximation theories, we systematically demonstrate that the lack of smoothness properties of the spectral abscissa heavily affects polynomial approximation methods. The insights on the behavior of the spectral abscissa, which can be generalized to the behavior of the spectral radius of periodic time-delay systems, also play a role in the development of novel stability optimization methods.

The novel stability optimization method, handling the uncertainty, considers as objective function the mean of the spectral abscissa with a variance penalty. Compared to the optimization of the spectral abscissa for the nominal model, the novel approach shows better robustness properties, and, in contrast to worst-case analysis, furnishes more realistic results, exploiting a probabilistic description of the uncertainties.

Moreover, we develop novel stability assessment and stabilization methods for time-delay systems, whose delays and period are commensurate numbers. For these systems, the spectral radius can be not only inferred from the monodromy operator but also from the eigenvalues of a characteristic matrix, whose evaluation involves solving an initial value problem. The exploitation of this characteristic matrix provides three main contributions. Firstly, we propose a novel two-stage approach for the stability assessment, which iteratively refines the accuracy of the spectral radius obtained by the discretization of the monodromy operator. Secondly, we prove a characterization of left eigenvectors of the characteristic matrix in terms of right eigenfunctions of the monodromy operator associated with a dual periodic time-delay system. As a third contribution, we derive from the characteristic matrix a formula to compute the derivatives of the spectral radius with respect to parameters, which is adopted in a spectrum-based stabilization method.