Abstract. This paper is concerned with the stabilisation of linear time-delay systems by tuning a finite number of parameters. Such problems typically arise in the design of fixed-order controllers. As time-delay systems exhibit an infinite amount of characteristic roots, a full assignment of the spectrum is impossible. However, if the system is stabilisable for the given parameter set, stability can in principle always be achieved through minimising the real part of the rightmost characteristic root, or spectral abscissa, in function of the parameters to be tuned. In general, the spectral abscissa is a nonsmooth and nonconvex function, precluding the use of standard optimisation methods. Instead, we use a recently developed bundle gradient optimisation algorithm which has already been successfully applied to fixed-order controller design problems for systems of ordinary differential equations. In dealing with systems of time-delay type, we extend the use of this algorithm to infinite-dimensional systems. This is realised by combining the optimisation method with advanced numerical algorithms to efficiently and accurately compute the rightmost characteristic roots of such time-delay systems. Furthermore, the optimisation procedure is adapted, enabling it to perform a local stabilisation of a nonlinear time-delay system along a branch of steady state solutions. We illustrate the use of the algorithm by presenting results for some numerical examples.Mathematics Subject Classification. 65Q05, 65K10, 90C26

We aim at the efficient computation of the rightmost, stability-determining characteristic roots of a system of delay differential equations. The approach we use is based on the discretization of the time integration operator by a linear multistep (LMS) method. The size of the resulting algebraic eigenvalue problem is inversely proportional to the steplength. We summarize theoretical results on the location and numerical preservation of roots. Furthermore, we select nonstandard LMS methods, which are better suited for our purpose. We present a new procedure that aims at computing efficiently and accurately all roots in any right half-plane. The performance of the new procedure is demonstrated for small-and large-scale systems of delay differential equations.

This paper highlights the use of advanced numerical tools to study the stability of large-scale systems of delay differential equations (DDEs). Specifically, we consider a model describing a semiconductor laser subject to conventional optical feedback and lateral carrier diffusion. The symmetry of the governing rate equations allows external cavity mode solutions (ECMs) to be computed as steady state solutions. Using the software package DDE-BIFTOOL, branches of ECMs are computed as a function of varying feedback strength. The stability along these branches is computed by solving eigenvalue problems, the size of which is governed by a step-length heuristic. In this paper, we employ an improved heuristic which substantially reduces the size of these eigenvalue problems. This approach makes the stability analysis of large-scale systems of DDEs computationally feasible.

This paper presents a collocation method with an iterative linear system solver to compute periodic solutions of a system of autonomous delay differential equations (DDEs). We show that the linearized collocation system is equivalent to a discretization of the linearized periodic boundary value problem (BVP). This linear BVP is solved using the Newton-Picard single shooting method ([Int. J. Bifurcation Chaos, 7 (1997), pp. 2547-2560]). The Newton-Picard method combines a direct method in the subspace of the weakly stable and unstable modes with an iterative solver in the orthogonal complement. As a side effect, we also obtain good estimates for the dominant Floquet multipliers. We have implemented the method in the DDE-BIFTOOL environment to test our algorithm.

Summary. An overview of eigenvalue based tools for the stability analysis of linear periodic systems with delays is presented. It is assumed that both the system matrices and the delays are periodically varying. First the situation is considered where the time-variation of the periodic terms is fast compared to the system's dynamics. Then averaging techniques are used to relate the stability properties of the time-varying system with these of a time-invariant one, which opens the possibility to use frequency domain tools. As a special characteristic the averaged system exhibits distributed delays if the delays in the original system are time-varying. Both analytic and numerical tools for the stability analysis of the averaged system are discussed. Special attention is paid to the characterization of situations where a variation of a delay has a stabilizing effect. Second, the assumption underlying the averaging approach is dropped. It is described how exact stability information of the original, periodic system can be directly computed. The two approaches are briefly compared with respect to generality, applicability and computational efficiency. Finally the results are illustrated by means of two examples from mechanical engineering. The first example concerns a model of a variable speed rotating cutting tool. Based on the developed theory and using the described computational tools, both a theoretical explanation and a quantitative analysis are provided of the beneficial effect of a variation of the machine speed on enhancing stability properties, which was reported in the literature. The second example concerns the stability analysis of an elastic column, subjected to a periodic force.