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We address the stabilisation problem of linear time-invariant systems with pointwise and distributed delays. We first introduce the concept of the smoothed spectral abscissa, a robust stability measure previously studied for pointwise delay systems. We show that the new stability measure for distributed systems is smooth with respect to the system parameters, and robust in the sense that it provides a trade-off between the decay rate of the solution and thenorm of the system. Next, we reformulate the stabilisation problem as an unconstrained minimisation problem considering the smoothed spectral abscissa as the objective function,  and show that its properties enable solving the stabilisation problem by using standard gradient-based optimisation techniques. We show the potential of the proposed approach by using it for the stabilisation of a fleet of vehicles, and an oscillator with a class of time-varying delays in the input.

Stabilisation of distributed time-delay systems: a smoothed spectral abscissa optimisation approach