Specified-time coordination control algorithms of multiple harmonic oscillators over directed graphs

This paper investigates distributed specified-time coordination consensus and containment control problems for multiple harmonic oscillators over directed graphs. First, by using Pontryagin's maximum principle, a class of specified-time control algorithms are designed for multiple harmonic oscillators, which solve the consensus problems within a specified settling time if the directed graph of the communication topology has a directed spanning tree. Furthermore, by considering the case with multiple leaders in systems, a specified-time containment control algorithm is then developed for the followers in networks such that their positions and velocities can converge to the convex hulls formed by those of the leaders in a specified settling time over directed graphs. Compared with the existing algorithms, one of main contributions of this paper is that by using the proposed specified-time control algorithms, the consensus and containment control problems are solved over directed graphs and the settling time can be off-line pre-specified according to task requirements. Further, when the frequency of the oscillators is zero, the proposed specified-time control algorithms can also solve the coordination consensus and containment control problems for multiple double-integrator dynamics over directed graphs. Finally, some numerical simulations are provided to demonstrate the effectiveness of the proposed algorithms.