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This paper studies the controllability of a leaderfollower network of dynamic agents in the presence of communication delays. The network dynamics is governed by nearest neighbor rules over a fixed communication topology. The leader is a particular agent acting as an external input to steer the other member agents. We derive sufficient conditions of the controllability of the dynamic network and give an example to illustrate the main results.

For a Boltzmann-Hamel equation of nonholonomic mechanical system, when it meets certain conditions, the Boltzmann-Hamel equation can be transformed into a Birkhoffian system. By constructing the generating function, the system is investigated numerically using the generalized symplectic geometric algorithm of the nonautonomous Birkhoffian system. Compared with the above-mentioned algorithm with the classical Runge-Kutta method, Birkhoffian symplectic scheme is very accurate in a long-term tracing.

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In this paper, we firstly analyse the position relation on four points from four cases, that is, (i) both x and y are on the outside side of m and n, (ii) either x or y is on the inner side of m and n, (iii) both x and y are on the inner side of m and n, (iv) both x and y are on the same side of m and n. Then we acquire eight kinds of transitivity-like postulates from the four cases. What is more, we discuss the interrelationships between eight transitivity-like postulates and get some important deductibility theorems. Finally, we give a few counter-examples to show that the inverse of the deductibility theorems do not hold and get a summary diagram to show the relationships among these transitivity-like postulates.