Abstract: In the circumstance of developing information technology, various and complicated information are ubiquitous at any time on the social networks, of which the negative messages are also increasingly growing at a fast rate, having a serious impact on the network safety and social stability. As a result, it is pressing to design an effective model to control the spread of the negative information. Time lag is a common thing in nature. In the process of spreading public opinions on the social network, some users may give a response after consideration. Among the existing studies, most do not take these thinkers’ time lag into account. Because of this, an optimal control model of public sentiment spread is proposed, with time lag considered. First of all, a model of public sentiment spread, which involves thinkers, has been built based on the classical epidemic model. A systematic dynamics analysis is made in terms of steady state and local stability. At the same time, Hopf bifurcation under the extreme threshold is discussed. Secondly, sensitivity analysis is conducted for the parameters in the model. With Pontryagin maximum principle and the attack mechanism of opposing public opinion introduced, the optimal control model concerning public sentiment spread is designed according to the optimizing control theory, aiming to minimize the density of people following the negative public sentiment and the controlling cost. The experiment indicates that this model can effectively control the spread of public sentiments in social networks. The system is in a stable state when thinkers need more time to give a response as well as when the attack of opposing public opinions spread at the fastest pace via the media and experts. Besides, the optimal control model is able to let the attack information of opposite public opinion spread effectively in a short time with the minimum cost when there is an event of public opinion.

Key words: social network; spread control of public opinion; thinking time-lag; Hopf bifurcation; optimal control theory