Control of complex systems

We propose that for many complex systems, e.g. economy, it is more important to control the statistical behaviour than the precise trajectory. Thus we present the beginnings of a theory for control of probability distributions for complex systems. We work in the class of probabilistic cellular automata (PCA), though results generalise easily to continuous-time interacting particle systems and some to coupled map lattices. For weakly dependent PCA, the effect on the statistical behaviour of time-dependent control is unique and we give a formula for it. For strongly dependent PCA, we demonstrate numerically that the effects of small parameter changes and small nudges can be large, leading to tipping from one phase to another. This work was supported by the Alfred P. Sloan Foundation New York.