l1-gain analysis and model reduction problem for Boolean control networks

In this paper, the weighted l1-gain analysis and l1 model reduction problem for Boolean control networks are proposed and investigated via semi-tensor product method. First, the input energy and output energy are described by pseudo-Boolean functions, based on which the definition of the weighted l1-gain is introduced. By constructing a co-positive Lyapunov function, a sufficient condition is established to ensure that a Boolean control network is not only internally asymptotically stable, but also has an l1-gain no more than a given scalar. Along this line, by virtue of the properties of semi-tensor product, the l1 model reduction problem of a Boolean control network is defined and converted to the l1-gain problem of another Boolean control network with more nodes. A sufficient condition for the l1 model reduction problem is then derived immediately, and an algorithm is presented to compute the matrices in the reduced order model. Finally, two examples, including the Boolean model for biochemical oscillators in the cell cycle, are displayed to show the feasibility of the theoretical results.