Controlling invariant density: an l∞ solution to the inverse Frobenius-Perron problem

In a previous paper we gave a new formalism to solve the inverse Frobenius-Perron problem (IFPP), to produce a dynamical system, uniformly nearby a given dynamical system, but with a drastically different and desirable invariant density. Our previous algorithm reduced the problem of producing the dynamical system G with desirable statistics β(x), into a constrained optimization problem, which we solved in l². However, we pointed out that if this l² solution does not correspond to a useful solution, one could not conclude nonexistence. The l^∞ solution to the same optimization problem allows for a sharp existence-nonexistence theorem. In this paper, we present for the first time an l^∞ algorithm which produces solutions to our IFPP, and conclude our nonexistence theorem which is pertinent to this solution. Then we discuss applications in control of chaos, both by open-loop control strategies for maps, and we discuss future applications to feedback control of flows