Attractors of control systems. (Q1432195)

Let \({\mathcal M}\) be a metric compact space with a metric \(d(\cdot, \cdot)\), and let \(f_i:{\mathcal M}\to {\mathcal M}\), \(i=1,2,\dots,k\) be contraction maps. On \({\mathcal M}\), we consider the discrete dynamical system \[ x_{n+1}= f_{m(n)}(x_n), \quad n=0,1,\dots, \tag{1} \] where \[ m(\cdot):\mathbb{N} \cup\{0\} \to\{1,2, \dots,k\} \tag{2} \] is a function. In what follows, the function \(m(\cdot)\) involved in the definition of the discrete system (1) is treated as an open-loop control that, together with a chosen initial condition \(x_0\), generates the trajectory \(\{x_n\}\) of system (1). The set of all controls \(m(\cdot)\) defined by (2) is denoted by \({\mathfrak M}\). Here, we examine the qualitative behavior of trajectories of all discrete systems (1) rather than particular problems arising in the theory of discrete-system control. It turns out that each control \(m(\cdot) \in{\mathfrak M}\) is associated with a compact attractor \({\mathcal A}\subset {\mathcal M}\) of the corresponding discrete system, i.e., with a set to which all trajectories of this system approach. This attractor does not depend on the choice of \(m(\cdot)\) and is common for all systems (1). Moreover, for almost all controls \(m(\cdot)\in {\mathfrak M}\), the trajectories of the corresponding systems (1) are dense in \({\mathcal A}\).