What is the finiteness conjecture for linear continuous time inclusions?

We give various continuous time generalizations of the discrete time finiteness "conjecture", formulated as problems about a priori norms. Ultimate answer is given for polytope norms, some low dimensional answers are given for Hubert norms.

1 Main Definitions and Problems

Let us consider a set : . I.e., is a set of complex matrices, is an index set. Recall that a linear discrete inclusion LDI(S) is a set of discrete dynamical systems $$x_{i+1}=A_{R(i)}x_{i};i\geq 0, R:N\rightarrow K.\eqno{\hbox{(1)}}$$