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In this paper we study affine generalized iterated function systems (for short AGIFSs) which are particular cases of the concept of generalized iterated function system introduced by R. Miculescu and A. Mihail. Using a technique introduced by F. Strobin and J. Swaczyna, we associate to each n ∈ N * and each AGIFS F a new AGIFS Fn. Our main result states that the following statements are equivalent: a) F has attractor. b) There exists n ∈ N * such that Fn has attractor. c) There exists n ∈ N * such that Fn is hyperbolic. d) There exists n ∈ N * such that Fn is topologically contractive.

In this paper, we focus on a new class of optimal control problems governed by a simple integral cost functional and isoperimetric-type constraints (constant level sets of some simple integral functionals). By using the notions of a variational differential system and adjoint equation, necessary optimality conditions are established for a feasible solution in the considered optimization problem. More precisely, under simplified hypotheses and using a modified Legendrian duality, we establish a maximum principle for the considered optimization problem.

We provide a new algorithm (called the grid algorithm) designed to generate the image of the attractor of a generalized iterated function system on a finite dimensional space and we compare it with the deterministic algorithm regarding generalized iterated function systems presented by P. Jaros, L. Maślanka and F. Strobin in [Algorithms generating images of attractors of generalized iterated function systems, Numer. Algorithms, 73 (2016), 477-499].