PurposeThis paper aims at applying the proven Adomian and Alienor methods to solve the problem of optimal chemotherapy of HIV model.Design/methodology/approachThe combination of the Adomian decomposition method and the Alienor reduction method allows us to solve the control problem as if it were a classical one‐dimensional minimization problem. The methodology is applied to a HIV model and simulation results given.FindingsA general abstract framework for the control of a non‐linear evolution system has been developed. It was shown that it is possible to control a system by using the powerful techniques of Adomian and Alienor, and produce results comparable with those obtained by classical methods where other cost functions are used.Research limitations/implicationsThe benefit of this work is based on the CD4 healthy cells being maximized and the cost based on a drug dose being minimized. The new methodology could be used after further research for solving many control problems in biology and in other areas such as those involving industrial processes.Practical implicationsNew methodology is cost‐effective in controlling the drug dose affecting rate of infection of cells by HIV virus.Originality/valueNew application of proven methodology will solve many control problems in biocybernetics/biomedicine/industry and other fields.

A set [Formula: see text] of vertices in a graph [Formula: see text] is a [Formula: see text]-dominating set of [Formula: see text] if every vertex of [Formula: see text] is adjacent to at least two vertices in [Formula: see text] The [Formula: see text]-domination number is the minimum cardinality of a [Formula: see text]-dominating set of [Formula: see text]. A graph is [Formula: see text]-domination dot-stable if the contraction of any arbitrary edge has no effect on the [Formula: see text]-domination number. On the other hand, a graph is [Formula: see text]-domination dot-critical if the contraction of any arbitrary edge decreases the [Formula: see text]-domination number. We present some properties of these graphs and we provide a characterization of all trees that are [Formula: see text]-domination dot-stable or [Formula: see text]-domination dot-critical.