The small parameter method in the optimisation of a quasi-linear dynamical system problem

We consider an optimisation problem for the transient process in a quasi-linear dynamical system (contains a small parameter at non-linearities) with a performance index that is a linear combination of energy costs and the duration of the process. An algorithm for constructing asymptotic approximations of a given order to the solution of this problem is proposed. The algorithm is based on the asymptotic decomposition by integer powers of a small parameter of the initial values of adjoint variables and the duration of the process that are finite-dimensional elements, according to which the solution of the problem is easily restored. The computational procedure of the algorithm includes solving the problem of optimising the transient process in a linear dynamical system, integrating systems of linear differential equations, and finding the roots of non-degenerate linear algebraic systems. We also show how the constructed asymptotic approximations can be used to construct optimal control in the problem under consideration for a given value of a small parameter.