Open AccessOn Resolution of an Extremum Norm Problem for the Terminal State of a Linear System
Author(s) -
В. А. Срочко,
E. V. Aksenyushkina
Publication year - 2020
Publication title -
izvestiâ irkutskogo gosudarstvennogo universiteta. seriâ "matematika"/izvestiâ irkutskogo gosudarstvennogo universiteta. seria matematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.411
H-Index - 3
eISSN - 2541-8785
pISSN - 1997-7670
DOI - 10.26516/1997-7670.2020.34.3
Subject(s) - mathematics , piecewise linear function , norm (philosophy) , mathematical optimization , optimization problem , finite set , linear system , regular polygon , convex optimization , piecewise , maximization , convex function , mathematical analysis , geometry , political science , law
We study extremum norm problems for the terminal state of a linear dynamical system using methods of parameterization of admissible controls. Piecewise continuous controls are approximated in the class of piecewise linear functions on a uniform grid of nodes of the time interval by linear combinations of special support functions. In this case, the restriction of a control of the original problem to the interval induces the same restrictions for the variables of the finite-dimensional problems. The finite-dimensional version of a minimum norm problem can effectively be resolved with the help of modern convex optimization programs. In the case of two variables, we propose an analytical method of resolution that uses a one-dimensional minimization problem for a parabola over a segment. For a non-convex norm maximization problem, the finite-dimensional version is resolved globally by exhaustive search over the vertices of a hypercube. The proposed approach provides further insights into global resolution of non-convex optimal control problems and is exemplified by some illustrative problems.