Geodesic
On resolution of an extremum norm problem for the terminal state of a linear system
The Bulletin of Irkutsk State University. Series Mathematics, Tome 34 (2020), pp. 3-17

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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.
Keywords: linear control system, extremum norm problems for the terminal state, piecewise linear approximation, finite-dimensional problems. 
@article{IIGUM_2020_34_a0,
     author = {V. A. Srochko and E. V. Aksenyushkina},
     title = {On resolution of an extremum norm problem for the terminal state of a linear system},
     journal = {The Bulletin of Irkutsk State University. Series Mathematics},
     pages = {3--17},
     publisher = {mathdoc},
     volume = {34},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IIGUM_2020_34_a0/}
}
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V. A. Srochko; E. V. Aksenyushkina. On resolution of an extremum norm problem for the terminal state of a linear system. The Bulletin of Irkutsk State University. Series Mathematics, Tome 34 (2020), pp. 3-17. http://geodesic.mathdoc.fr/item/IIGUM_2020_34_a0/