V. I. BOLDYREV
Russia, 630090, Novosibirsk,
Ave. of acad. Koptyug, 4,
S.L.Sobolev Institute of Mathematics, Siberian
Branch of the Russian Academy of Sciences,
The paper is aimed to present systematically the author's late
publications on numerical methods of solving optimal control problems.
The basis for the study is the special mapping piecewise linear
approximation method (simplex method) appropriate to the problems
in which necessary conditions are imposed on the control to make
it optimal according to Pontrjagin principle of the maximum.
For linear systems it involves the following. Let it be required to
minimize the pseudoconvex functional defined on the set of final
states of the controlled system. On the strength of Pontrjagin
principle of the maximum an "extremal" relation of boundary
condition of the adjoint system to extreme points of the system
attainability set is established. Let there be (n+1)-th extreme
point of the attainability set. In that case special mapping of this
sort is linearly interpolated at the convex hull of the extreme
points (at the simplex). With the simplex method running, the
sequence of simplices is generated, final state function minimizator,
local relative to the simplex, being found for every simplex.
The simplex method produces strictly decreasing sequence of the
functional values. For nonlinear systems the simplex algorithm
is based on piecewise linear approximation of the function on
triangulation which makes the algorithm suitable for solving boundary
problems.
The paper consists of preface and three sections.
In Section 1 dealing with the problem of minimizing pseudoconvex
function on convex compact set, belonging to the class of problems
of nonlinear programming, its solution (exact) simplex algorithm
(algorithm 1.1) is described.
Algorithm 1.1 generates sequence of simplices. For every of them
the minimum (exact) of objective function is computed, the procedure
being an auxiliary task (Algorithm 1.2). Two versions of approximate
simplex algorithm (Algorithm 1.3) are described in which instead
of seeking local minimum its reasonably rough approximation is
determined. Proving convergence of the simplex algorithm is given.
For the problem of minimizing pseudoconvex functional defined on the
set of final states of linear controlled system an algorithm is
described that is used for computing optimal control in the form of
convex combination of extremal controls in Mayer linear auxiliary tasks.
In Section 2 the problem of minimizing convex functional on the final
state set is solved for linear controlled system under linear terminal
constraints. The simplex algorithm is used to obtain efficient
solutions to auxiliary extremal tasks which present themselves
when the method of modified Lagrangian function and the method of
parametrizing objective function should be applied. The procedure of
forming promptly realizable controls is given by means of which linear
system is moved from an initial point to the final one under additional
constraints or without them. These controls are formed as linear
combinations of the controls computed before the control process.
In Section 3 an algorithm is described that represents combination of
the fixed point method and the quasi-Newtonian method. The algorithm
is used for solving boundary nonlinear problems when Pontrjagin
principle of the maximum is required to be applied. Use of piecewise
linear approximation of function on triangulations enables the
range of the algorithm efficiency to be widened for boundary problems
to be solved.
In each section the data resulted from computing are summarazed in
the tables.