Russian Mathematics (Iz. VUZ)
Vol. 48, No. 12, pp.1{14, 2004
Izvestiya VUZ. Matematika
UDC 517.977
PROGRAM AND POSITIONAL SOLUTIONS OF A TERMINAL
LINEAR-CONVEX OPTIMAL CONTROL PROBLEM
R. Gabasov, F.M. Kirillova, and T.G. Khomitskaya
1. Introduction
Linear-convex optimal control (OC) problems represent a direct generalization of linear problems. In applications, they are often formulated as original problems. In a constructive theory,
they may be also used as auxiliary ones necessary for solving nonlinear OC problems 1]. The
qualitative theory of such problems is thoroughly developed 2], as well as the analogous theory of
linear problems. Many algorithms of their numerical solution are proposed. However, the known
algorithms deal mainly with the program optimization of dynamic systems. They are not eective in the sense of the synthesis of the feedback type OC. The only exception is the class of the
Letov{Kalman linear-quadratic problems and the connected problems of the H1 -control theory,
where the OC can be constructed in the form of linear feedbacks. These results attract much attention because of the simplicity of their implementation. The Letov{Kalman problems are smooth
and deal more with the classical variational calculus rather than with the OC theory, because the
control actions are not bounded by direct (geometrical) restrictions there. Such constraints make
extremal problems nonsmooth. They represent the prevalent (nonlinear) component of up-to-date
applied problems. They were the reason that stimulated (in its time) the development of the theory
of variational problems of a nonclassical type, which was called the OC theory. Within fty years,
the OC theory made the noticeable progress. However, its central problem (the synthesis of the
feedback type OC) remains unsolved even for linear systems.
One special case of problems considered in this paper is formed by the linear-quadratic OC
problems. They play an important role in the thriving stabilization theory which is based on the
Model Predictive Control (MPC) methodology 3]. The applied signicance of this theory has
strongly increased 4] due to the consideration of the OC problems with direct constraints imposed
on control actions. The implementation of the MPC methodology depends on the possibility of
the real-time program solution of special optimal control problems. This is the only case when
the control actions can be formed during the concrete stabilization processes (online control). The
known bibliography on MPC contains no references on special fast methods of the program solving
of the corresponding OC problems. For this purpose, the standard quadratic programming methods
are mainly used. In these methods, time discretization periods are large and control time-frames
are short, which, in turn, yield the slow transient processes and limit the MPC ability. In this
paper, we describe the special methods for solving linear-convex OC problems. The eectiveness
of these methods is insensible to the time discretization period. The rst methods of such a type
for linear OC problems were proposed in 5], 6], then they were developed in 7]{9]. The results
The work was supported by the Byelorussian Republican Foundation for Basic Research (project no. F04R002) and the State Program of Basic Research (\Mathematical Structures-16").
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