ISSN 1990-4789, Journal of Applied and Industrial Mathematics, 2018, Vol. 12, No. 2, pp. 201–212.
Pleiades Publishing, Ltd., 2018.
Original Russian Text
V.M. Aleksandrov, 2018, published in Sibirskii Zhurnal Industrial’noi Matematiki, 2018, Vol. XXI, No. 2, pp. 3–16.
Optimal Resource Consumption Control with Interval Restrictions
V. M. Aleksandrov
Sobolev Institute of Mathematics, pr. Akad. Koptyuga 4, Novosibirsk, 630090 Russia
Received February 13, 2017; in ﬁnal form, January 29, 2018
Abstract—Some method is developed for calculating the optimal resource consumption control
with interval restrictions on the components of the control vector. The approach is based on the
sequential adjustment of the values of a quasioptimal control actions up to their limit values. The
connection is found between the deviations of the initial conditions of the adjoint system and the
deviations of the values of the quasioptimal control from the limit values. The rule for specifying the
initial approximation is given, and the speciﬁc features of the rule are noted. An iterative algorithm is
developed, and an example is given.
Keywords: optimal control, resource consumption, interval restrictions, transfer time, switch-
ing times, adjoint system, iterative process, phase trajectory
Development of the theory of optimal control is of a considerable theoretical and practical interest .
Domestic and foreign specialists have considered the various aspects of the theory [2–6] among which
we mention the problem of calculating an optimal control with interval constraints [7–11]. The problems
of optimal control are usually considered under various kinds of restrictions such as inequalities and
equalities, integral constraints, phase constraints, mixed constraints, etc. A new class of constraints on
the control, the interval constraints, is introduced into the theory of optimal control; and, for the ﬁrst
time in the literature, the problem of fast-response optimal control was considered for linear dynamical
systems with interval restrictions on the controls .
In [13–15], the resource consumption optimization problem is under consideration, and some method
is proposed for calculating the resource consumption optimal control with interval constraints on the
components of the control vector based on the transformation of a quasioptimal control. A quasioptimal
control is a sequence of control actions whose values are proportional to the initial conditions. The
quasioptimal control has a few important properties that include: (1) easy implementation, (2) transfer
of the system to the origin of the coordinate system from every initial state belonging to the controllability
domain, as well as (3) independence of the switching times of the initial conditions. The presence of the
modulus of the control actions in the functional makes the resource consumption minimization problem
essentially nonlinear even for linear dynamic systems.
A quasioptimal resource consumption control is formed using a conjugate system; i.e., using the ﬁrst
necessary condition for optimality in the classical variational calculus (the Euler–Lagrange equations).
Therefore, the structure of quasioptimal control coincides with the structure of the sought optimal
resource consumption control with interval constraints. The use of approximate relations leads to
an iterative computational process. A method for specifying the initial approximation is presented.
An iterative algorithm is developed and its special features are considered. The results of simulation
and numerical calculations are presented too.