Problems of Information Transmission, Vol. 41, No. 1, 2005, pp. 45–58. Translated from Problemy Peredachi Informatsii, No. 1, 2005, pp. 53–67.
Original Russian Text Copyright
2005 by Tertychnyi-Dauri.
METHODS OF SIGNAL PROCESSING
Solution of Variational Dynamic Problems
under Parametric Uncertainty
V. Yu. Tertychnyi-Dauri
St. Petersburg State University of Information Technologies, Mechanics, and Optics
Received July 26, 2004
Abstract—The paper deals with a number of variational dynamic problems with parameters
subject to unknown smooth drift in time. Solution schemes are considered using both the
classical variational method and reduction of the original problem to a conditional nonholonomic
adaptive optimal control problem. In the second case, a solution is found with the help of
the dynamic programming method and a specially chosen adjustment algorithm for unknown
Variational dynamic problems, notwithstanding their classical ﬂanking and a certain complete-
ness in the framework of the Euler–Lagrange theory, are of intense interest, ﬁrst of all, in respect
of dynamic systems with drifting parameters (see, for instance, [1–3] and references therein), and
in many aspects they remain unexplored and unsolved in adaptive problems for dynamic systems
whose parameters vary in time in an unknown way [4, 5].
From the informational viewpoint, functionals with unknown parameters can be regarded as
functionals with some new (additional) variables, which should be determined. In order to solve
the original optimization problem, it is convenient to introduce control variables into analysis which
are higher-order derivatives of the state of the corresponding dynamic system or, if the case in point
is controlling a concrete object in a phase space, control variables subject to the motion equation
of the object in question.
Apparently, analytical potentialities of the variational and optimization approaches in control
theory, as well as the existence of their interrelation, were ﬁrst studied by Young  in the analysis
of equivalence of two problems: the Lagrange (conditional variational) problem and the optimal
control problem. Note that the authoritative investigation of Young was made in a rather general
form and absolutely did not touch upon the adaptive parametric case. The present paper is an
attempt to ﬁll this gap.
2. REDUCTION TO A CONDITIONAL NONHOLONOMIC PROBLEM
First, we suggest to rewrite the original variational problem with the functional
J = V (x(t
F (x(t), ˙x(t),τ(t),t) dt −→ extr (2.1)
as a terminal control problem with a conditional ﬁdelity functional. Here, V (x(t
)) is a
positive deﬁnite scalar function of the ﬁnite state (position) and system parameters. Note that,
2005 Pleiades Publishing, Inc.