1. Introduction
Mathematical models described by ordinary differential equations are considered. The equations are linear with respect to unknown constant parameters. Inaccurate measurements of the basic trajectory of the model are given with known restrictions on admissible small errors.
The history of study of identification problems is rich and wide. See, for example, [1] [2]. Nevertheless, the problems stay to be actual.
In the paper a new approach is suggested to solve them. The identification problems are reduced to auxiliary optimal control problems where unknown parameters take the place of controls. The integral discrepancy cost functionals with a small regularization parameter are implemented. It is obtained that applications of dynamic programming to the optimal control problems provide approximations of the solution of the identification problem.
See [3] [4] to compare different close approaches to the considered problems.
2. Statement
We consider a mathematical model of the form
(1)
where
is the state vector,
,
is the vector of unknown para- meters satisfying the restrictions
(2)
Let the symbol
denote the Euclidean norm of the vector
.
It is assumed that a measurement
of a realized (basic) solution
of Equation (1) is known, and
(3)
We consider the problem assuming that the elements
of the
matrix
are twice continuously differentiable functions in
. The coordinates
of the measurement
are twice continuously differentiable functions in
, too. The coordinates
of the vector- function
are continuous functions on the interval
.
We assume also that the following conditions are satisfied
There exists such constants
and
that for all
the inequalities
(4)
are true.
There exist such constant
(
from
) and such compact set
that for any
the following conditions are held
;
.
Here
.
The identification problem is to create parameters
such, that
(5)
where
is the solution of Equation (1), as ![]()
3. Solution
3.1. An Auxiliary Optimal Control Problem
Let us introduce the following auxiliarly optimal control problem for the system
(6)
where
is a control papameter satisfying the restrictions
(7)
for a large constant
.
Admissible controls are all measurable functions
. For any initial state
, the goal of the optimal control problem is to reach the state
and minimize the integtal discrepancy cost functional
(8)
Here
is the given measurment;
is a small regularization parameter,
is the trajectoty of the system (6), (7) generated under an admissible control
out the initial point
. The sign minus in the integrand allows to get solutions which are stable to perturbations of the input data.
N o t e 1. A solution
of the optimal control problem (6), (7), (8) allows us to construct the averaging value ![]()
(9)
which can be considered as an approximstion of the solution of the identification problem (1), (2).
3.2. Necessary Optimality Conditions: The Hamiltonian
Recall necessary optimality conditions to problem (6), (7), (8) in terms of the hami- ltonian system [5] [6].
It is known that the Hamiltonian
to problem (6), (7), (8) has the form
![]()
where
is an ajoint variable, the symbol
denotes the transpose operation.
It is not difficult to get
![]()
where ![]()
![]()
Here the vector-column
has the form
(10)
3.3. The Hamiltonian System
Necessary optimality conditions can be expressed in the hamiltonian form. An optimal trajectory
generating by an optimal admissible control
in problem (6), (7), (8) have to satisfy the hamiltonian system of differential inclusions
(11)
and the boundary conditions
(12)
where symbols
denote Clarke’s subdifferentials [7] and
,
.
Parameters
belong to the intervals
where values
and
are choosen from the conditions
(13)
We introduce the last important assumption.
There exists a constant
such that restrictions on controls in problem (6), (7), (8) satisfy the relations
(14)
![]()
where
are from (10).
N o t e 2. Using definition (10) one can check that constant K, satisfying assumtion
,
can be taken as
,
where
![]()
Here
are components of matrix
and
are components of matrix
.
If
and
the Hamiltonian has the simple form
![]()
and the differential inclusions (11) transform into the ODEs.
(15)
Let us introduce the discrepancies
, and obtain from (15) the following equations
(16)
and the boundary conditions
(17)
where
saisfy (13).
3.4. Main Result: Dynamic Programming
Using skims of proof for similar results in papers [8] [9] [10] we have provided the following assertion.
Theorem 1 Let assumptions
be satisfied and the concordance of para- meters
:
takes place, then solutions of problem (11), (12), (13)
are extendable and unique on
for any
saisfying (13) and
(18)
It follows from theorem 1, that the average values
(9) obtained with the help of dynamic programmig satisfy the desired relation
(19)
4. Numerical Example
A series of numerical experiments, realizing suggested method, has been carried out. As an example a simple mechanical model has been taken into consideration.
This simplified model describes a vertical rocket launch after engines depletion. The dynamics are described as
(20)
where
is a vertical coordinate of the rocket,
is an unknown windage coefficient and
=9.8 is a free fall acceleration.
A function
is known and satisfies assumption
. This function was obtained by random perturbing of the basic solution
for
=0.3.
The suggested method is applied to solve the identification problem for
= 0.3.
We introduce new variables
and transform Equation (20) into
(21)
where
and
is a fictitious control, which was introduced in order to get
matrix
in (1) satisfying dimentions restriction
.
We put
.
The corresponding hamiltonian system (16) for problem (21),(8) has the form
(22)
with initial conditions
(23)
The solutions were obtained numerically. On the Figure 1 and Figure 2 the graphs
![]()
Figure 1. k(t) graph for δ = 5; k(α, δ) = 0.375.
![]()
Figure 2. k(t) graph for δ = 2; k(α, δ) = 0.325.
of functions
are exposed. The graphs illustrate convergence of the suggested method. The calculated corresponding average values (9) are exposed as well.
Acknowledgements
This work was supported by the Russian Foundation for Basic Research (projects no. 14-01-00168 and 14-01-00486) and by the Ural Branch of the Russian Academy of Sciences (project No. 15-16-1-11).