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An identification problem is considered as inaccurate measurements of dynamics on a time interval are given. The model has the form of ordinary differential equations which are linear with respect to unknown parameters. A new approach is presented to solve the identification problem in the framework of the optimal control theory. A numerical algorithm based on the dynamic programming method is suggested to identify the unknown parameters. Results of simulations are exposed.

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, [

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 [

We consider a mathematical model of the form

where

Let the symbol

It is assumed that a measurement

We consider the problem assuming that the elements

We assume also that the following conditions are satisfied

are true.

Here

The identification problem is to create parameters

where

Let us introduce the following auxiliarly optimal control problem for the system

where

for a large constant

Admissible controls are all measurable functions

Here

N o t e 1. A solution

which can be considered as an approximstion of the solution of the identification problem (1), (2).

Recall necessary optimality conditions to problem (6), (7), (8) in terms of the hami- ltonian system [

It is known that the Hamiltonian

where

It is not difficult to get

where

Here the vector-column

Necessary optimality conditions can be expressed in the hamiltonian form. An optimal trajectory

and the boundary conditions

where symbols

Parameters

We introduce the last important assumption.

where

N o t e 2. Using definition (10) one can check that constant K, satisfying assumtion

where

Here

If

and the differential inclusions (11) transform into the ODEs.

Let us introduce the discrepancies

and the boundary conditions

where

Using skims of proof for similar results in papers [

Theorem 1 Let assumptions

It follows from theorem 1, that the average values

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

where

A function

The suggested method is applied to solve the identification problem for

We introduce new variables

where

We put

The corresponding hamiltonian system (16) for problem (21),(8) has the form

with initial conditions

The solutions were obtained numerically. On the

of functions

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).

Subbotina, N.N. and Krupennikov, E.A. (2016) Dynamic Program- ming to Identification Problems. World Jour- nal of Engineering and Technology, 4, 228-234. http://dx.doi.org/10.4236/wjet.2016.43D028