Atlantis Press Paper Details: Two-Point Boundary Optimization Problem for Bilinear Control Systems

This paper presents a new approach to the optimization problem for the bilinear system x = {x, } (1) based on the well-known method of continuous parametric group reconstruction using of its structure constants defined by the Brockett equation z = {z, }. (2) Here x is the system state vector, {·, ·} are the Lie brackets, z = {x, y}, y is the vector of cojoint variables, = A-1 z is the control vector, A is the inertion matrix. The quadratic control functional has to reach an extremum at the optimal solution of the equation (2) and the boundary optimization problem is to find such z0 that solution (2) makes evolution from the state x(t0) = x0 up to the final state x(t1) = x1 during the time delay T = t1 -t0. Therefore it is necessary to define a transformation group of the state space which is parametrized by components of the vector and then to solve the Cauchy problem for an arbitrary smooth curve joining x(t0) with x(t0). Key words. Bilinear system, Lie group, optimization, boundary problem, structure constants.

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title:		Two-Point Boundary Optimization Problem for Bilinear Control Systems
publication:		JNMP
volume-issue:		4 - 1-2
pages:		209 - 213
ISSN:		1402-9251
DOI:		doi:10.2991/jnmp.1997.4.1-2.33 (how to use a DOI)
author(s):		Alla V. VINOGRADSKAYA
publication date:		May 1997
abstract:		This paper presents a new approach to the optimization problem for the bilinear system x = {x, } (1) based on the well-known method of continuous parametric group reconstruction using of its structure constants defined by the Brockett equation z = {z, }. (2) Here x is the system state vector, {·, ·} are the Lie brackets, z = {x, y}, y is the vector of cojoint variables, = A-1 z is the control vector, A is the inertion matrix. The quadratic control functional has to reach an extremum at the optimal solution of the equation (2) and the boundary optimization problem is to find such z0 that solution (2) makes evolution from the state x(t0) = x0 up to the final state x(t1) = x1 during the time delay T = t1 -t0. Therefore it is necessary to define a transformation group of the state space which is parametrized by components of the vector and then to solve the Cauchy problem for an arbitrary smooth curve joining x(t0) with x(t0). Key words. Bilinear system, Lie group, optimization, boundary problem, structure constants.
copyright:		© Atlantis Press. This article is distributed under the terms of the Creative Commons Attribution License, which permits non-commercial use, distribution and reproduction in any medium, provided the original work is properly cited.
full text:		4-1_2art33.pdf (124 K)