Asymptotic properties of an infinite horizon partial cheap control problem for linear systems with known disturbances

Valery Y. Glizer; Oleg Kelis

doi:10.3934/naco.2018013

Article Contents

2018, Volume 8, Issue 2: 211-235. Doi: 10.3934/naco.2018013

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Asymptotic properties of an infinite horizon partial cheap control problem for linear systems with known disturbances

Valery Y. Glizer^1, , and
Oleg Kelis^1,2,

1.
Department of Applied Mathematics, ORT Braude College of Engineering, 51 Snunit Str., P.O.B. 78, Karmiel 2161002, Israel
2.
Department of Mathematics, University of Haifa, Mount Carmel, Haifa 3498838, Israel

* Corresponding author
* Corresponding author

Received: February 28, 2017

Revised: November 30, 2017

Published: May 2018

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Abstract

An infinite horizon quadratic control of a linear system with known disturbance is considered. The feature of the problem is that the cost of some (but in general not all) control coordinates in the cost functional is much smaller than the costs of the other control coordinates and the state cost. Using the control optimality conditions, the solution of this problem is reduced to solution of a hybrid set of three equations, perturbed by a small parameter. One of these equations is a matrix algebraic Riccati equation, while two others are vector and scalar differential equations subject to terminal conditions at infinity. For this set of the equations, a zero-order asymptotic solution is constructed and justified. Using this asymptotic solution, a relation between solutions of the original problem and the problem, obtained from the original one by replacing the small control cost with zero, is established. Based on this relation, the best achievable performance in the original problem is derived. Illustrative examples are presented.

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Mathematics Subject Classification: Primary: 49N10, 93C70; Secondary: 93C73.

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Table 1. Optimal value of the cost functional (105)

$\varepsilon $	0.4	0.2	0.1	0.05	0.025	0.0125	0.00625
${\mathcal J}_{\varepsilon}^{*}$	10.8004	9.2653	8.6776	8.4775	8.4076	8.3809	8.3697

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