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18-01-2022

Min-Max Robust Control in LQ-Differential Games

Published in: Dynamic Games and Applications

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Abstract

In this paper, we consider the design of equilibrium linear feedback control policies in an uncertain process (e.g., an economy) affected by either one or more players. We consider a process which nominal (commonly believed) development in time is described by a linear system. Assuming every player is risk averse and has his own expectation about a worst-case development of the nominal process we model this problem using a linear quadratic differential game framework. Conditions under which equilibrium policies exist are studied. Assuming players have an infinite planning horizon, we provide a complete description in case the system is scalar, whereas for the multi-variable case, we provide existence results for some important classes of systems.

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Footnotes
1

This should be interpreted in the usual (almost everywhere, a.e.) sense.

 
2

See, e.g., [9][Proposition 7.21] where this zero-sum solution is derived under some additional stabilization constraint. Following the lines of the above reasoning, it is, however, clear that this additional constraint is in fact void here. Furthermore, it follows from, e.g., [19][Corollary 9.1.6] that the associated Riccati equation (7.4.19), (15) here has an appropriate solution provided (17) has a solution.

 
3

\(\mathbf {N}=\{1,\ldots ,N\}.\)

 
4

At some places also extended algebraic Riccati equations occurred that also appear in the study of absolute stability and \(H^{\infty }\) control problems, which solvability is, e.g., studied in [2].

 
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Metadata
Title
Min-Max Robust Control in LQ-Differential Games
Publication date
18-01-2022
Published in
Dynamic Games and Applications
Print ISSN: 2153-0785
Electronic ISSN: 2153-0793
DOI
https://doi.org/10.1007/s13235-021-00421-z

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