Available online at www.sciencedirect.com Automatica 39 (2003) 533–541 www.elsevier.com/locate/automatica Brief Paper Multiple-objective risk-sensitive control and its small noise limit  Andrew E.B. Lim a , Xun Yu Zhou b; *; 1 , John B. Moore c a Department of Industrial Engineering and Operations Research, University of California, Berkeley, CA 94720, USA b Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, NT, Hong Kong c Department of Systems Engineering, Australian National University, Canberra, ACT 0200, Australia Received 3 July 2001; received in revised form 6 August 2002; accepted 4 November 2002 Abstract This paper is concerned with a (minimizing) multiple-objective risk-sensitive control problem. Asymptotic analysis leads to the introduction of a new class of two-player, zero-sum, deterministic dierential games. The distinguishing feature of this class of games is that the cost functional is multiple-objective in nature, being composed of the risk-neutral integral costs associated with the original risk-sensitive problem. More precisely, the opposing player in such a game seeks to maximize the most ‘vulnerable’ member of a given set of cost functionals while the original controller seeks to minimize the worst ‘damage’ that the opponent can do over this set. It is then shown that the problem of nding an ecient risk-sensitive controller is equivalent, asymptotically, to solving this dierential game. Surprisingly, this dierential game is proved to be independent of the weights on the dierent objectives in the original multiple-objective risk-sensitive problem. As a by-product, our results generalize the existing results for the single-objective risk-sensitive control problem to a substantially larger class of nonlinear systems, including those with control-dependent diusion terms. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Risk-sensitive control; Multiple-objective optimization; Dierential games; Hamilton–Jacobi–Bellman equations; Upper/lower Isaacs equations; Viscosity solutions 1. Introduction The distinguishing feature of risk-sensitive control prob- lems is that cost functionals involve the expectation of an exponential where the exponent of this exponential is the cost functional of a standard (risk-neutral) stochastic con- trol problem. One consequence of the exponential term is that larger values of the exponent are weighted more heav- ily. For this reason, robust (or risk-averse) controllers can be obtained by minimizing the risk-sensitive cost. Another  This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor R. Srikant under the direction of Editor Tamer Basar. * Corresponding author. Tel.: +852-2609-8320; fax: +852-2603-5505. E-mail addresses: lim@ieor.berkeley.edu (A.E.B. Lim), xyzhou@se.cuhk.edu.hk (X.Y. Zhou), john.moore@anu.edu.au (J.B. Moore). 1 The research of this author was supported by the RGC Earmarked Grants CUHK 4054/98E and CUHK 4234/01E. important property of the risk-sensitive control problem is its relationship with the class of two-player, zero-sum, de- terministic dierential games associated with the so-called H ∞ control problem. In this setting, the controller (for the risk-sensitive problem) takes the part of the minimizing player in the dierential game while the opponent may be interpreted as a worst case disturbance. As a consequence, robustness issues for linear, nonlinear and stochastic systems can be studied in the framework of risk-sensitive control as well as dierential games. For further details of such inter- pretations of the H ∞ control problem, we refer the reader to Ba sar and Bernhard (1995). In this paper, we study (minimizing) risk-sensitive con- trol problems with multiple objectives. Due to the expo- nential form of the individual cost functionals, however, it is dicult to nd an elegant solution to this problem, and we will focus instead on its asymptotic properties (small noise limits). In this regard, we explore the con- nection between multiple-objective risk-sensitive control problems and a new class of deterministic dierential 0005-1098/03/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0005-1098(02)00270-4