Appl Math Optim 42:91–101 (2000)
2000 Springer-Verlag New York Inc.
Conditions for No Breakdown and Bellman Equations of
and H. Nagai
Centre National d’Etudes Spatiales,
2 Place Maurice Quentin, 75039 Paris, France
Department of Mathematical Science, Faculty of Engineering Science,
Osaka University, Toyonaka, 560 Osaka, Japan
Abstract. Inthe treatmentof therisk-sensitive control problem, it is knownthat the
criterion function may not have a ﬁnite value. The risk parameter cannot be arbitrary.
Conditions have been presented by the authors in previous papers to guarantee the
no breakdown.In the present article, we present a framework in which the conditions
can be greatly relaxed.
Since the relationship between risk-sensitive control and H
-control was noticed in the
works of Whittle ,  and Glover and Doyle  many authors have been interested
in the study of the relationship (e.g., , , , , , and ). In considering the
relationship the LEQG (Linear Exponential Quadratic Gaussian) model has played a
particularly important role as a typical model where computations can be done explicitly
(see  and ). We note that in the LEQG model for the risk-averse case it may
occur that the criterion function never has a ﬁnite value, namely breakdown may occur,
provided the risk-sensitive parameter is large. So, there arises naturally the problem of
knowing the conditions on the size of the risk-sensitive parameter for no breakdown to
occur in more general settings of risk-sensitive control problems.
In  and  we have shown that under the conditions which assure the existence
of the solution of the Bellman equation no breakdown occurs. The present paper is a
The research of the second author was supported in part by Grant-in-Aid for Scientiﬁc Research
No. 8640280, the Ministry of Education, Science and Culture.