IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 42, NO. 8, AUGUST 1997 1163 REFERENCES [1] M. A. Rotea and P. P. Khargonekar, “Stabilization of uncertain sys- tems with norm bounded uncertainty: A control Lyapunov function approach,” SIAM J. Contr. Optimization, vol. 27, pp. 1462–1476, Nov. 1989. [2] M. Fu and B. R. Barmish, “Maximal unidirectional perturbation bounds for stability of polynomials and matrices,” Syst. Contr. Lett., vol. 11, pp. 173–179, 1980. [3] P. P. Khargonekar, I. R. Petersen, and Z. Zhou, “Robust stabilization of uncertain linear systems: Quadratic stabilizability and H control theory,” IEEE Trans. Automat. Contr., vol. 35, Mar. 1990. [4] S. Boyd, L. El Ghaoui, E. Feronard, and V. Balakrishnan, Linear Matrix Inequalities in Systems and Control Theory. Philadelphia, PA: SIAM, 1994. [5] P. Gahinet, A. Nemirovskii, A. Laub, and M. Chilali, LMI Control Toolbox. Natick, MA: Mathworks, 1994. The Role of Information State and Adjoint in Relating Nonlinear Output Feedback Risk-Sensitive Control and Dynamic Games Charalambos D. Charalambous Abstract— This paper employs logarithmic transformations to estab- lish relations between continuous-time nonlinear partially observable risk-sensitive control problems and analogous output feedback dynamic games. The first logarithmic transformation is introduced to relate the stochastic information state to a deterministic information state. The second logarithmic transformation is applied to the risk-sensitive cost function using the Laplace–Varadhan lemma. In the small noise limit, this cost function is shown to be logarithmically equivalent to the cost function of an analogous dynamic game. Index Terms—Dynamic games, risk sensitive, small-noise limit, stochas- tic control. I. INTRODUCTION The deterministic minimax dynamic games arising in - or robust control design deal with controlling plants that are sub- ject to deterministic exogenous inputs (disturbances). On the other hand, exponential-of-integral control design (risk-sensitive) deals with plants that are subject to random disturbances usually modeled as Wiener processes. For linear-exponential-quadratic-Gaussian (LEQG) problems, with complete or partial information, the optimal control policies are shown to be equivalent to that of minimax dynamic games described above (see [1]–[3]). Similar results are also established for nonlinear partially observable problems which are estimation solvable in [4]. For general nonlinear completely observable risk-sensitive prob- lems, relations to dynamic games are derived in [5]–[7], using logarithmic transformations and small noise limits. Similar relations are also established for analogous continuous-time partially observ- Manuscript received February 24, 1995; revised February 5, 1996 and October 1, 1996. This work was completed during the author’s post-doctoral appointment with the Measurement and Control Research Center, Idaho State University, 1993–1995. The author is with the Department of Electrical Engineering, McGill Uni- versity, Montreal, P.Q., H3A 2A7 Canada (e-mail: chadcha@cim.mcgill.ca). Publisher Item Identifier S 0018-9286(97)05057-5. able problems using a certainty-equivalence principle (which is not always applicable), by Whittle in [8]. An alternative approach to the certainty-equivalence principle is to use an information state (suf- ficient statistic) to convert the partially observable control problem to an equivalent completely observable control problem having as new the information state. This idea was originally implemented by Mortensen (see [9]) to show that the value function satisfies Mortensen’s Hamilton–Jacobi (HJ) equation which has an infinite- dimensional state space. Mortensen’s HJ equation has been formally investigated in the small noise limit in [10] to obtain relations with dynamic games, and it is made precise in the analogous discrete- time case in [11]. The contributions in [10] and [11] are important in introducing the concept of information state to nonlinear output feedback dynamic games. In this paper, we formulate the small noise version of the continuous-time nonlinear partially observable exponential-of- integral control problem, and then we explore various relations, in the limit as the noise terms tend to zero, with dynamic games. Our approach is different from the one considered in [10] and [11] in that we avoid the use of Mortensen’s infinite-dimensional HJ equation and thus the various technical difficulties associated with nonlinear infinite dimensional equations. Although in the current paper only the continuous-time case is considered, the discrete-time case can be treated in an analogous manner. We now wish to describe the results anticipated in this paper. Con- sider in Euclidean spaces and , two random processes, named the state process and observation process , respectively, satisfying the stochastic differential equations (1) (2) Here , , are, respectively, -dimensional standard Wiener processes, is the control process, is the initial value of the state process, and is the parameter scaling the randomness in (1), (2) (the coefficients of (1)–(3) satisfy Assumptions 1.1). The cost function to be minimized over the controls , which are nonanticipating functionals of the observations , is the exponential- of-integral cost criterion (3) In Section I-A, system (1)–(3) is formulated using a reference proba- bility measure. In Section II, the partially observable system (1)–(3) is converted to an equivalent completely observable stochastic control problem in function space by introducing an information state which satisfies a modified version of the Duncan–Mortensen–Zakai (DMZ) equation. The invariant property of the resulting total cost function is next introduced, followed by gauge transformations to derive path- wise versions of the modified DMZ equation. The first logarithmic transformation is employed to give stochastic control interpretations to the pathwise DMZ equation and its adjoint version (Lemma 3.4) in terms of second-order HJ equations. In the limit, as , these HJ equations converge to first-order HJ equations (Lemma 3.6); their solutions are deterministic information states associated 0018–9286/97$10.00  1997 IEEE