IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 1, JANUARY 2000 77 q.m. energy as in (11) q.m. energy operator, obtained by substituting for in classical Lagrangian q.m. Lagrangian mass classical momentum q.m. momentum q.m. momentum operator classical velocity q.m. velocity classical potential energy obtained by substituting for in as in (1) classical action integral with reversed sign as in (6) q.m. action integral density of physical particles on the real axis wave function satisfying Schrödinger’s equation stationary value obtained by varying the functions stationary value obtained by varying the functions variation of the succeeding expression with respect to variation of the succeeding expression with respect to . REFERENCES [1] C. Lanczos, The Variational Principles of Mechanics: Toronto Univer- sity Press, 1964. [2] H. H. Rosenbrock, “A stochastic variational principle for quantum me- chanics,” Phys. Lett., vol. 110A, pp. 343–346, 1986. [3] , “A variational treatment of quantum mechanics,” Proc. R. Soc. A, vol. 450, pp. 417–437, 1995. [4] , “A correction to a stochastic variational treatment of quantum me- chanics,” Proc. R. Soc. A, vol. 453, pp. 983–986, 1997. [5] , “The definition of state in the stochastic variational treatment of quantum mechanics,” Phys. Lett. A, vol. 254, pp. 307–313, 1999. [6] R. E. Bellman and S. E. Dreyfus, Applied Dynamic Programming: Princeton University Press, 1964. [7] L. Arnold, Stochastic Differential Equations. New York, NY: Wiley, 1974. [8] M. Hesse, Revolutions and Reconstructions in the Philosophy of Sci- ence: Harvester, 1980. [9] B. Barnes, Scientific Knowledge and Sociological Theory: Routledge and Kegan Paul, 1974. [10] S. Drake, Discoveries and Opinions of Galileo. New York, NY: Dou- bleday, 1957, p. 164. [11] R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Inte- grals. New York, NY: McGraw-Hill, 1965. [12] L. S. Schulman, Techniques and Applications of Path Integration. New York, NY: Wiley-Interscience, 1981. [13] B. D. Bramson, “Stochastic optimal control and command,” DERA, Malvern, U.K., Rep. DERA/LSB/2CR980 635/1.0, 1999. [14] J. Monod, Chance and Necessity: Collins, 1971. [15] W. Yourgrau and S. Mandelstam, Variational Principles in Dynamics and Quantum Theory: Pitman, 1968, pp. 173–174. [16] A. Einstein, Out of My Later Years. New York, NY: Thames and Hudson, 1950, p. 114. [17] R. Dawkins, The Blind Watchmaker. White Plains, NY: Longman, 1986. [18] H. H. Rosenbrock, Machines with a Purpose. New York, NY: Oxford University Press, 1990. Robust Control of Uncertain Markovian Jump Systems with Time-Delay Yong-Yan Cao and James Lam Abstract—This correspondence is concerned with the robust stochastic stabilizability and robust disturbance attenuation for a class of uncer- tain linear systems with time delay and randomly jumping parameters. The transition of the jumping parameters is governed by a finite-state Markov process. Sufficient conditions on the existence of a robust stochastic stabi- lizing and -suboptimal state-feedback controller are presented using the Lyapunov functional approach. It is shown that a robust stochastically stabilizing state-feedback controller can be constructed through the numerical solution of a set of coupled linear matrix inequalities. Index Terms—Jumping parameters, linear matrix inequality (LMI), linear uncertain systems, robust control, time-delay systems. I. INTRODUCTION A great deal of attention has recently been devoted to the Markovian jump linear systems. This family of systems is modeled by a set of linear systems with the transitions between the models determined by a Markov chain taking values in a finite set. It was introduced by Krasovskii and Lidskii in 1961 [13] and may represent a large variety of processes, including those in production systems and economic problems. Developments in control engineering regarding applications, stability conditions, and optimal control problems for jump linear systems are reported in [1], [3], [8]–[10], [12], and [17]. On the other hand, time-delay systems have been studied extensively on the subject of stability and control over the years; see [4], [5], and [16] for instance. The problem of robust control of linear uncertain systems with time delay has gathered much attention, and some sufficient conditions have been presented [6], [7], [11], [15]. In this correspondence, we study the robust stochastic stabilizability and robust disturbance attenuation for a class of uncertain linear Manuscript received October 5, 1998; revised March 15, 1999. Recom- mended by Associate Editor, J. C. Spall. This work was supported in part by the University of Hong Kong, under a CRCG Grant, and the National Natural Science Foundation of China under Grant 69604007. Y.-Y. Cao is with the National Lab of Industrial Control Technology, Institute of Industrial Process Control, Zhejiang University, Hangzhou, 310027 China (e-mail: yycao@iipc.zju.edu.cn). J. Lam is with the Department of Mechanical Engineering, University of Hong Kong, Hong Kong (e-mail: jlam@hku.hk). Publisher Item Identifier S 0018-9286(00)00807-2. 0018–9286/00$10.00 © 2000 IEEE