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SOOCHOW JOURNAL OF MATHEMATICS

Volume 33, No. 4, pp. 529-542, October 2007




DISSIPATIVE CONTROL SYSTEM FOR

THE STOCHASTIC $ H^{\infty}$ PROBLEMS


BY


YONG ZHOU, ZHENGMEI HOU AND SANYANG LIU




Abstract. We characterize functions satisfying a dissipative inequality associated with a stochastic control problem. Such a characterization is provided in terms of an upper generalized Gaussian solution, or a viscosity supersolution to a partial differential equation called Hamilton-Jacobi equation(H-J). Links between upper generalized Gaussian solutions and viscosity supersolutions to Hamilton-Jacobi equation are studied. Finally it shows that generalized Gaussian solutions is identical to viscosity solutions to Hamilton-Jacobi equation.

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Received June 7, 2004; revised June 15, 2005; November 18, 2005; January 11, 2007.

AMS Subject Classification. 93E20, 93C15, 60K05.

Key words. H-J-B equation, stochastic $ H^{\infty}$ control, generalized Gaussian and viscosity solutions to PDEs.

This research is supported by the key item of scientific and study of Xinjiang high-university scientific and study plan under Grant No. XJEDU2005I26, P.R. China.