摘要
This paper aims to deal with the indefinite linear quadratic regulator(ILQR) for stochastic systems. It provides both the analytic solution to the ILQR and the sufficient and necessary condition under which the ILQR is solvable. Different from the existed literature, we obtain the results in a novel way. In order to obtain a tighter necessary condition, we investigate the solution structure of a two-point boundary value problem for a stochastic differential equations directly but involve no complicated probabilistic derivation. Consider that there are close relationships between the linear game problem and H+∞ control and ILQR,the idea in the paper can also be extended to solve these problems.
This paper aims to deal with the indefinite linear quadratic regulator(ILQR) for stochastic systems. It provides both the analytic solution to the ILQR and the sufficient and necessary condition under which the ILQR is solvable. Different from the existed literature, we obtain the results in a novel way. In order to obtain a tighter necessary condition, we investigate the solution structure of a two-point boundary value problem for a stochastic differential equations directly but involve no complicated probabilistic derivation. Consider that there are close relationships between the linear game problem and H+∞ control and ILQR,the idea in the paper can also be extended to solve these problems.
引文
[1]J.Bismut,“Linear quadratic optimal stochastic control with random coefficients,”SIAM Journal on Control and Optimization,vol.14,no.3,pp.419–444,1976.
[2]M.Rami,J.Moore,and X.Zhou,“Indefinite stochastic linear quadratic control and generalized differential Riccati equation,”SIAM Journal on Control and Optimization,vol.40,no.4,pp.1296–1311,2002.
[3]S.Chen,X.Li,and X.Zhou,“Stochastic linear quadratic regulators with indefinite control weight costs,”SIAM Journal on Control and Optimization,vol.36,no.5,pp.1685–1702,1998.
[4]M.A.Rami and X.Y.Zhou,“Linear matrix inequalities,riccati equations,and indefinite stochastic linear quadratic controls,”IEEE Transactions on Automatic Control,vol.45,no.6,pp.1131–1143,2000.
[5]M.Ait Rami,X.Chen,J.B.Moore,and X.Y.Zhou,“Solvability and asymptotic behavior of generalized riccati equations arising in indefinite stochastic lq controls,”Automatic Control IEEE Transactions on,vol.46,no.3,pp.428–440,2001.
[6]W.M.Wonham,“On a matrix riccati equation of stochastic control,”SIAM Journal on Control,vol.6,no.4,pp.681–697,1968.
[7]A.Bensoussan,“Lecture on stochastic control,part I,”Lecture Notes in Math,vol.972,no.1,pp.1–39,1983.
[8]M.H.A.Davis,“Linear estimation and stochastic control,”Technometrics,vol.76,no.2,pp.192–193,1984.
[9]S.Chen and J.Yong,“Stochastic linear quadratic optimal control problems,”Applied Mathematics and Optimization,vol.43,no.1,pp.21–45,2001.
[10]J.Yong,“Linear Forward-Backward Stochastic Differential Equations,”Applied Mathematics and Optimization,vol.39,no.1,pp.93–119,1999.
[11]S.Peng and Z.Wu,“Fully coupled forward-backward stochastic differential equations and applications to optimal control,”SIAM Journal on Control&Optimization,vol.37,no.3,pp.825–843,1999.