We show that some HJB equations arising from both finite and infinite horizon stochastic optimal control problems have a regular singular point at the origin. This makes them amenable to solution by power series techniques. This extends the work of Al’brecht who showed that the HJB equations of an infinite horizon deterministic optimal control problem can have a regular singular point at the origin, Al’brekht solved the HJB equations by power series, degree by degree.
In particular, we show that the infinite horizon stochastic optimal control problem with linear dynamics, quadratic cost and bilinear noise leads to a new type of algebraic Riccati equation which we call the Stochastic Algebraic Riccati Equation (SARE). If SARE can be solved then one has a complete solution to this infinite horizon stochastic optimal control problem.
We also show that a finite horizon stochastic optimal control problem with linear dynamics, quadratic cost and bilinear noise leads to a Stochastic Differential Riccati Equation (SDRE) that is well known. If these problems are the linear-quadratic-bilinear part of a nonlinear finite horizon stochastic optimal control problem then we show how the higher degree terms of the solutions can be computed degree by degree. To our knowledge this computation is new.
We extend these methods to discrete time stochastic optimal control problems over infinite and finite horizons.
Arthur J. Krener received the PhD in Mathematics from the University of California, Berkeley in 1971. From 1971 to 2006 he was at the University of California, Davis. He retired in 2006 as a Distinguished Professor of Mathematics. Currently he is a Research Professor in the Department of Applied Mathematics at the Naval Postgraduate School.
His research interests are in developing methods for the control and estimation of nonlinear dynamical systems and stochastic processes.
Professor Krener is a Life Fellow of IEEE, a Fellow of AMS, IFAC and SIAM. His 1981 IEEE Transactions on Automatic Control paper with Isidori, Gori-Giorgi and Monaco won a Best Paper Award. The IEEE Control Systems Society chose his 1977 IEEE Transactions on Automatic Control paper with Hermann as one of 25 Seminal Papers in Control in the last century. He was a Fellow of the John Simon Guggenheim Foundation for 2001-2. In 2004 he received the W. T. and Idalia Reid Prize from SIAM for his contributions to control and system theory. He was the Hendrik Bode Prize Lecturer at 2006 IEEE CDC and in 2010 he received a Certificate of Excellent Achievements from IFAC. He received the Richard Bellman Control Heritage Award from AACC in 2012 and in 2016 he received the IEEE Field Award in Control Systems.