New fundamental solution for time-varying differential Riccati equations

Using the tools of semiconvex duality and max-plus algebra, this work derives a new fundamental solution for the matrix differential Riccati equation (DRE) with time-varying coefficients. Such a fundamental solution, is the counterpart of the state transition matrix in linear time-varying differential equations, and can solve the DRE analytically starting with any initial condition. By parametrizing the exit cost of the underlying optimal control problem using an additional variable, a bivariate DRE is derived. Any particular solution of such bivariate time-varying DRE, can generate the fundamental solution, and hence the general solution, analytically. The fundamental solution is equivalently represented by three matrices, and the solution for any initial condition is obtained by a few matrix operations on the initial condition. It covers the special case of time invariant DRE, and derives the kernel matching conditions for transforming the DRE into the semiconvex dual DRE. As a special case, this dual DRE can be made linear, and is thus solvable analytically. Using this, the paper rederives the analytical solutions previously obtained by Leipnik [4] and Rusnak [6]. It also suggests a modification to the exponentially fast doubling algorithm described in [1], used to solve the time invariant DRE , and makes it more stable and accurate numerically for the propagation at small time step. This work is inspired from the previous work by McEneaney and Fleming [1],[2]. ∗Ameet Deshpande is with Dept. of Mech. and Aero. Eng., University of California San Diego, San Diego, CA 92093-0411, USA. [email protected] †Research partially supported by NSF grant DMS-0307229 and AFOSR grant FA9550-061-0238. ar X iv :0 90 3. 04 05 v1 [ m at h. O C ] 2 M ar 2 00 9