Using semiconvex duality and max-plus analysis to obtain a new fundamental solution for the differential riccati equation

Semiconvex duality and max-plus analysis are used to obtain a surprising new fundamental solution for the ubiquitous matrix differential Riccati equation (DRE). We consider the DRE as a finite-dimensional solution to a deterministic linear/quadratic control problem. Taking the semiconvex dual of the associated semigroup, one obtains the solution operator as a max-plus integral operator with quadratic kernel. The kernel is equivalently represented as a matrix. Using the semigroup property of the dual operator, one obtains a matrix operation whereby the kernel matrix propagates as a semigroup. The propagation forward is through some simple matrix operations. This time-indexed family of matrices forms a new fundamental solution for the DRE. Solution for any initial condition is obtained by a few matrix operations on the fundamental solution and the initial condition. This fundamental solution has a particularly nice control interpretation.