Max-Plus Representation for the Fundamental Solution of the Time-Varying Differential Riccati Equation

TL;DR

This paper introduces a unified max-plus algebra-based representation for solving time-varying matrix differential Riccati equations, enabling analytical solutions from arbitrary initial conditions through duality and kernel methods.

Contribution

It develops a max-plus fundamental solution framework for time-varying DREs, extending previous methods and providing analytical solutions via duality and kernel relationships.

Findings

Unified max-plus representation for time-varying DREs

Analytical solutions from any initial condition

Dual DREs satisfy matrix compatibility conditions

Abstract

Using the tools of optimal control, semiconvex duality and \maxp algebra, this work derives a unifying representation of the solution for the matrix differential Riccati equation (DRE) with time-varying coefficients. It is based upon a special case of the \maxp fundamental solution, first proposed in \cite{FlemMac}. Such fundamental solution can extend a special solution of certain bivariate DRE into the general solution, and can analytically solve the DRE starting from any initial condition. This paper also shows that under a fixed duality kernel, the semiconvex dual of a DRE solution satisfies another dual DRE, whose coefficients satisfy the matrix compatibility conditions involving Hamiltonian and certain symplectic matrices. For the time invariant DRE, this allows us to make dual DRE linear and thereby solve the primal DRE analytically. This paper also derives various kernel/duality relationships between the primal and time shifted dual DREs, which leads to an array of DRE solutions. Time invariant analogue of one of these methods was first proposed in \cite{Funda}.