The extended symplectic pencil and the finite-horizon LQ problem with two-sided boundary conditions

TL;DR

This paper presents a novel analytic method for solving finite-horizon optimal control problems with two-sided boundary conditions, using a new decomposition of the extended symplectic pencil that requires weaker assumptions than previous methods.

Contribution

It introduces a new decomposition of the extended symplectic pencil enabling explicit solutions under weaker conditions, expanding applicability in finite-horizon control problems.

Findings

Provides a parametric expression for optimal control sequences.

Develops ancillary results on generalized Riccati equations.

Analyzes eigenstructure of the extended symplectic pencil.

Abstract

This note introduces a new analytic approach to the solution of a very general class of finite-horizon optimal control problems formulated for discrete-time systems. This approach provides a parametric expression for the optimal control sequences, as well as the corresponding optimal state trajectories, by exploiting a new decomposition of the so-called extended symplectic pencil. Importantly, the results established in this paper hold under assumptions that are weaker than the ones considered in the literature so far. Indeed, this approach does not require neither the regularity of the symplectic pencil, nor the modulus controllability of the underlying system. In the development of the approach presented in this paper, several ancillary results of independent interest on generalised Riccati equations and on the eigenstructure of the extended symplectic pencil will also be presented.