# Minimum energy for linear systems with finite horizon: a non-standard Riccati equation

## TL;DR

This paper investigates a non-standard Riccati equation in infinite-dimensional linear-quadratic control, establishing existence, partial uniqueness, and solutions in special cases for minimum energy control over finite horizons.

## Contribution

It introduces a novel Riccati equation with an opposite sign linear part, proves its connection to the value function, and analyzes solution uniqueness and special cases.

## Key findings

- The operator P(t) solves the non-standard Riccati equation.
- Partial uniqueness of solutions is established for invertible operators.
- Special case analysis when involved operators commute.

## Abstract

This paper deals with a non-standard infinite dimensional linear-quadratic control problem arising in the physics of non-stationary states (see e.g. [6]): finding the minimum energy to drive a fixed stationary state x = 0 into an arbitrary non-stationary state x. The Riccati Equation (RE) associated to this problem is not standard since the sign of the linear part is opposite to the usual one, thus preventing the use of the known theory. Here we consider the finite horizon case. We prove that the linear selfadjoint operator P(t), associated to the value function, solves the above mentioned RE (Theorem 4.12). Uniqueness does not hold in general but we are able to prove a partial uniqueness result in the class of invertible operators (Theorem 4.13). In the special case where the involved operators commute, a more detailed analysis of the set of solutions is given (Theorems 4.14, 4.15 and 4.16). Examples of applications are given.