Minimum energy with infinite horizon: from stationary to non-stationary states

TL;DR

This paper investigates a non-standard infinite horizon linear-quadratic control problem in infinite dimensions, focusing on minimum energy to transition from stationary to non-stationary states, and characterizes solutions to the associated algebraic Riccati equation.

Contribution

It introduces a novel approach to solving a non-standard algebraic Riccati equation arising in non-stationary state control, proving maximality of the solution and characterizing all solutions in special cases.

Findings

The value function's operator solves the non-standard ARE.

The maximal solution to the ARE is identified.

All solutions are characterized when operators commute.

Abstract

We study a non standard infinite horizon, infinite dimensional linear-quadratic control problem arising in the physics of non-stationary states (see e.g. \cite{BDGJL4,BertiniGabrielliLebowitz05}): finding the minimum energy to drive a given stationary state $\bar x=0$ (at time $t=-\infty$) into an arbitrary non-stationary state $x$ (at time $t=0$). This is the opposite to what is commonly studied in the literature on null controllability (where one drives a generic state $x$ into the equilibrium state $\bar x=0$). Consequently, the Algebraic Riccati Equation (ARE) associated to this problem is non-standard since the sign of the linear part is opposite to the usual one and since it is intrinsically unbounded. Hence the standard theory of AREs does not apply. The analogous finite horizon problem has been studied in the companion paper \cite{AcquistapaceGozzi17}. Here, similarly to such paper, we prove that the linear selfadjoint operator associated to the value function is a solution of the above mentioned ARE. Moreover, differently to \cite{AcquistapaceGozzi17}, we prove that such solution is the maximal one. The first main result (Theorem \ref{th:maximalARE}) is proved by approximating the problem with suitable auxiliary finite horizon problems (which are different from the one studied in \cite{AcquistapaceGozzi17}). Finally in the special case where the involved operators commute we characterize all solutions of the ARE (Theorem \ref{th:sol=proj}) and we apply this to the Landau-Ginzburg model.