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

摘要

This paper deals with a non-standard infinite dimensional linear-quadraticcontrol 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 anarbitrary non-stationary state x. The Riccati Equation (RE) associated to thisproblem is not standard since the sign of the linear part is opposite to theusual one, thus preventing the use of the known theory. Here we consider thefinite 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 partialuniqueness result in the class of invertible operators (Theorem 4.13). In thespecial case where the involved operators commute, a more detailed analysis ofthe set of solutions is given (Theorems 4.14, 4.15 and 4.16). Examples ofapplications are given.