Riccati Differential Equations with Unbounded Coefficients and Nonsmooth Terminal Condition—The Case of Analytic Semigroups

Abstract

This paper provides, constructively, an explicit solution to the (operator) Differential Riccati Equation in Hilbert space with unbounded coefficients on a fixed time interval $[ {0,T} )$, $T < \infty $, which arises in the optimal control problem with nonsmoothing terminal condition at $t = T$ for an abstract dynamics modeled by an analytic semigroup. The results are sharp as illustrated by counterexamples. Regularity properties of all the quantities involved are also given. Uniqueness of the solution is asserted under some additional assumptions on the terminal condition. Applications include parabolic equations with Dirichlet, or Neumann (Robin) boundary control, or else with point control, as well as plate-like equations with a high degree of damping, etc.