Extended algebraic Riccati equations in the abstract hyperbolic case
The paper is concerned with the optimal quadratic cost problem over an infinite (time) horizon, and the related algebraic Riccati equation, associated with the abstract equation \[ y_t= Ay+ Bu\in [{\mathcal D}(A^*)]',\quad y(0)= y_0\in Y, \] subject to the hypothesis of the abstract hyperbolic dynamics and with a fully unbounded control operator \(B\). The cost function is \[ J(u,y)= \int^\infty_0 [\|Ry(t)\|^2_Y+ \|u(t)\|^2_U] dt,\quad R\in{\mathcal L}(Y), \] where \(U\) (control) and \(Y\) (state) are two separable Hilbert spaces, and the optimal control problem is \[ \text{minimize} J(u,y)\text{ over all }u\in L_2(0,\infty; U). \] The paper studies this optimal control problem and the related algebraic Riccati equation on \({\mathcal D}(A)\) in the abstract hyperbolic case and provides a quantitative analysis, based on a regularization/approximation approach, of how to achieve a suitable extension of \(B\) so that the algebraic Riccati equation is satisfied on \({\mathcal D}(A)\). The theory presented is illustrated by a computable example.
scientific article; zbMATH DE number 1472210