Exact Riccati equation solution for optimal control of a flexible beam with a quadratic-like cost functional.

The authors derive the algebraic operator Riccati equation for the beam equation with a quadratic like index consisting of the first-order partial differential operator in spatial variables. The derivation is based on the maximum principle and the costate system equation. The authors find the exact analytical solution of the algebraic operator Riccati equation for a class of operators which are equivalent to the integal operators having infinite sum kernels (they may not be summable). Under the assumption that all the operators in the equation have the same eigenfunctions, the exact analytical solution for the equation is obtained and some aspects of the solution are investigated.