Finite-horizon optimal control for linear and nonlinear systems relying on constant optimal costate

A class of finite-horizon optimal control problems, the solution of which relies on a time-varying change of coordinates that incorporates the transition matrix of the system linearized along the current estimate of the optimal process, is studied. The transformed dynamics exhibit a constant optimal costate. Differently from existing methods that hinge upon similar tools, the proposed strategy does not require at each step the (numerical) solution of a two-point boundary value problem or of a time-varying Riccati equation, and only the solution of a linear initial value problem is needed. The method is firstly illustrated in the setting of linear dynamics and quadratic cost for which the construction permits the identification of a class of problems in which the solution to the underlying (quadratic) Differential Riccati Equation exhibit a separation between homogeneous and particular contributions.