Finding initial costates in finite-horizon nonlinear-quadratic optimal control problems

Abstract

A procedure for obtaining the initial value of the costate in a regular, finite-horizon, nonlinear-quadratic problem is devised in dimension one. The optimal control can then be constructed from the solution to the Hamiltonian equations, integrated on-line. The initial costate is found by successively solving two first-order, quasi-linear, partial differential equations (PDEs), whose independent variables are the time-horizon duration T and the final-penalty coefficient S. These PDEs need to be integrated off-line, the solution rendering not only the initial condition for the costate sought in the particular (T, S)-situation but also additional information on the boundary values of the whole two-parameter family of control problems, that can be used for design purposes. Results are tested against exact solutions of the PDEs for linear systems and also compared with numerical solutions of the bilinear-quadratic problem obtained through a power-series' expansion approach. Bilinear systems are specially treated in their character of universal approximations of nonlinear systems with bounded controls during finite time-periods.