A Fast Sequential Linear Quadratic Algorithm for Solving Unconstrained Nonlinear Optimal Control Problems

We develop a numerically efficient algorithm for computing controls for nonlinear systems that minimize a quadratic performance measure. We formulate the optimal control problem in discrete-time, but many continuous-time problems can also be solved after discretization. Our approach is similar to sequential quadratic programming for finite-dimensional optimization problems in that we solve the nonlinear optimal control problem using sequence of linear quadratic subproblems. Each subproblem is solved efficiently using the Riccati difference equation. We show that each iteration produces a descent direction for the performance measure and that the sequence of controls converges to a solution that satisfies the well-known necessary conditions for the optimal control. We also show that the algorithm is a Gauss-Newton method, which means it inherits excellent convergence properties. We demonstrate the convergence properties of the algorithm with two numerical examples.