Numerical-Analytical Algorithms for Nonlinear Optimal Control Problems on a Large Time Interval

Abstract

Some nonlinear and discrete optimal control problems with phase constraints on a fixed but sufficiently large time interval are considered as singularly perturbed problems. In continuous-time case the state equations are reduced to singularly perturbed equations on a finite time interval and, in discrete-time case, the state equations have the form of systems with a small step. Using the technique for singularly perturbed systems, the formal asymptotic expansions by the corresponding small parameter are constructed which contain the structural information about the solution. That is usually sufficient for most applications to obtain an initial approximation to control in the global optimum neighborhood. The obtained algorithms can be applied to mathematical economics and technical objects control problems with phase and control constraints, and with turnpike effects in the trajectories, where the turnpike trajectories can be discontinuous. The use of traditional algorithms for these problems is inefficient due to the large increase of computational difficulty.