Abstract
An integral representation for the second variation of trajectory of a dynamical system under control is obtained. This representation contains some tensor of the third rank introduced here. A differential equation for this tensor is presented. A second order method for solution of the optimal control problem based on the second variation of a trajectory is proposed.
Highlights
The article deals with an optimal control problem with a fixed endpoint
We found solution of this differential equation in integral form
The number of its components is cube of dimension of the phase space that is sufficiently smaller than in methods using the matrix momenta. By this reason we expect that the numerical methods based on using of the second variation of the trajectory will be effective in problems that require a great number of calculations, and when the control function can be parametrized by a great number of parameters
Summary
The article deals with an optimal control problem with a fixed endpoint. The optimal control problem can be formulated as follows. Various variational methods are developed for finding solution of an optimal control problem [Pytlak, 1999] These methods use the first variation of the trajectory or the conjugate momentum Ψ(t) satisfying to the equation dΨ. The main goals of this article is analysis of the second variation of the trajectory and application of it in numerical methods of finding solutions for the optimal control problem. A new method for numerical solution of the optimal control problem based on expression for the second variation of the cost functional including the second variation the trajectory is presented. Let us write components of the expression for the second variation (14), of the tensor D, and of the tensor equation (23): δ2xi(t)
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