Variational Approach to Construction of Piecewise-Constant Approximations of the Solution of Dynamic Reconstruction Problem

Abstract

In the paper, the problem of dynamic reconstruction of controls and trajectories for deterministic control-affine systems is considered. The reconstruction is performed in real time using known discrete inaccurate measurements of the observed trajectory of the system. This trajectory is generated by an unknown measurable control that satisfies known geometric constraints. A well-posed statement of the problem is given. A solution is proposed using the variational approach developed by the authors. This approach uses auxiliary variational problem with regularized integral residual functional. The integrant of the functional is a d.c. function. The suggested algorithm reduces the reconstruction problem to integration of Hamiltonian systems of ordinary differential equations. This paper offers a method for construction of piecewise-constant approximations that satisfy the given geometric control constraints. The approximations converge almost everywhere to the desired control, and the reconstructed trajectories of the dynamical system converge uniformly to the observed trajectory.