Weak* Solution to a Dynamic Reconstruction Problem

We consider a dynamic control reconstruction problem for deterministic affine control systems. The reconstruction is performed in real time on the basis of known discrete inaccurate measurements of the observed trajectory of the system generated by an unknown measurable control with values in a given compact set. We formulate a well-posed reconstruction problem in the weak* sense and propose its solution obtained by the variational method developed by the authors. This approach uses auxiliary variational problems with a convex–concave Lagrangian regularized by Tikhonov's method. Then the solution of the reconstruction problem reduces to the integration of Hamiltonian systems of ordinary differential equations. We present matching conditions for the approximation parameters (accuracy parameters, the frequency of measurements of the trajectory, and an auxiliary regularizing parameter) and show that under these conditions the reconstructed controls are bounded and the trajectories of the dynamical system generated by these controls converge uniformly to the observed trajectory.