Designing a continuous reconstruction operator in observation problems for ordinary linear systems (Q1768984)

For the observation system \[ \dot x (t) = A (t) x(t), \quad y(t) = c (t) x (t), \;\;\;\;\;t \in T = [t_0, t_1] \tag{1} \] a linear operator reconstructing the current state \(x (t_1)\) of system (1) is construced in the form of the Stieltjes integral \[ \varphi(y) = \int\limits_{t_0}^{t_1} y (\tau)\, d V (\tau) . \] The reconstruction operator is designed without using the principal solution matrix of the homogeneous system.