Solution of inverse problems for linear dynamical systems by the cascade method (Q1945178)

The author poses inverse problems with multi-point conditions on state and control functions for several types of linear dynamical systems. Consider the systems: \[ \dot{x}(t) = B(t)x(t) + D(t)u(t) + f(t) \tag{1} \] where \(x(t) \in \mathbb R^n\), \(u(t) \in \mathbb R^m\); \(B(t) \in L(\mathbb R^n)\) and \( D(t) \in L(\mathbb R^m, \mathbb R^n) \), and the stationary systems \[ A \dot{x}(t) = Bx(t) + D u(t), \tag{2} \] \[ A \dot{x}(t) = Bx(t) + Du(t) + f(t), \tag{3} \] where \(x(t) \in \mathbb R^l\), \(u(t) \in \mathbb R^m\); \(A,B \in L(\mathbb R^l, \mathbb R^n)\); \(D\in L(\mathbb R^m, \mathbb R^n)\); \( f(t) \in \mathbb R^n\) and \(t\in I \equiv [t_0, t_k]\). A pair \((A, B) \) is said to be regular if \((A - \lambda B)\) is invertible, for some \(\lambda \in \mathcal{C}\). (this is also a criterion for the total controllability of system (2)). The main contribution of the paper is to state the total controllability criteria for systems (2), (3), for irregular pairs \((A, B)\), and to prove the solvability of the problems for totally controlled systems of the form (1)--(3) by describing a method for constructing \(x(t) \) and \(u(t)\). Four theorems are stated.