From the introduction: Consider the linear controlled system
\[ \dot x = A(t)x + B(t)u,\quad x\in\mathbb R^n, \quad u\in\mathbb R^m, \quad t\in\mathbb R,\tag{1} \]
with bounded piecewise continuous operator coefficients \(A:\mathbb R \to \text{End}(\mathbb R^n)\) and \(B:\mathbb R \to \text{Hom}(\mathbb R^m,\mathbb R^n)\). Let the control be defined on the basis of the linear feedback principle, i.e. \(u = U(t)x\); moreover, the operator function \(U : \mathbb R \to \text{Hom}(\mathbb R^m,\mathbb R^n)\) is also assumed to be bounded and piecewise continuous. Then the original system (1) becomes the closed system
\[ \dot x = (A(t) + B(t)U)x,\quad x\in\mathbb R^n, \quad t\in\mathbb R. \tag{2} \]
Since system (2) is homogeneous and has bounded piecewise continuous coefficients, we see that it has well-defined Lyapunov characteristic exponents. The main problem solved in the present paper is to clarify sufficient conditions under which, by an appropriate choice of the feedback coefficient \(U\) (treated as a control), one can deliberately vary the Lyapunov exponents of system (2) in some range, i.e., control them, under the assumption that system (1) is uniformly completely controllable.