Canonical forms, control of Lyapunov exponents, and stabilizability of nonstationary linear systems

It is shown that any uniformly controllable single-input linear system on the half-line R+ whose coefficients satisfy some boundedness and smoothness hypotheses is reduced to a scalar differential equation of the nth order with variable coefficients in the general case by using the Lyapunov transformation. This fact allows one to prove the solvability of the control problem of Lyapunov exponents for each of these systems, and, therefore, to prove its stabilizability via a linear feedback. If the system is subject to uncontrolled disturbances, stabilization via the mentioned feedback is impossible. Therefore, under the existence of a disturbance, impulse controls that are linear combinations of delta-functions and their derivatives are used. The necessary and sufficient conditions for stabilizability of nonautonomous systems by impulse control actions are proved. Examples of physically meaningful systems are presented; these systems are embedded into the frameworks of the proved general results, and, therefore, can be stabilized by using the classical or impulse controls.