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| In this talk we consider nonautonomous linear control systems of the form
\[ \dot{x}(t) = A(t) x(t) + B(t) u(t) \,.\]
For such linear time-varying systems with bounded system matrices we discuss the problem of stabilizability by linear state feedback,
that is, the existence of a time-varying feedback $F$ such that
\[ \dot{x}(t) = A(t) x(t) + B(t) F(t) x(t) \]
is stable in a suitable sense.
For example, it is shown that complete controllability implies the existence of a feedback so that the closed-loop system is asymptotically stable. We also show that the system is completely controllable if, and only if, the Lyapunov exponent is arbitrarily assignable by a suitable feedback. For uniform exponential stabilizability and the assignability of the Bohl exponent this property is known. Also, it is shown that dynamic feedback does not provide more freedom to address the stabilization problem. The unifying tools for our results are two finite $L^2-L^2$ cost conditions. The distinction of exponential and uniform exponential stabilizability is then a question of whether the finite cost condition is uniform in the initial time or not. |
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