Manifolds, tensor analysis, and applications: 2nd edition
Manifolds, tensor analysis, and applications: 2nd edition
Mathematical control theory: deterministic systems
Mathematical control theory: deterministic systems
The Dynamics of Control
Invariance Entropy for Control Systems
SIAM Journal on Control and Optimization
Topological and Invariance Entropy for Infinite-Dimensional Linear Systems
Journal of Dynamical and Control Systems
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Invariance entropy for continuous-time control systems measures how often open-loop controls have to be updated in order to render a controlled invariant subset of the state space invariant. A special type of a controlled invariant set is a control set, i.e., a maximal set of approximate controllability. In this paper, we investigate the properties of the invariance entropy of such sets. Our main result gives an upper bound of this quantity in terms of the positive Lyapunov exponents of a periodic solution in the interior of the control set. Moreover, for one-dimensional control-affine systems with a single control vector field we provide an analytical formula for the invariance entropy of a control set in terms of the drift vector field, the control vector field, and their derivatives. As an application, we study a controlled bilinear oscillator.