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Zbl 0899.93004
Colonius, F.; Johnson, R.
Local and global null controllability of time varying linear control systems. (English)
[J] ESAIM, Control Optim. Calc. Var. 2, 329-341 (1997). ISSN 1292-8119; ISSN 1262-3377/e

A linear control system with coefficients determined by a dynamical system (flow) is given by $$x'= A(T_t(\omega))x+ B(T_t(\omega))u,\quad x\in\bbfR^n,\quad u\in U\subset\bbfR^m,$$ where $\omega\in\Omega$ and $(\Omega,\{T_t\}_{t\in \bbfR})$ is a flow on a compact metric space $\Omega$. The authors prove earlier known results about null controllability under a more general assumption on the flow -- namely that the flow admits a finite, invariant ergodic measure whose support is all of $\Omega$. The earlier results were proved under a stronger assumption of minimality of the flow. The results proved are about (a) uniform local null controllability over all of $\Omega$, (b) global null controllability for `almost all $\omega$'' (given local null controllability and non-positivity of the Lyapunov exponents) and (c) global null controllability for a dense set of $\omega$'s under the assumption of `irreducibility'', even if some of the Lyapunov exponents are positive. A certain `measurable selection theorem' is the key ingradient in extending these results from the minimal case to the case when the flow admits a supported finite, invariant ergodic measure.
[M.G.Nerurkar (Camden)]

MSC 2000:: *93B05 Controllability
93C99 Control systems, guided systems
93C05 Linear control systems

Keywords: time varying linear control systems; measurable selection theorem; null controllability; invariant ergodic measure; Lyapunov exponents

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