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An open-plus-closed-loop (OPCL) control of complex dynamic systems. (English) Zbl 0888.93034

Summary: A new method of controlling arbitrary nonlinear dynamic systems, \(dx/dt= F(x,t)\) (\(x \in{}\mathbb{R}{}^{n}\)), is presented. It is proved that there exists solutions, \(x(t)\), in the neighborhood of any arbitrary ‘goal’ dynamics \(g(t)\) that are entrained to \(g(t)\), through the use of an additive controlling action, \(K(g,x,t) = H(dg/dt,g) + C(g,t) (g(t)-x)\), which is the sum of the open-loop (Hübler) action, \(H(dg/dt,g)\), and a suitable linear closed-loop (feedback) action \(C(g,t)\). Examples of some newly obtained entrainment capabilities are given for the Duffing and Van der Pol systems. For these and the Lorenz, and Rössler systems proofs are given for global basins of entrainment for all goal dynamics that can be exponentially bounded in time. The basin of entrainment is also established for the Chua system, as well as the possibility of a coexisting basin of attraction to another fixed point.

MSC:

93C15 Control/observation systems governed by ordinary differential equations
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
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References:

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