By L. Garrido
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Extra resources for Dynamical System and Chaos
72) where ε0 , θ0 , and x0 are arbitrary constants (not violating the condition ε(x) > 0 on [0, 1]) and a = ε0 θ0 . 71). 72). 72) always has one maximum or one minimum. The value and the location of the maximum (minimum) can be arbitrarily set by ε0 and x0 , correspondingly. The sign of θ0 determines whether it is a maximum or a minimum, and the value of θ0 can set to an arbitrary “sharpness” of the extremum (Fig. 3). By selecting the extremum outside the region [0,1] and changing its value and sharpness, we can almost perfectly match any linear function as well (Fig.
24) helps to stabilize, so the worst case is a peak somewhere in the middle of the domain. 1. 41) was obtained using the method of separation of variables. One can show that the most general λ(x) for which the controller can be found using the method of separation of variables is λ(x) = 2(α 2 − β 2 ) α 2 h2 (x) + (β 2 − γ 2 ) sinh2 (αx) , (α cosh(αx)h(x) − sinh(αx)h (x))2 where α, β, and γ are arbitrary constants and γ h(x) = cosh(βx) + sinh(βx). 43) The control gain kernel for the plant with this λ(x) is given by k(x, y) = − (α 2 − β 2 )h(y) sinh(βx) .
116) The decay rate of the closed-loop v-system is equal to the decay rate of the 2 target system, that is, e−(c+π )t . 117) t→∞ or c > −π 2 if λ ∈ L1 (0, ∞) ∪ L2 (0, ∞). 91) for any λ(t), it is most suitable for the cases when minimum and maximum values of λ(t) are close, for example when it is a constant plus sinusoid with a small amplitude. ). 105) is advantageous, as indicated by the simulation results in Fig. 7. 5 can be combined to obtain the explicit results for even more complex systems.