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- W4256150907 abstract "A.1 Differential EquationsConsider the finite-dimensional nonlinear systemx˙(s)=ƒ(x(s))+g(x(s))υ(s),x(t)=x,(A.1)with inputs υ∈L2[0,∞) . We shall assume that ƒ and g are globally Lipschitz continuous :|ƒ(x1)−ƒ(x2)|≤Cƒ|x1−x2|,|g(x1)−g(x2)|≤Cg|x1−x2|.(A.2)The constants Cƒ,Cg≥0 do not depend on x1,x2∈Rn . We also assume that ƒ(0)=0 so that x=0 is an equilibrium. In view of these assumptions, ƒ and g grow at most linearly in x:|ƒ(x)|≤Cƒ|x|,|g(x)|≤Cgg(1+|x|).(A.3)Let Φs,tυ(x) denote the transition operator associated with (A.1). Our convention is that Φs,tυ(x) denotes the solution at time s from x at time t, so that Φt,tυ(x)=x for any t. In particular, if t≥0 , then Φt,0υ(x) denotes the solution at time t starting from x at time 0 (Figure 2.1), and Φ0,tυ(x) denotes the solution at time t starting from x at time 0 (Figure 2.2). From the theory of ODEs, Φs,tυ(x) is uniquely defined, is smooth in x (at least C1 ) and s, t (at least absolutely continuous), and is a diffeomorphism in x.The following estimate measures the influence of the input υ on the state trajectory for finite time durations." @default.
- W4256150907 created "2022-05-12" @default.
- W4256150907 date "1999-01-01" @default.
- W4256150907 modified "2023-09-27" @default.
- W4256150907 title "Appendix A: Differential Equations and Stability" @default.
- W4256150907 doi "https://doi.org/10.1137/1.9780898719840.appa" @default.
- W4256150907 hasPublicationYear "1999" @default.
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