Properties of Nonlinear System (Related to Theorem)

Theorem 1.1 (Local Existence and Uniqueness)

With the presumption that f from \dot{x}\left(t \right)=f\left[t,x\left(t \right) \right];Vt\geq 0is continuous on t and x, and finite constant h, r, k and T exist, so that:

\left|\left|f\left(t,x \right)-f\left(t,y \right) \right| \right|\leq k\left|\left|x-y \right|\right|;V x,y\in B,V t\in \left[0,T \right]
\left|\left|f\left(t,{x}_{0} \right) \right| \right|\leq h;V t\in \left[0,T \right]

where B is a ball in {R}^{n}of the form:

B ={x\in {R}^{n};\left|\left|x-{x}_{0} \right| \right|\leq r}

then \dot{x}\left(t \right)=f\left[t,x\left(t \right) \right];Vt\geq 0 has just one solution in the interval \left[0,\tau  \right] where the number \tau is sufficiently small to satisfy the inequalities:

h\tau {e}^{k\tau }\leq r
\tau \leq min{ T,\frac{\rho }{k},\frac{r}{h+kr}}

for some constant \rho <1

Theorem 1.2 (Global Existence and Uniqueness)

Suppose that for each T\in [0,\sim > there exist finite constant {h}_{T} and {k}_{T} such that

\left|\left|f\left(t,x \right)-f\left(t,y \right) \right| \right|\leq {k}_{T}\left|\left|x-y \right|\right|;V x,y\in {R}^{n},V t\in \left[0,T \right]
\left|\left|f\left(t,{x}_{0} \right) \right| \right|\leq {h}_{T};V t\in \left[0,T \right]

Then \dot{x}\left(t \right)=f\left[t,x\left(t \right) \right];Vt\geq 0 has only one solution in the interval \left[0,\sim\right].

Note:

\dot{x}\left(t \right)=f\left[t,x\left(t \right) \right];Vt\geq 0 is taken from Definition 1.6 (System’s Trajectory-Solution)

Source:

Vukic, Zoran, Kuljaca, Ljubomir, Donlagic, Dali, Tesnjak, Sejid, Nonlinear Control Systems, Control Engineering Series, Marcel Dekker, New York USA, 2003.

2 Responses to Properties of Nonlinear System (Related to Theorem)

  1. Assalaamu’alaikum…

    bagus nih pa….

    hmm.. belajar fisika On Line

    semoga lancar kang S2 nya,… mudah2an Qita2 senior Fisika Unpad banyak yang S2 juga…

    nuhun

  2. Irwan Purnama says:

    @Titisan Qolbu
    Wa’alaykum salam,
    bukan belajar fisika online, tapi matematika online:-))
    sekedar catatan kuliah aja kok…biar ga lupa..
    jangan patah semangat buat S-2 juga….ditunggu di Taipei…..

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