Consider \(x_i'=F_i(x_1,\dots,x_n)\) with \(F:\mathbb R^n\to\mathbb R^n\) of class \(C^1\).
If \(\langle DF(x)z,z\rangle\le0\) for all \(x,z\), show that \(\|U(t)-V(t)\|^2\) is decreasing for any two solutions \(U,V\).
If \(W(t)\) is a solution for \(t>0\) and \(\langle DF(x)z,z\rangle\le-\|z\|^2\), show that \(W(t)\) converges to some \(C\in\mathbb R^n\) as \(t\to\infty\).
Solution
1. Express the difference \(F(U)-F(V)\) by integrating the derivative of \(F\) along a segment.
Fix a time at which two solutions \(U,V\) are defined, and put \[ Z=U-V. \] For \[ \Phi(s)=F(V+sZ), \qquad 0\le s\le1, \] the fundamental theorem of calculus gives \[ F(U)-F(V) =\int_0^1 DF(V+sZ)Z\,ds. \]
2. Under the hypothesis of part (a), the squared distance between two solutions is nonincreasing.
Since \[ Z'=U'-V'=F(U)-F(V), \] step 1 gives \[ \begin{aligned} \frac d{dt}\|Z\|^2 &=2\langle Z',Z\rangle\\ &=2\int_0^1 \left\langle DF(V+sZ)Z,Z\right\rangle ds. \end{aligned} \] If \[ \langle DF(x)z,z\rangle\le0 \] for all \(x,z\), every integrand is nonpositive. Hence \[ \frac d{dt}\|U(t)-V(t)\|^2\le0. \] Thus \(\|U(t)-V(t)\|^2\) is nonincreasing, proving part (a).
3. Under the stronger hypothesis, the velocity of a solution decays exponentially.
Assume now \[ \langle DF(x)z,z\rangle\le-\|z\|^2. \] Because \(F\) is \(C^1\) and \(W'=F(W)\), the solution \(W\) is \(C^2\) and \[ W''(t)=DF(W(t))W'(t). \] Therefore \[ \begin{aligned} \frac d{dt}\|W'(t)\|^2 &=2\langle W''(t),W'(t)\rangle\\ &=2\left\langle DF(W(t))W'(t),W'(t)\right\rangle\\ &\le-2\|W'(t)\|^2. \end{aligned} \] Fix \(t_0>0\). Multiplying by \(e^{2t}\), or integrating this differential inequality, gives \[ \|W'(t)\|^2 \le e^{-2(t-t_0)}\|W'(t_0)\|^2 \] for every \(t\ge t_0\). Hence \[ \|W'(t)\| \le e^{-(t-t_0)}\|W'(t_0)\|. \]
4. The trajectory is Cauchy and therefore converges.
If \(s>t\ge t_0\), then step 3 yields \[ \begin{aligned} \|W(s)-W(t)\| &\le\int_t^s\|W'(r)\|\,dr\\ &\le\|W'(t_0)\|\int_t^s e^{-(r-t_0)}\,dr\\ &\le\|W'(t_0)\|e^{-(t-t_0)}. \end{aligned} \] The final expression tends to \(0\) as \(t\to\infty\), independently of \(s>t\). Thus \(W(t)\) is a Cauchy trajectory at infinity. Since \(\mathbb R^n\) is complete, there is some \[ C\in\mathbb R^n \] such that \[ \boxed{W(t)\longrightarrow C\qquad(t\to\infty).} \]