Consider \[ x'(t)=A(t)x(t), \] where \(A\) is a smooth real \(n\times n\) matrix-valued function. Assume that \[ \langle A(t)y,y\rangle\le c\|y\|^2 \] for all \(y\in\mathbb R^n\) and all \(t\), where \(c\in\mathbb R\) is fixed. Prove that every solution satisfies \[ \|x(t)\|\le e^{ct}\|x(0)\| \qquad(t>0). \]
Solution
1. The function\[ u(t)\coloneqq\norm{x(t)}^2 \]satisfies\[ u'(t)\le2c\,u(t). \]
Using the differential equation, \[ \begin{aligned} u'(t) &=2\langle x'(t),x(t)\rangle\\ &=2\langle A(t)x(t),x(t)\rangle\\ &\le2c\norm{x(t)}^2\\ &=2c\,u(t). \end{aligned} \]
2. The function\[ v(t)\coloneqq e^{-2ct}u(t) \]is nonincreasing for \(t\ge0\).
By step 1, \[ \begin{aligned} v'(t) &=e^{-2ct}\bigl(u'(t)-2c\,u(t)\bigr)\\ &\le0. \end{aligned} \]
3. For every \(t>0\),\[ \norm{x(t)}^2\le e^{2ct}\norm{x(0)}^2. \]
By step 2, \[ e^{-2ct}\norm{x(t)}^2=v(t)\le v(0)=\norm{x(0)}^2. \] Multiply by \(e^{2ct}>0\).
4. Therefore\[ \boxed{\norm{x(t)}\le e^{ct}\norm{x(0)}} \qquad(t>0). \]
Both sides in step 3 are nonnegative. Taking square roots gives the stated inequality.
5. Q.E.D.
Step 4 is the required estimate.