Majorization of ODEs: comparison of solutions

Majorization of ODE Systems: Comparing Solutions via Right-Hand Sides

In the analysis of dynamical systems, exact solutions to ODEs are often unattainable. The majorization theorem allows us to estimate the behavior of a "complex" system through the solution of a "simple" one with known properties. If the right-hand side and initial conditions of one system are component-wise greater than another, then the solution trajectories preserve this inequality.

For vectors $x=(x^1,\ldots,x^m)$, $y=(y^1,\ldots,y^m) \in \mathbb{R}^m$, denote $x \le y$ if $x^i \le y^i$ for all $i=1,\dots,m$. The cube $[x,y]:=\{z\in\mathbb{R}^m\mid x\le z\le y\}$.

Consider a domain $D \subset \mathbb{R}^m$ such that for $x,y\in D$, $x\le y$ implies $[x,y]\subset D$. Functions $f_1,f_2\in C(I\times D,\mathbb{R}^m)$ with continuous Jacobians $\frac{\partial f_i}{\partial x}\in C(I\times D,\mathbb{R}^{m\times m})$, where $I=[t_0,t_1]$.

Two Cauchy problems:

$\dot x = f_i(t,x)$, $x(t_0)=\hat x_i$, $i=1,2$.

Formulation of the Main Theorem

Let $f_1(t,x)\le f_2(t,x)$ for all $(t,x)\in I\times D$ and $\hat x_1\le\hat x_2$. There exist diagonal matrices $U_i(t,x)$ with non-negative elements and matrices $B_i(t,x)$ with non-negative elements:

$\frac{\partial f_i}{\partial x}=U_i + B_i$, $i=1,2$.

Statement: If $x_i(t)$ are solutions to the Cauchy problems on $I$, then $x_1(t)\le x_2(t)$ for all $t\in I$.

This property allows us to majorize an unsolvable system with a simple one whose properties are known.

Proof Structure

Introduce a homotopy $y=y(t,s)$ for $s\in[0,1]$:

$y_t=(1-s)f_1(t,y)+s f_2(t,y)$, $y(t_0,s)=(1-s)\hat x_1 + s\hat x_2$.

At $s=0$, we get $x_1(t)$; at $s=1$, we get $x_2(t)$. It suffices to show that $s\mapsto y(t,s)$ is non-decreasing for fixed $t$, i.e., $y_s\ge 0$.

Differentiation with Respect to Parameter

Differentiate with respect to $s$:

$$\begin{aligned} y_{st} &= f_2 - f_1 + (u + b)y_s, \\ u(t,s) &= sU_2(t, y(t,s)) + (1-s)U_1(t, y(t,s)), \\ b(t,s) &= sB_2(t, y(t,s)) + (1-s)B_1(t, y(t,s)), \\ y_s|_{t=t_0} &= \hat x_2 - \hat x_1 \ge 0.\end{aligned}$$

Monotonicity Region

$M=A=\{t\in I\mid y_s(\xi,s)\ge0,\ \xi\in[t_0,t],\ s\in[0,1]\}$. The set is non-empty ($t_0\in A$), closed. Let $\tau=\sup A$.

Assume $\tau<t_1$. In integral form:

$$y_s(t,s)=e^{\int_\tau^t u\,d\lambda}y_s(\tau,s)+\int_\tau^t e^{\int_\xi^t u\,d\lambda}(f_2-f_1+by_s)\,d\xi=:\Phi(y_s).$$

Contraction Method

By the contraction mapping principle, the equation $\Phi(y_s)=y_s$ is solved for small $t-\tau>0$. Zero approximation $Y_0=e^{\int_\tau^t u\,d\lambda}y_s(\tau,s)\ge0$. Inductively $Y_{k+1}=\Phi(Y_k)\ge0$, since the operator $\Phi$ preserves non-negativity. The limit $y_s\ge0$ on $[\tau,\tau+\varepsilon]\times[0,1]$, contradicting the definition of $\tau$.

Application: Gronwall-Bellman Inequality

For non-negative $w(t)\le C+\int_{t_0}^t v(\xi)w(\xi)\,d\xi$, $v\ge0$.

Auxiliary $\phi(t)=C+\int_{t_0}^t v w\,d\xi$, $w\le\phi$, $\dot\phi=vw\le v\phi$.

System 1: $\dot x=f_1(t,x):=vx-(v\phi-\dot\phi)$, $x(t_0)=C$. The parentheses are non-negative.

Majorizing: $\dot x=f_2(t,x):=vx$, $x(t_0)=C$, solution $x(t)=Ce^{\int_{t_0}^tv\,d\xi}$.

By the theorem $w\le\phi\le Ce^{\int_{t_0}^tv\,d\xi}$.

Key Points:

• Majorization works when $f_1\le f_2$ and $\hat x_1\le\hat x_2$.
• Jacobians decompose into diagonal $U$ (non-negative) and $B$ (non-negative).
• Proof via homotopy and contraction method.
• Direct connection to Gronwall's inequality for integral estimates.
• Application: estimating solutions of nonlinear ODEs with simple exponentials.

Conclusion

The theorem provides a tool for qualitative analysis of dynamics without explicit integration. Conditions on the Jacobians ensure monotonicity of the homotopy, which is critical for preserving the inequality.

— Editorial Team