# A Game-Theoretic Rent-Seeking Framework for Improving Multipath TCP Performance +
## 3 Modeling Framework
### Diagonal Strict Concavity
#### 3.3.3 Stability of the Dynamical System  
We make use of a Lyapunov function for Lasalle's invariance principle to prove the global asymptotic stability of the dynamical system. Let \(\partial\mathbf{p}:=\mathbf{p}-\mathbf{p}^{*}\) and \(\partial\mathbf{\lambda}:=\mathbf{\lambda}-\mathbf{\lambda}^{*}\). The function  
\[L(\mathbf{\lambda},\mathbf{p})=\sum_{i}\Big{(}\int_{\lambda_{i}^{*}}^{\lambda_{i}} \big{(}\frac{z-\lambda_{i}^{*}}{k_{i}(z)}\big{)}\text{d}z+\gamma_{i}\partial p _{i}^{2}\Big{)} \tag{18}\]  
is a Lyapunov function. To prove Lemma 2, we extend the ideas from [23], and therefore, we include the essential steps only.  
Proof.: Differentiating \(L(\mathbf{x},\mathbf{p})\) with respect to \(t\),  
\[\dot{L}(\mathbf{\lambda},\mathbf{p})=\sum_{i}\Big{(}\frac{\partial\lambda_{i}\dot{ \lambda}_{i}}{\dot{k}_{i}(\lambda_{i})}+\gamma_{i}\partial p_{i}\dot{p}_{i}\Big{)}.\]  
If \(\partial\mathbf{\lambda}\
eq 0\), then, using (4),  
\[\frac{\partial\lambda_{i}\dot{\lambda}_{i}}{\dot{k}_{i}(\lambda_{ i})} = \partial\lambda_{i}(\phi_{i}(\mathbf{\lambda})-p_{i})_{\lambda_{i}}^{+} \leq \partial\lambda_{i}(\phi_{i}(\mathbf{\lambda})-p_{i}) \tag{19}\] \[= \partial\lambda_{i}(\phi_{i}(\mathbf{\lambda})-\phi_{i}(\mathbf{\lambda} ^{\star})-\partial p_{i}).\]  
Furthermore,  
\[\sum_{i}\Big{(}\partial\lambda_{i}\dot{\lambda}_{i}/\dot{k}_{i}( \lambda_{i})\Big{)} \tag{20}\] \[\leq \partial\mathbf{\lambda}^{T}(\mathbf{\phi}(\mathbf{\lambda})-\mathbf{\phi}(\mathbf{ \lambda}^{\star}))-\partial\mathbf{\lambda}^{T}\partial\mathbf{p}<-\partial\mathbf{ \lambda}^{T}\partial\mathbf{p}.\]  
Similarly,  
\[\sum_{i}\gamma_{i}\partial p_{i}\dot{p}_{i}= \sum_{i}\partial p_{i}(\lambda_{i}-\theta_{i})_{p_{i}}^{+}\leq \sum_{i}\partial p_{i}(\lambda_{i}-\theta_{i}) \tag{21}\] \[\leq \sum_{i}\partial p_{i}\partial\lambda_{i}=\partial\mathbf{p}^{T} \partial\mathbf{\lambda}.\]  
Now,  
\[\text{if }\partial\mathbf{\lambda}\
eq 0\text{, then }\dot{L}(\mathbf{ \lambda},\mathbf{p})<\partial\mathbf{p}^{T}\partial\mathbf{\lambda}-\partial\mathbf{\lambda}^{T }\partial\mathbf{p}=0;\] \[\text{if }\partial\mathbf{\lambda}=0\text{, then }\dot{L}(\mathbf{ \lambda},\mathbf{p})=0. \tag{22}\]  
Therefore, \(\dot{L}(\mathbf{\lambda},\mathbf{p})\leq 0\) and \(L(\mathbf{\lambda},\mathbf{p})\)_is indeed a Lyapunov function_.  
Let \(\Gamma:=(\mathbf{\lambda}(t),\mathbf{p}(t))\) be a set of trajectories on which \(\dot{L}(\mathbf{\lambda}(t),\mathbf{p}(t))=0\) for all \(t\geq 0\). The only trajectory in \(\Gamma\) is the trivial trajectory \((\mathbf{\lambda},\mathbf{p})\equiv(\mathbf{\lambda}^{\star},\mathbf{p}^{\star})\). Therefore, [PERSON]'s invariance principle implies that \((\mathbf{\lambda}^{\star},\mathbf{p}^{\star})\) is globally asymptotically stable.  
Observe that \(\dot{L}\equiv 0\) implies \(\partial\mathbf{\lambda}\equiv 0\), i.e., any trajectory in \(\Gamma\) must have \(\mathbf{\lambda}(t)=\mathbf{\lambda}^{\star}\) for all \(t\geq 0\). Hence, \((\mathbf{\lambda},\mathbf{p})\equiv(\mathbf{\lambda}^{\star},\mathbf{p}^{\star})\) is the only trajectory in \(\Gamma\).