## 2 Method  
### General Formulation of the Problem  
We consider a road network in which intersections are controlled by adaptive traffic signals. The day-to-day evolution of states of such a network can be described by means of a discrete time, deterministic process model formulated in terms of link variables as follows [8]:  
\[\mathbf{C}^{t}=\beta\mathbf{K}\Big{(}\mathbf{F}^{t-1},\mathbf{G}^{t-1}\Big{)}+(1-\beta)\mathbf{C} ^{t-1} \tag{1}\]  
\[\mathbf{F}^{t}=\alpha\mathbf{S}\big{(}\mathbf{C}^{t}\big{)}+(1-\alpha)\mathbf{F}^{t-1} \tag{2}\]  
\[\mathbf{G}^{t}=\mathbf{H}\big{(}\mathbf{F}^{t}\big{)} \tag{3}\]  
where  
\(t\) represents the generic period (\"day\") of the dynamic process;  
\(\mathbf{C}^{t}\) is the vector of mean perceived link costs at the start of day \(t\);  
\(\mathbf{F}^{t}\) is the vector of link flows on day \(t\);  
\(\mathbf{G}^{t}\) is the vector of signal settings on day \(t\);  
\(\mathbf{K}\) is the vector representing average link costs actually experienced by users as a function of link flows and signal settings;  
\(\mathbf{S}\) is the vector function representing the relationship between link flows and mean perceived link costs based on an assumed probabilistic route choice model;  
\(\mathbf{H}\) is the vector function representing the relationship between signal settings and link flows based on an assumed signal control policy;  
\(\beta\) (\(0<\beta\leq 1\)) is a parameter quantifying the extent to which the most recent travel experience (\"yesterday's trip\") contributes to the formation of the cost perceived by users at the start of day \(t\);  
\(\alpha\) (\(0<\alpha\leq 1\)) is a parameter representing the fraction of travelers who actually reconsider their previous day's choice. Note that parameter \(\beta\) controls the rate at which perceived costs are updated as a consequence of the previous day's experience, and hence it allows us to represent the effect of memory and learning on drivers' route choice behavior. In the limit, when \(\beta=1\), travelers are \"memoryless\", as they base their next choice solely on the most recently experienced costs. On the other hand, parameter \(\alpha\) controls the extent to which route choices are adjusted in response to updated costs. It allows us to represent the effect of habit and inertia on drivers' behavior, as \((1-\alpha)\) represents the fraction of users who repeat their previous day's choice, regardless of the cost actually experienced. Note that when \(\alpha=1\), all drivers are \"inertia-free\", as they reconsider their route choices on a daily basis without any habit effect. Substituting Equation (1) for \(\mathbf{C}^{t}\) in Equation (2) and expressing the signal settings through the signal control policy (3), model (1)-(3) can be reformulated as the following nonlinear dynamical system:  
\[\mathbf{C}^{t}=\beta\mathbf{K}\Big{[}\mathbf{F}^{t-1},\mathbf{H}\Big{(}\mathbf{F}^{t-1}\Big{)}\Big{]} +(1-\beta)\mathbf{C}^{t-1} \tag{4}\]  
\[\mathbf{F}^{t}=\alpha\mathbf{S}\Big{\{}\beta\mathbf{K}\Big{[}\mathbf{F}^{t-1},\mathbf{H}\Big{(}\bm {F}^{t-1}\Big{)}\Big{]}+(1-\beta)\mathbf{C}^{t-1}\Big{\}}+(1-\alpha)\mathbf{F}^{t-1} \tag{5}\]  
whose fixed points \((\widetilde{\mathbf{C}},\widetilde{\mathbf{F}})\) are found by setting  
\[\mathbf{C}^{t}=\mathbf{C}^{t-1}=\widetilde{\mathbf{C}} \tag{6}\]  
\[\mathbf{F}^{t}=\mathbf{F}^{t-1}=\widetilde{\mathbf{F}} \tag{7}\]  
which yields  
\[\widetilde{\mathbf{C}}=\mathbf{K}\Big{[}\widetilde{\mathbf{F}},\mathbf{H}\widetilde{\mathbf{F}} \Big{]} \tag{8}\]  
\[\widetilde{\mathbf{F}}=\mathbf{S}(\widetilde{\mathbf{C}}) \tag{9}\]  
and the fixed-point signal settings are given by  
\[\widetilde{\mathbf{G}}=\mathbf{H}(\widetilde{\mathbf{F}}) \tag{10}\]  
According to a basic result of nonlinear dynamical systems theory, a fixed point \((\widetilde{\mathbf{C}},\widetilde{\mathbf{F}})\) is locally stable if all eigenvalues of the Jacobian matrix of (4) and (5), evaluated at \((\widetilde{\mathbf{C}},\widetilde{\mathbf{F}})\), have a modulus smaller than 1.