###### Abstract
[25; 26] showed how equilibrium stability conditions can be embedded as a constraint in a day-to-day signal setting-route choice framework and demonstrated the applicability of the resulting model to a realistic size network. [PERSON] [9] showed that the frequency of signal updating may significantly affect the duration of the day-to-day dynamic process needed to achieve a network flow-control equilibrium and provided examples of bifurcations arising as a result of model parameters crossing certain critical thresholds. [PERSON] and [PERSON] [10] investigated the behavior of a joint route choice-signal control dynamical system and illustrated the computation of fixed points through a simple three-link network example. They also demonstrated how the knowledge of the attraction domains of different fixed points can be instrumental in defining an appropriate adaptive signal control scheme. [PERSON] [27] proposed a dynamical system model of the day-to-day evolution of route flows, bottleneck delays, and signal green times based on the proportional switch adjustment process [12] and on an extension of the so-called \(P_{0}\) control policy [28; 29]. He showed that, under the assumption of vertical queuing, the dynamical system is convergent and network capacity is maximized. [PERSON] and [PERSON] [30] analyzed the stability of a day-to-day dynamic model of route choice and signal control based on flow swaps between paired alternative segments [31] and used red times as signal control variables. [PERSON] et al. [32] studied the properties of some \"pressure driven\" responsive control policies within dynamical models of route choice and signal control, while [PERSON] et al. [33] demonstrated the potential of a coordinated implementation of signal control and pricing for eliminating queues and maximizing network throughput. [PERSON] et al. [34] proposed a discrete time optimal control formulation of the day-to-day signal retiming problem for single-destination networks using a traffic flow evolution model based on splitting rates at nodes [35]. [PERSON] et al. [36] applied adaptive signal control in combination with day-to-day route choice dynamics and deep reinforcement learning as a traffic management strategy aimed at improving the resilience of urban road networks under disruptions in emergency situations. Among the many modeling and implementation aspects of the day-to-day dynamics of route choice and signal control, the issue of equilibrium stability deserves special attention. A stable network equilibrium represents a consistent reference state, whose prediction can provide a sound and reliable basis for design and management activities. The relevance of this concept provides the motivation for the work described in this paper, whose main aim is to show how a properly calibrated signal control policy can be used, in a day-to-day dynamic framework, to protect the system from instabilities triggered by other factors and conditions. The remainder of the paper is organized as follows. In Section 2, the general formulation of the dynamic process model describing the joint evolution of route choices and signal settings is presented, and explicit fixed-point stability conditions are derived for a two-link network. The results of numerical tests of the proposed model are illustrated in Section 3 for a two-link network and in Section 4 for a grid network. Concluding remarks and suggestions for possible developments of the proposed approach are offered in Section 5.