While most approaches in formal methods address system correctness, ensuring robustness has remained a challenge. In this article, we present and study the logic rLTL, which provides a means to formally reason about both correctness and robustness in system design. Furthermore, we identify a large fragment of rLTL for which the verification problem can be efficiently solved, i.e., verification can be done by using an automaton, recognizing the behaviors described by the rLTL formula φ, of size at most O(3 |φ |), where |φ | is the length of φ. This result improves upon the previously known bound of O(5|φ |) for rLTL verification and is closer to the LTL bound of O(2|φ |). The usefulness of this fragment is demonstrated by a number of case studies showing its practical significance in terms of expressiveness, the ability to describe robustness, and the fine-grained information that rLTL brings to the process of system verification. Moreover, these advantages come at a low computational overhead with respect to LTL verification.

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This paper focuses on the possibility that financial markets require risk premia on holding sovereign debt of countries that appear vulnerable from a fiscal sustainability perspective. Both the level of debt as well as the rate of change of debt are assumed to impact on the risk premium. We analyze the impact of such an endogenous risk premium in a simple debt game between a monetary and a fiscal player, as introduced by [Tabellini (1986) Journal of Economic Dynamics and Control 10, 427–442]. The risk premium term adds a nonlinearity to the linear model in case risk premia are absent. We analyze outcomes in case of noncooperative open-loop Nash strategies and in case of cooperative strategies and consider the workings of the risk premium as a market-based disciplining device (in case of high debt) and adjustment rewarding device (in case of a declining debt trajectory).

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In this paper we revisit the problem of computing robust controlled invariant sets for discrete-time linear systems. The key idea is that by considering controllers that exhibit eventually periodic behavior, we obtain a closed-form expression for an implicit representation of a robust controlled invariant set in the space of states and finite input sequences. Due to the derived closed-form expression, our method is suitable for high dimensional systems. Optionally, one obtains an explicit robust controlled invariant set by projecting the implicit representation to the original state space. The proposed method is complete in the absence of disturbances, with a weak completeness result established when disturbances are present. Moreover, we show that a specific controller choice yields a hierarchy of robust controlled invariant sets. To validate the proposed method, we present thorough case studies illustrating that in safety-critical scenarios the implicit representation suffices in place of the explicit invariant set.