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Register a new userHow to manage the post pandemic opening? A Pontryagin Maximum Principle approach
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R. Mansilla
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The COVID-19 pandemic has completely disrupted the operation of our societies. Its elusive transmission process, characterized by an unusually long incubation period, as well as a high contagion capacity, has forced many countries to take quarantine and social isolation measures that conspire against the performance of national economies. This situation confronts decision makers in different countries with the alternative of reopening the economies, thus facing the unpredictable cost of a rebound of the infection. This work tries to offer an initial theoretical framework to handle this alternative.
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This paper gives a brief contact-geometric account of the Pontryagin maximum principle. We show that key notions in the Pontryagin maximum principle---such as the separating hyperplanes, costate, necessary condition, and normal/abnormal minimizers---have natural contact-geometric interpretations. We then exploit the contact-geometric formulation to give a simple derivation of the transversality condition for optimal control with terminal cost.
We consider distributed-order non-local fractional optimal control problems with controls taking values on a closed set and prove a strong necessary optimality condition of Pontryagin type. The possibility that admissible controls are subject to pointwise constraints is new and requires more sophisticated techniques to include a maximality condition. We start by proving results on continuity of solutions due to needle-like control perturbations. Then, we derive a differentiability result on the state solutions with respect to the perturbed trajectories. We end by stating and proving the Pontryagin maximum principle for distributed-order fractional optimal control problems, illustrating its applicability with an example.
In this paper we consider a measure-theoretical formulation of the training of NeurODEs in the form of a mean-field optimal control with $L^2$-regularization of the control. We derive first order optimality conditions for the NeurODE training problem in the form of a mean-field maximum principle, and show that it admits a unique control solution, which is Lipschitz continuous in time. As a consequence of this uniqueness property, the mean-field maximum principle also provides a strong quantitative generalization error for finite sample approximations. Our derivation of the mean-field maximum principle is much simpler than the ones currently available in the literature for mean-field optimal control problems, and is based on a generalized Lagrange multiplier theorem on convex sets of spaces of measures. The latter is also new, and can be considered as a result of independent interest.
Natural and social multivariate systems are commonly studied through sets of simultaneous and time-spaced measurements of the observables that drive their dynamics, i.e., through sets of time series. Typically, this is done via hypothesis testing: the statistical properties of the empirical time series are tested against those expected under a suitable null hypothesis. This is a very challenging task in complex interacting systems, where statistical stability is often poor due to lack of stationarity and ergodicity. Here, we describe an unsupervised, data-driven framework to perform hypothesis testing in such situations. This consists of a statistical mechanical approach - analogous to the configuration model for networked systems - for ensembles of time series designed to preserve, on average, some of the statistical properties observed on an empirical set of time series. We showcase its possible applications with a case study on financial portfolio selection.
The benefits, both in terms of productivity and public health, are investigated for different levels of engagement with the test, trace and isolate procedures in the context of a pandemic in which there is little or no herd immunity. Simple mathematical modelling is used in the context of a single, relatively closed workplace such as a factory or back-office where, in normal operation, each worker has lengthy interactions with a fixed set of colleagues. A discrete-time SEIR model on a fixed interaction graph is simulated with parameters that are motivated by the recent COVID-19 pandemic in the UK during a post-peak phase, including a small risk of viral infection from outside the working environment. Two kinds of worker are assumed, transparents who regularly test, share their results with colleagues and isolate as soon as a contact tests positive for the disease, and opaques who do none of these. Moreover, the simulations are constructed as a ``playable model in which the transparency level, disease parameters and mean interaction degree can be varied by the user. The model is analysed in the continuum limit. All simulations point to the double benefit of transparency in maximising productivity and minimising overall infection rates. Based on these findings, public policy implications are discussed on how to incentivise this mutually beneficial behaviour in different kinds of workplace, and simple recommendations are made.
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