
Addison, B. C., Wright, D. J., Nicholson, B. A., Cale, B., Mocnik, T., Huber, D., et al. (2021). TOI257b (HD 19916b): a warm subsaturn orbiting an evolved Ftype star. Mon. Not. Roy. Astron. Soc., 502(3), 3704–3722.
Abstract: We report the discovery of a warm subSaturn, TOI257b (HD 19916b), based on data from NASA's Transiting Exoplanet Survey Satellite (TESS). The transit signal was detected by TESS and confirmed to be of planetary origin based on radial velocity observations. An analysis of the TESS photometry, the MINERVAAustralis, FEROS, and HARPS radial velocities, and the asteroseismic data of the stellar oscillations reveals that TOI257b has a mass of MP = 0.138 +/ 0.023M(J) (43.9 +/ 7.3 Mcircle plus), a radius of RP = 0.639 +/ 0.013 RJ (7.16 +/ 0.15 Rcircle plus), bulk density of 0.65(0.11)(+0.12) (cgs), and period 18.38818(0.00084)(+0.00085) days. TOI257b orbits a bright (V = 7.612 mag) somewhat evolved late Ftype star with M* = 1.390 +/ 0.046(Msun), R* = 1.888 +/ 0.033 Rsun, Teff = 6075 +/ 90 K, and vsin i = 11.3 +/ 0.5 kms(1). Additionally, we find hints for a second nontransiting subSaturn mass planet on a similar to 71 day orbit using the radial velocity data. This system joins the ranks of a small number of exoplanet host stars (similar to 100) that have been characterized with asteroseismology. Warm subSaturns are rare in the known sample of exoplanets, and thus the discovery of TOI257b is important in the context of future work studying the formation and migration history of similar planetary systems.



Aledo, J. A., Goles, E., MontalvaMedel, M., Montealegre, P., & Valverde, J. C. (2023). Symmetrizable Boolean networks. Inf. Sci., 626, 787–804.
Abstract: In this work, we provide a procedure that allows us to transform certain kinds of deterministic Boolean networks on minterm or maxterm functions into symmetric ones, so inferring that such symmetrizable networks can present only periodic points of periods 1 or 2. In particular, we deal with generalized parallel (or synchronous) dynamical systems (GPDS) over undirected graphs, i. e., discrete parallel dynamical systems over undirected graphs where some of the selfloops may not appear. We also study the class of antisymmetric GPDS (which are nonsymmetrizable), proving that their periodic orbits have period 4. In addition, we introduce a class of nonsymmetrizable systems which admit periodic orbits with arbitrary large periods.



Almenara, J. M., Bonfils, X., Bryant, E. M., Jordan, A., Hebrard, G., Martioli, E., et al. (2024). TOI4860 b, a shortperiod giant planet transiting an M3.5 dwarf. Astron. Astrophys., 683, A166.
Abstract: We report the discovery and characterisation of a giant transiting planet orbiting a nearby M3.5V dwarf (d = 80.4pc, G = 15.1 mag, K=11.2mag, R* = 0.358 +/ 0.015 Rcircle dot, M* = 0.340 +/ 0.009 Mcircle dot). Using the photometric time series from TESS sectors 10, 36, 46, and 63 and nearinfrared spectrophotometry from ExTrA, we measured a planetary radius of 0.77 +/ 0.03 RJ and an orbital period of 1.52 days. With highresolution spectroscopy taken by the CFHT/SPIRou and ESO/ESPRESSO spectrographs, we refined the host star parameters ([Fe/H] = 0.27 +/ 0.12) and measured the mass of the planet (0.273 +/ 0.006 MJ). Based on these measurements, TOI4860 b joins the small set of massive planets (>80 ME) found around mid to late M dwarfs (<0.4 Rcircle dot), providing both an interesting challenge to planet formation theory and a favourable target for further atmospheric studies with transmission spectroscopy. We identified an additional signal in the radial velocity data that we attribute to an eccentric planet candidate (e = 0.66 +/ 0.09) with an orbital period of 427 +/ 7 days and a minimum mass of 1.66 +/ 0.26 MJ, but additional data would be needed to confirm this.



AlvarezMiranda, E., CamposValdes, C., Quiroga, M. M., MorenoFaguett, M., & Pereira, J. (2020). A MultiCriteria Pen for Drawing Fair Districts: When Democratic and Demographic Fairness Matter. Mathematics, 8(9), 27 pp.
Abstract: Electoral systems are modified by individuals who have incentives to bias the rules for their political advantage (i.e., gerrymandering). To prevent gerrymandering, legislative institutions can rely on mathematical tools to guarantee democratic fairness and territorial contiguity. These tools have been successfully used in the past; however, there is a need to accommodate additional meanings of the term fairness within the electoral systems of modern democracies. In this paper, we present an optimization framework that considers multiple criteria for drawing districts and assigning the number of representatives. Besides some typical districting criteria (malapportionment and contiguity), we introduce novel criteria for ensuring territorial equilibrium and incentives for candidates to deploy their representation efforts fairly during their campaign and period in office. We test the method, which we denote as Multicriteria Pen, in a recent and a forthcoming reform of the Chilean electoral system. The results show the potential of our tool to improve the current territorial design and offers insights on the motivations, objectives, and deficiencies of both reform plans.



Antilen, J., Casassus, S., Cieza, L. A., & GonzalezRuilova, C. (2023). Gas distribution in ODISEA sources from ALMA longbaseline observations in (CO)C12(21). Mon. Not. Roy. Astron. Soc., 522(2), 2611–2627.
Abstract: The (CO)C12 rotational lines in protoplanetary discs are good tracers of the total spatial extension of the gas component, and potentially planetdisc interactions. We present ALMA long baseline observations of the (CO)C12(21) line of 10 protoplanetary discs from the Ophiuchus DIsc Survey Employing ALMA (ODISEA) project, aiming to set constraints on the gas distribution of these sources. The position angle of the gaseous disc can be inferred for five sources using highvelocity channels, which trace the gas in the inner part of the disc. We compare the highvelocity PAs to the orientations inferred from the continuum, representative of the orientation over similar to 53 to 256 au in these resolved discs. We find a significant difference in orientation for DoAr 44, which is evidence of a tilted inner disc. Eight discs show evidence of gas inside inner dust cavities or gaps, and the disc of ISOOph 196 is not detected in (CO)C12(21), except for the compact signal located inside its dust cavity. Our observations also point out a possible outflow in WLY 263.



AstudilloDefru, N., Cloutier, R., Wang, S. X., Teske, J., Brahm, R., Hellier, C., et al. (2020). A hot terrestrial planet orbiting the bright M dwarf L 1689 unveiled by TESS. Astron. Astrophys., 636, 13 pp.
Abstract: We report the detection of a transiting superEarthsized planet (R = 1.39 +/ 0.09 Rcircle plus) in a 1.4day orbit around L 1689 (TOI134), a bright M1V dwarf (V = 11, K = 7.1) located at 25.15 +/ 0.02 pc. The host star was observed in the first sector of the Transiting Exoplanet Survey Satellite (TESS) mission. For confirmation and planet mass measurement purposes, this was followed up with groundbased photometry, seeinglimited and highresolution imaging, and precise radial velocity (PRV) observations using the HARPS and Magellan/PFS spectrographs. By combining the TESS data and PRV observations, we find the mass of L 1689 b to be 4.60 +/ 0.56 Mcircle plus and thus the bulk density to be 1.74(0.33)(+0.44) times higher than that of the Earth. The orbital eccentricity is smaller than 0.21 (95% confidence). This planet is a level one candidate for the TESS mission's scientific objective of measuring the masses of 50 small planets, and it is one of the most observationally accessible terrestrial planets for future atmospheric characterization.



Attard, M., & Balbontin, C. (2024). Workshop 6 report: Micromobility movement in urban transport. Res. Transp. Econ., 103, 101399.
Abstract: The theme of micromobility was introduced for the first time in Thredbo 17 as the growth of shared and privatelyowned escooters, bicycles and ebicycles continue to affect the nature and structure of urban transport systems worldwide. And whilst in some cases they challenge the priority afforded to the private car, in others they complement already existing and wellestablished greener transport modes such as cycling and walking. The discussion in this workshop focused on a number of questions looking at the benefits of micromobility and discussing the main incentives for their use as an urban mode of transport, questioning the role of government and describing the potential threats, if any, to public transport systems, in what we expect for the future of micromobility. Five papers showcased evidence on the use of micromobility, from escooters in Norway and Australia, to public bicycles in South Korea and The Netherlands. Each of the questions discussed in the workshop is reported in this paper. The need for multimodal integration is evident and remains essential to ensure complementarity across transport in cities. However, other concerns such as the need for regulation, education/ enforcement structures, stronger business models and more effective tendering procedures have been identified and discussed. A long list of future research topics in the area of micromobility is provided and some themes for Thredbo 18 are recommended.



Barrera, J., Carrasco, R. A., & Moreno, E. (2020). Realtime fleet management decision support system with security constraints. TOP, 28(3), 728–748.
Abstract: Intelligent transportation, and in particular, fleet management, has been a forefront concern for a plethora of industries. This statement is especially true for the production of commodities, where transportation represents a central element for operational continuity. Additionally, in many industries, and in particular those with hazardous environments, fleet control must satisfy a wide range of security restrictions to ensure that risks are kept at bay and accidents are minimum. Furthermore, in these environments, any decision support tool must cope with noisy and incomplete data and give recommendations every few minutes. In this work, a fast and efficient decision support tool is presented to help fleet managers oversee and control ore trucks, in a mining setting. The main objective of this system is to help managers avoid interactions between ore trucks and personnel buses, one of the most critical security constraints in our case study, keeping a minimum security distance between the two at all times. Furthermore, additional algorithms are developed and implemented, so that this approach can work with reallife noisy GPS data. Through the use of historical data, the performance of this decision support system is studied, validating that it works under the reallife conditions presented by the company. The experimental results show that the proposed approach improved truck and road utilization significantly while allowing the fleet manager to control the security distance required by their procedures.



Benavides, C., Diaz, M., O' Ryan, R., Gwinner, S., & Sierra, E. (2021). Methodology to analyse the impact of an emissions trading system in Chile. Clim. Policy, 21(8), 1099–1110.
Abstract: In the context of updating the 2015 Nationally Determined Contribution (NDC), the government of Chile has updated its estimates of compliance costs for a series of mitigation actions with an emphasis on the energy sector as the main source of its greenhouse gas emissions. Using the information developed in this process, we assess the impact on compliance costs of increasing the flexibility for sources by introducing different emissions trading schemes. For this we develop a detailed optimization model that represents the operational and investment decisions that could be taken by the energy generation, industrial and mining sectors if an Emissions Trading System (ETS) was implemented. An ETS with two cap and trade options is analysed together with an offset mechanism for sources not included in the ETS. Also, two policy goals are considered: a stringent 76% sectoral reduction goal in 2050 similar to Chile's current strict NDC, and a more lax 46% goal similar to Chile's initial 2015 NDC proposal. The results show that (i) cost reductions from increased flexibility for Chile's current strict NDC are significant, and that offsets can play an important role; (ii) the stringency of the reduction goal affects the magnitude of the cost savings related to flexibility and, surprisingly, total abatement costs are negative (i.e. there are benefits) for the 46% reduction goal. In this latter case, the most significant cost reductions result from compelling firms to comply with their allowances in each sector, not increased flexibility. These results highlight the policy relevance of case by case analysis using a modelling approach similar to the one we develop here. Key policy insights ETS implementation can help Chile meet its mitigation commitment for 2050. The compliance costs can vary significantly depending on the flexibility implemented in the emissions trading schemes. Optimization models can help decisionmakers define the attributes of an ETS, such as the sectors that should participate, the cap, and the percentage of offsets. The proposed methodology also highlights and quantifies the offsets that can be acquired from sectors that are not part of an ETS, such as forestry, agriculture, and the waste sector. The possibility to acquire of offsets could reduce significantly the cost for industries that participate of an ETS.



Bustamante, M., & Contreras, M. (2016). Multiasset BlackScholes model as a variable second class constrained dynamical system. Physica A, 457, 540–572.
Abstract: In this paper, we study the multiasset BlackScholes model from a structural point of view. For this, we interpret the multiasset BlackScholes equation as a multidimensional Schrodinger one particle equation. The analysis of the classical Hamiltonian and Lagrangian mechanics associated with this quantum model implies that, in this system, the canonical momentums cannot always be written in terms of the velocities. This feature is a typical characteristic of the constrained system that appears in the highenergy physics. To study this model in the proper form, one must apply Dirac's method for constrained systems. The results of the Dirac's analysis indicate that in the correlation parameters space of the multi assets model, there exists a surface (called the Kummer surface Sigma(K), where the determinant of the correlation matrix is null) on which the constraint number can vary. We study in detail the cases with N = 2 and N = 3 assets. For these cases, we calculate the propagator of the multiasset BlackScholes equation and show that inside the Kummer Sigma(K) surface the propagator is well defined, but outside Sigma(K) the propagator diverges and the option price is not well defined. On Sigma(K) the propagator is obtained as a constrained path integral and their form depends on which region of the Kummer surface the correlation parameters lie. Thus, the multiasset BlackScholes model is an example of a variable constrained dynamical system, and it is a new and beautiful property that had not been previously observed. (C) 2016 Elsevier B.V. All rights reserved.



Canessa, E., & Chaigneau, S. (2015). Calibrating AgentBased Models Using a Genetic Algorithm. Stud. Inform. Control, 24(1), 79–90.
Abstract: We present a Genetic Algorithm (GA)based tool that calibrates Agentbased Models (ABMs). The GA searches through a userdefined set of input parameters of an ABM, delivering values for those parameters so that the output time series of an ABM may match the real system's time series to certain precision. Once that set of possible values has been available, then a domain expert can select among them, the ones that better make sense from a practical point of view and match the explanation of the phenomenon under study. In developing the GA, we have had three main goals in mind. First, the GA should be easily used by nonexpert computer users and allow the seamless integration of the GA with different ABMs. Secondly, the GA should achieve a relatively short convergence time, so that it may be practical to apply it to many situations, even if the corresponding ABMs exhibit complex dynamics. Thirdly, the GA should use a few data points of the real system's time series and even so, achieve a sufficiently good match with the ABM's time series to attaining relational equivalence between the real system under study and the ABM that models it. That feature is important since social science longitudinal studies commonly use few data points. The results show that all of those goals have been accomplished.



Canessa, G., Gallego, J. A., Ntaimo, L., & Pagnoncelli, B. K. (2019). An algorithm for binary linear chanceconstrained problems using IIS. Comput. Optim. Appl., 72(3), 589–608.
Abstract: We propose an algorithm based on infeasible irreducible subsystems to solve binary linear chanceconstrained problems with random technology matrix. By leveraging on the problem structure we are able to generate good quality upper bounds to the optimal value early in the algorithm, and the discrete domain is used to guide us efficiently in the search of solutions. We apply our methodology to individual and joint binary linear chanceconstrained problems, demonstrating the ability of our approach to solve those problems. Extensive numerical experiments show that, in some cases, the number of nodes explored by our algorithm is drastically reduced when compared to a commercial solver.



Carvallo, C., JalilVega, F., & Moreno, R. (2023). A multienergy multimicrogrid system planning model for decarbonisation and decontamination of isolated systems. Appl. Energy, 343, 121143.
Abstract: Decarbonising and decontaminating remote regions in the world presents several challenges. Many of these regions feature isolation, dispersed demand in large areas, and a lack of economic resources that impede the development of robust and sustainable networks. Furthermore, isolated systems in the developing world are mostly based on diesel generation for electricity, and firewood and liquefied petroleum gas for heating, as these options do not require a significant infrastructure cost. In this context, we present a stochastic multienergy multimicrogrid system planning model that integrates electricity, heat and hydrogen networks in isolated systems. The model is stochastic to capture uncertainty in renewable generation outputs, particularly hydro and wind, and thus design a multienergy system proved secured against such uncertainty. The model also features two distinct constraints to limit the emissions of CO2 (for decarbonisation) and particulate matter (for decontamination), and incorporates firewood as a heating source. Moreover, given that the focus is on lowvoltage networks, we introduce a fully linear AC power flow equations set, allowing the planning model to remain tractable. The model is applied to a realworld case study to design a multienergy multimicrogrid system in an isolated region in Chilean Patagonia. In a case with a zero limit over direct CO2 emissions, the total system's cost increases by 34% with respect to an unconstrained case. In a case with a zero limit over particulate matter emissions, the total system's cost increases by 189%. Finally, although an absolute zero limit over both, particulate matter and direct CO2 emissions, leads to a total system's cost increase of 650%, important benefits in terms of decarbonisation and decontamination can be achieved at marginal cost increments.



ClaveroLeon, C., Ruiz, D., Cillero, J., Orlando, J., & Gonzalez, B. (2021). The multi metalresistant bacterium Cupriavidus metallidurans CH34 affects growth and metal mobilization in Arabidopsis thaliana plants exposed to copper. PeerJ, 9, e11373.
Abstract: Copper (Cu) is important for plant growth, but high concentrations can lead to detrimental effects such as primary root length inhibition, vegetative tissue chlorosis, and even plant death. The interaction between plantsoil microbiota and roots can potentially affect metal mobility and availability, and, therefore, overall plant metal concentration. Cupriavidus metallidurans CH34 is a multi metalresistant bacterial model that alters metal mobility and bioavailability through ion pumping, metal complexation, and reduction processes. The interactions between strain CH34 and plants may affect the growth, metal uptake, and translocation of Arabidopsis thaliana plants that are exposed to or not exposed to Cu. In this study, we looked also at the specific gene expression changes in C. metallidurans when cocultured with Cuexposed A. thaliana. We found that A. thaliana's rosette area, primary and secondary root growth, and dry weight were affected by strain CH34, and that beneficial or detrimental effects depended on Cu concentration. An increase in some plant growth parameters was observed at copper concentrations lower than 50 mM and significant detrimental effects were found at concentrations higher than 50 mM Cu. We also observed up to a 90% increase and 60% decrease in metal accumulation and mobilization in inoculated A. thaliana. In turn, copperstressed A. thaliana altered C. metallidurans colonization, and cop genes that encoded copper resistance in strain CH34 were induced by the combination of A. thaliana and Cu. These results reveal the complexity of the plantbacteriametal triad and will contribute to our understanding of their applications in plant growth promotion, protection, and phytoremediation strategies.



Cofre, C., Campos, J. L., ValenzuelaHeredia, D., Pavissich, J. P., Camus, N., Belmonte, M., et al. (2018). Novel system configuration with activated sludge likegeometry to develop aerobic granular biomass under continuous flow. Bioresour. Technol., 267, 778–781.
Abstract: A novel continuous flow system with “flat geometry” composed by two completely mixed aerobic tanks in series and a settler was used to promote the formation of aerobic granular sludge. Making similarities of this system with a typical sequencing batch reactor (SBR), for aerobic granules cultivation, the value of the tank 1/tank 2 vol ratio and the biomass recirculation rate would correspond with the feast/famine length ratio and the length of the operational cycle, respectively, while the settler upflow liquid velocity imposed would be related to the settling time. From the three experiments performed the best results were obtained when the tank 1/tank 2 vol ratio was of 0.28, the sludge recycling ratio of 0.25 and the settler upflow velocity of 2.5 m/h. At these conditions the aggregates had settling velocities between 29 and 113 m/h, sludge volume index at 10 min (SVI10) of 70 mL/g TSS and diameters between 1.0 and 5.0 mm.



ConchaVega, P., Goles, E., Montealegre, P., RiosWilson, M., & Santivanez, J. (2022). Introducing the activity parameter for elementary cellular automata. Int. J. Mod Phys. C, 33(09), 2250121.
Abstract: Given an elementary cellular automaton (ECA) with local transition rule R, two different types of local transitions are identified: the ones in which a cell remains in its current state, called inactive transitions, and the ones in which the cell changes its current state, which are called active transitions. The number of active transitions of a rule is called its activity value. Based on latter identification, a rule R1 is called a subrule of R2 if the set of active transitions of R1 is a subset of the active transitions of R2.
In this paper, the notion of subrule for elementary cellular automata is introduced and explored: first, we consider a lattice that illustrates relations of nonequivalent elementary cellular automata according to nearby subrules. Then, we introduce statistical measures that allow us to compare rules and subrules. Finally, we explore the possible similarities in the dynamics of a rule with respect to its subrules, obtaining both empirical and theoretical results.



Contreras, G. M. (2014). Stochastic volatility models at rho = +/ 1 as second class constrained Hamiltonian systems. Physica A, 405, 289–302.
Abstract: The stochastic volatility models used in the financial world are characterized, in the continuoustime case, by a set of two coupled stochastic differential equations for the underlying asset price S and volatility sigma. In addition, the correlations of the two Brownian movements that drive the stochastic dynamics are measured by the correlation parameter rho (1 <= rho <= 1). This stochastic system is equivalent to the FokkerPlanck equation for the transition probability density of the random variables S and sigma. Solutions for the transition probability density of the Heston stochastic volatility model (Heston, 1993) were explored in Dragulescu and Yakovenko (2002), where the fundamental quantities such as the transition density itself, depend on rho in such a manner that these are divergent for the extreme limit rho = +/ 1. The same divergent behavior appears in Hagan et al. (2002), where the probability density of the SABR model was analyzed. In an option pricing context, the propagator of the bidimensional BlackScholes equation was obtained in Lemmens et al. (2008) in terms of the path integrals, and in this case, the propagator diverges again for the extreme values rho = +/ 1. This paper shows that these similar divergent behaviors are due to a universal property of the stochastic volatility models in the continuum: all of them are second class constrained systems for the most extreme correlated limit rho = +/ 1. In this way, the stochastic dynamics of the rho = +/ 1 cases are different of the rho (1 <= rho <= 1) case, and it cannot be obtained as a continuous limit from the rho not equal +/ 1 regimen. This conclusion is achieved by considering the FokkerPlanck equation or the bidimensional BlackScholes equation as a Euclidean quantum Schrodinger equation. Then, the analysis of the underlying classical mechanics of the quantum model, implies that stochastic volatility models at rho = +/ 1 correspond to a constrained system. To study the dynamics in an appropriate form, Dirac's method for constrained systems (Dirac, 1958, 1967) must be employed, and Dirac's analysis reveals that the constraints are second class. In order to obtain the transition probability density or the option price correctly, one must evaluate the propagator as a constrained Hamiltonian pathintegral (Henneaux and Teitelboim, 1992), in a similar way to the high energy gauge theory models. In fact, for all stochastic volatility models, after integrating over momentum variables, one obtains an effective Euclidean Lagrangian path integral over the volatility alone. The role of the second class constraints is determining the underlying asset price S completely in terms of volatility, so it plays no role in the path integral. In order to examine the effect of the constraints on the dynamics for both extreme limits, the probability density function is evaluated by using semiclassical arguments, in an analogous manner to that developed in Hagan et al. (2002), for the SABR model. (C) 2014 Elsevier B.V. All rights reserved.



Contreras, M., & Hojman, S. A. (2014). Option pricing, stochastic volatility, singular dynamics and constrained path integrals. Physica A, 393, 391–403.
Abstract: Stochastic volatility models have been widely studied and used in the financial world. The Heston model (Heston, 1993) [7] is one of the best known models to deal with this issue. These stochastic volatility models are characterized by the fact that they explicitly depend on a correlation parameter p which relates the two Brownian motions that drive the stochastic dynamics associated to the volatility and the underlying asset. Solutions to the Heston model in the context of option pricing, using a path integral approach, are found in Lemmens et al. (2008) [21] while in Baaquie (2007,1997) [12,13] propagators for different stochastic volatility models are constructed. In all previous cases, the propagator is not defined for extreme cases rho = +/ 1. It is therefore necessary to obtain a solution for these extreme cases and also to understand the origin of the divergence of the propagator. In this paper we study in detail a general class of stochastic volatility models for extreme values rho = +/ 1 and show that in these two cases, the associated classical dynamics corresponds to a system with second class constraints, which must be dealt with using Dirac's method for constrained systems (Dirac, 1958,1967) [22,23] in order to properly obtain the propagator in the form of a Euclidean Hamiltonian path integral (Henneaux and Teitelboim, 1992) [25]. After integrating over momenta, one gets an Euclidean Lagrangian path integral without constraints, which in the case of the Heston model corresponds to a path integral of a repulsive radial harmonic oscillator. In all the cases studied, the price of the underlying asset is completely determined by one of the second class constraints in terms of volatility and plays no active role in the path integral. (C) 2013 Elsevier B.V. All rights reserved.



Contreras, M., & Pena, J. P. (2019). The quantum dark side of the optimal control theory. Physica A, 515, 450–473.
Abstract: In a recent article, a generic optimal control problem was studied from a physicist's point of view (Contreras et al. 2017). Through this optic, the Pontryagin equations are equivalent to the Hamilton equations of a classical constrained system. By quantizing this constrained system, using the right ordering of the operators, the corresponding quantum dynamics given by the Schrodinger equation is equivalent to that given by the HamiltonJacobiBellman equation of Bellman's theory. The conclusion drawn there were based on certain analogies between the equations of motion of both theories. In this paper, a closer and more detailed examination of the quantization problem is carried out, by considering three possible quantization procedures: right quantization, left quantization, and Feynman's path integral approach. The Bellman theory turns out to be the classical limit h > 0 of these three different quantum theories. Also, the exact relation of the phase S(x, t) of the wave function Psi(x, t) = e(i/hS(x,t)) of the quantum theory with Bellman's cost function J(+)(x, t) is obtained. In fact, S(x, t) satisfies a 'conjugate' form of the HamiltonJacobiBellman equation, which implies that the cost functional J(+)(x, t) must necessarily satisfy the usual HamiltonJacobiBellman equation. Thus, the Bellman theory effectively corresponds to a quantum view of the optimal control problem. (C) 2018 Elsevier B.V. All rights reserved.



Contreras, M., Pellicer, R., & Villena, M. (2017). Dynamic optimization and its relation to classical and quantum constrained systems. Physica A, 479, 12–25.
Abstract: We study the structure of a simple dynamic optimization problem consisting of one state and one control variable, from a physicist's point of view. By using an analogy to a physical model, we study this system in the classical and quantum frameworks. Classically, the dynamic optimization problem is equivalent to a classical mechanics constrained system, so we must use the Dirac method to analyze it in a correct way. We find that there are two secondclass constraints in the model: one fix the momenta associated with the control variables, and the other is a reminder of the optimal control law. The dynamic evolution of this constrained system is given by the Dirac's bracket of the canonical variables with the Hamiltonian. This dynamic results to be identical to the unconstrained one given by the Pontryagin equations, which are the correct classical equations of motion for our physical optimization problem. In the same Pontryagin scheme, by imposing a closedloop lambdastrategy, the optimality condition for the action gives a consistency relation, which is associated to the HamiltonJacobiBellman equation of the dynamic programming method. A similar result is achieved by quantizing the classical model. By setting the wave function Psi (x, t) = e(is(x,t)) in the quantum Schrodinger equation, a nonlinear partial equation is obtained for the S function. For the righthand side quantization, this is the HamiltonJacobiBellman equation, when S(x, t) is identified with the optimal value function. Thus, the HamiltonJacobiBellman equation in Bellman's maximum principle, can be interpreted as the quantum approach of the optimization problem. (C) 2017 Elsevier B.V. All rights reserved.

