This paper considers the stabilization of a second order ODE–heat system coupling at an intermediate point under natural and checkable assumptions, which is motivated by the thermoelastic coupling ...physics arising in microelectromechanical systems (MEMS). A novel backstepping transformation form is proposed in this work and the feedback gain for the lumped-parameter component is constructed using the eigenvector of coefficient matrix in the ODE system. Then, we prove the existence of smooth kernels with second order continuously derivative for the forward and inverse transformations in the backstepping feedback control law design. At the same time, we show that the forward and inverse transformations are mutually invertible transformation pair. Finally, the effectiveness of the stabilization feedback controller design is shown with some numerical examples.
We describe an extension of the Taylor method for the numerical solution of ODEs that uses Padé approximants to obtain extremely precise numerical results. The accuracy of the results is essentially ...limited only by the computer time and memory, provided that one works in arbitrary precision. In this method the stepsize is adjusted to achieve the desired accuracy (variable stepsize), while the order of the Taylor expansion can be either fixed or changed at each iteration (variable order).
As an application, we have calculated the periodic solutions (limit cycle) of the van der Pol equation with an unprecedented accuracy for a large set of couplings (well beyond the values currently found in the literature) and we have used these numerical results to validate the asymptotic behavior of the period, of the amplitude and of the Lyapunov exponent reported in the literature. We have also used the numerical results to infer the formulas for the asymptotic behavior of the fast component of the period and of the maximum velocity, which have never been calculated before.
•We introduce a Pade–Taylor method suitable for stiff ODEs.•Can obtain very accurate numerical solutions.•We have tested the method on the van der Pol equation (stiff and non-stiff).•We have obtained the most precise estimates of period, amplitude, Lyapunov exponents, etc.•We have used Richardson and Richardson–Pade extrapolation to derive the asymptotic behaviors.
In this work, a Lyapunov-based predictive control method utilizing deep learning techniques is proposed for driving continuous-time nonlinear processes towards the desired equilibrium point. ...Initially, a neural ordinary differential equation model is developed to learn the continuous-time dynamics from discrete-time sampling data. Subsequently, the theoretical conditions that the control Lyapunov function (CLF) developed on the learned continuous-time system is feasible to act on the actual system through a zero-order-holder are analyzed. Building upon the theoretical analysis and the constructed continuous-time dynamics, a deep neural network control Lyapunov function (DNN-CLF) controller is developed. Importantly, the DNN-CLF is independent of the affine system formulation, effectively addressing the challenge of designing the CLF. Benefiting from the stability guarantee provided by the established DNN-CLF, it is integrated into the model predictive control (MPC) framework to develop Lyapunov-based MPC to further improve the control performance. To ensure safety, considering error propagation and associated risks with enlarging the feasible region of the CLF, tubes are constructed and integrated into the MPC scheme. Finally, a numerical simulation of a chemical process is conducted to validate the efficiency of the proposed method.
•A Lyapunov-based predictive control method employs deep learning for nonlinear processes.•A neural ordinary differential equation model learns continuous-time dynamics from data.•Development of the deep neural network control Lyapunov function (DNN-CLF).•Enhancing control performance through Lyapunov-based MPC with DNN-CLF.•Integration of tubes into the MPC scheme for ensured safety.
During the PAMARCMiP 2018 campaign (March and April 2018) a proton-transfer-reaction mass spectrometer (PTR-MS) was deployed onboard the POLAR 5 research aircraft and sampled the high Arctic ...atmosphere under Arctic haze conditions. More than 100 compounds exhibited levels above 1 pmol/mol in at least 25% of the measurements. We used acetone mixing ratios, ozone concentrations, and back trajectories to identify periods with and without long-range transport from continental sources. During two flights, surface ozone depletion events (ODE) were observed that coincided with enhanced levels of acetone, and methylethylketone, and ice nucleating particles (INP).
Air masses with continental influence contained elevated levels of compounds associated with aged biogenic emissions and anthropogenic pollution (e.g., methanol, peroxyacetylnitrate (PAN), acetone, acetic acid, methylethylketone (MEK), proprionic acid, and pentanone). Almost half of all positively detected compounds (>100) in the high Arctic atmosphere can be associated with terpene oxidation products, likely produced from monoterpenes and sesquiterpenes emitted from boreal forests. We speculate that the transport of biogenic terpene emissions may constitute an important control of the High Arctic aerosol burden. The sum concentration of the detected aerosol forming vapours is ∼12 pmol/mol, which is of the same order than measured dimethylsulfide (DMS) mixing ratios and their mass density corresponds to approximately one fifth of the measured non-black-carbon particles.
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•A PTR-MS instrument onboard the POLAR 5 aircraft detected more than 100 compounds above 1 pmol/mol in at least 25% of the measurements.•During ozone depletion events enhanced levels of acetone, and methylethylketone, and ice nucleating particles (INP) were observed.•Acetone and ozone mixing ratios were used as metric for continental influence.•Continentally influenced air exhibited enhanced signals of biogenic and anthropogenic emissions.•∼50% of all positively detected compounds (>100) can be associated with terpene oxidation products.
State-space modeling (SSM) provides a general framework for many image reconstruction tasks. Error in a priori physiological knowledge of the imaging physics, can bring incorrectness to solutions. ...Modern deep-learning approaches show great promise but lack interpretability and rely on large amounts of labeled data. In this paper, we present a novel hybrid SSM framework for electrocardiographic imaging (ECGI) to leverage the advantage of state-space formulations in data-driven learning. We first leverage the physics-based forward operator to supervise the learning. We then introduce neural modeling of the transition function and the associated Bayesian filtering strategy. We applied the hybrid SSM framework to reconstruct electrical activity on the heart surface from body-surface potentials. In unsupervised settings of both in-silico and in-vivo data without cardiac electrical activity as the ground truth to supervise the learning, we demonstrated improved ECGI performances of the hybrid SSM framework trained from a small number of ECG observations in comparison to the fixed SSM. We further demonstrated that, when in-silico simulation data becomes available, mixed supervised and unsupervised training of the hybrid SSM achieved a further 40.6% and 45.6% improvements, respectively, in comparison to traditional ECGI baselines and supervised data-driven ECGI baselines for localizing the origin of ventricular activations in real data.
Given an isoparametric function f on the n-dimensional sphere, we consider the space of functions w∘f to reduce the Yamabe equation on the round sphere into a singular ODE on w in the interval 0,π, ...of the form w″+(h(r)/sinr)w′+λ(|w|4/n−2w−w)=0 and boundary conditions w′(0)=0=w′(π), where h is a monotone function with exactly one zero on (0,π) and λ>0 is a constant. For any positive integer k we obtain a solution with exactly k-zeroes yielding solutions to the Yamabe equation with exactly k connected isoparametric hypersurfaces as nodal set. The idea of the proof is to consider the initial value problems on the singularities, and then to solve the corresponding double shooting problem, matching the values of w and w′ at the unique zero of h. In particular we obtain solutions with exactly one zero, providing solutions of the Yamabe equation with low energy.
Pine wilt disease is caused by nematodes transmitted by pine sawyer beetles and is fatal for several pine species. The trees might be destroyed within a few months after being attacked, leads to ...major ecological and financial losses. In this article, we presented a model of pine wilt disease in the trees considering the interaction between nematodes, transmitting beetles with both asymptomatic and symptomatic pine trees. The disease dynamics is first displayed through a schematic diagram which is then transformed to non-linear coupled integer order ODEs through the law of mass action. The positivity, boundedness and equilibrium points has been analyzed and basic reproduction number is calculated through the next generation technique. Sensitivity analysis is also done for the most sensitive parameters which is also displayed through different figures and tables. The solution of the considered nonlinear fractal-fractal model has been obtained through numerical method via MATLAB software. Some numerical results have been obtained and it has been observed that the fractional model give us the more general results by considering different non-integer orders and the integer order results can be easily recovered. It is also noticed that reducing the interaction among the infected beetles and susceptible trees by killing the main source of the infection i.e., killing beetles can reduce the infection drastically, which are briefly discussed and conclusion has been drawn on the basis of the obtained results.
•A mathematical model for the pine wilt disease dynamics is considered.•All the possibilities of interaction of trees and beetles are considered.•Using law of mass action, the physical model is transformed into system of ODEs.•Fractal-Fractional differential operator is applied to integer order ODE system.•Simulation results are obtained by use of a numerical approach.
Stabilisation via adaptive feedback is investigated for an ODE system cascaded by coupled parabolic equations. The remarkable characteristic of the system is the presence of serious uncertainties ...since no known or bounded intervals are needed for the unknown parameters. Besides, strong coupling exists in the PDE subsystem which brings essential difference compared with those of the related literature. For this, a passive identifier is skillfully constructed firstly, and in turn adaptive laws are given to compensate the serious parametric unknowns. Then, by constructing a backstepping transformation and its inverse one, the identifier joint with the original system is transformed into a new system. From the new system, an adaptive state-feedback controller is designed which ensures that all the states of the closed-loop system are bounded and particularly those of the original system converge to zero. Finally, a numerical example is provided to validate the proposed algorithm.
In various applications in life and social sciences the agents of a complex system, which can be individuals of a sub–populations, can have one of the few inner states or can choose one of few ...strategies. Their states are effects of interactions with other agents. Behavior is described by a kinetic–like a nonlinear equation. In the present paper, we study the behavior of solutions in the case of three possible states and show that in some cases a kind of self–organization occurs but in others, a periodic behavior characterizes the system. We can observe a wide variety of dynamics that can relate to the behavior of real systems. The model contains a natural interaction intensity parameter γ≥1. Case γ>1 leads to “very” nonlinear structures. We prove that the system has no non–constant periodic solutions. We propose conditions guaranteeing the asymptotic stability of equilibrium points on the boundary that corresponds to the asymptotic extinction of two states. Moreover, conditions for the uniqueness and instability of an inner equilibrium point, corresponding to an asymptotic presence of all states, are formulated. The case of γ=1 with asymmetric interaction rate is studied as well. Possible complex behavior of solutions can reflect the possible complex performance of systems with asymmetric interactions — typical e.g. in Economy and some applications in Biology.
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