Costate Research Articles

Time-Fuel-Optimal Navigation of a Commercial Aircraft in Cruise With Heading and Throttle Controls Using Pontryagin’s Maximum Principle

In this research, we consider the commercial aircraft trajectory optimization problem for a general cruise model with arbitrary spatial wind fields to be solved using the Pontryagin’s maximum principle. The model features two fundamental controls, namely “throttle setting” (appearing as a singular control) and “heading angle” (appearing as a regular control). For a problem with state-inequality constraints and minimum time-fuel objective, we show that the optimal “heading angle” can be described through the classic Zermelo’s navigation identity. We also demonstrate, by analyzing the switching function, that the singular “throttle setting” can be characterized through a feedback function that relies on both the optimal states and “heading angle”. The switching-point algorithm is employed to solve a case study where we inspect the optimality conditions and graph the optimal controls together with the optimal state and co-state variables.

Two-grid methods of finite element approximation for parabolic integro-differential optimal control problems

<abstract><p>In this paper, we present a two-grid scheme of fully discrete finite element approximation for optimal control problems governed by parabolic integro-differential equations. The state and co-state variables are approximated by a piecewise linear function and the control variable is discretized by a piecewise constant function. First, we derive the optimal a priori error estimates for all variables. Second, we prove the global superconvergence by using the recovery techniques. Third, we construct a two-grid algorithm and discuss its convergence. In the proposed two-grid scheme, the solution of the parabolic optimal control problem on a fine grid is reduced to the solution of the parabolic optimal control problem on a much coarser grid; additionally, the solution of a linear algebraic system on the fine grid and the resulting solution maintain an asymptotically optimal accuracy. Finally, we present a numerical example to verify the theoretical results.</p></abstract>

Frequency Effects on the Vibrational States and Conversion of CO2 in Radio Frequency Discharges Under Martian Pressure

In recent years, with the progress of Mars exploration, the in situ utilization of CO2 resources in the Martian atmosphere has become a widely discussed subject. Nonequilibrium plasma technology as an efficient molecular activation approach has great potential for CO2 conversion. In this study, a fluid model is established to discuss driving frequency effects on the vibrational states and conversion of CO2 in radio frequency (RF) discharges at a given power density under Martian pressure. The calculated CO2 conversion and energy efficiency depending on the specific energy input are in good agreement with the experimental measurements. The simulation results found that a considerable fraction of electron energy is transferred to the asymmetric stretching modes of CO2, and the characteristic times for electron impact vibrational excitation and vibration–vibration (VV) relaxation are small, which results in a larger density of higher vibrational levels. The density of vibrational levels of CO2 increases with the driving frequency at a constant power density, which can be explained by the enhancement of vibrational excitation and VV relaxation. The computational data also show that more CO, O, and O2 are produced as the driving frequency increases at a constant power density, and the vibration-induced dissociation with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\textrm {CO}_{2}\textrm {v}_{8}$ </tex-math></inline-formula> gradually replaces the direct dissociation to be the main reaction for the production of CO and O. In addition, the increase in driving frequency causes more electron energy to be transferred to the asymmetrical stretch modes, and the vibrational energy of these vibrational states of CO2 can reduce the activation barrier of the reaction, improving the CO2 conversion and energy efficiency. This study contributes to a deeper insight into the frequency effects on plasma chemistry in RF CO2 discharges under Martian pressure and shows the optimized ways for the decomposition of CO2 on Mars.

Структура сплавов для постоянных магнитов типа (Sm,Zr)(Co,Cu,Fe)z и механизм формирования высококоэрцитивного состояния

Owing to the high temperature stability of hysteretic characteristics and corrosion resistance under external actions, (Sm,Zr)(Co,Cu,Fe)z-based magnets are widely used in devices operating at high temperatures, most critical constructions of aerospace engineering and defense technology, and in traction electric motors for transport engineering. A number of modern research methods are used to study in detail the structure and phase composition of the (Sm,Zr)(Co,Cu,Fe)z alloys with z = 5.5 – 7.0 in as-cast state and after heat treatment, which includes the solid solution treatment at 1150 – 1180 °С for 5 h, subsequent water quenching, isothermal aging at 800 °С for 20 h, and cooling (stepped aging) from 800 to 400 °С at an average cooling rate of 100 °С/h. The results obtained allowed us to suggest the schematic diagram of grain-boundary constituent transformation at different stages of heat treatment of (Sm,Zr)(Co,Cu,Fe)z permanent magnets prepared by powder metallurgy. The formation mechanism of high-coercivity state of (Sm,Zr)(Co,Cu,Fe)z permanent magnets in the course of complete heat treatment cycle is given as the sequence of phase transformations accompanied by redistribution of a number of alloy components.

A priori error estimates of finite volume element method for bilinear parabolic optimal control problem

&lt;abstract&gt;&lt;p&gt;In this paper, we study the finite volume element method of bilinear parabolic optimal control problem. We will use the optimize-then-discretize approach to obtain the semi-discrete finite volume element scheme for the optimal control problem. Under some reasonable assumptions, we derive the optimal order error estimates in $ L^2(J; L^2) $ and $ L^\infty(J; L^2) $-norm. We use the backward Euler method for the discretization of time to get fully discrete finite volume element scheme for the optimal control problem, and obtain some error estimates. The approximate order for the state, costate and control variables is $ O(h^{3/2}+\triangle t) $ in the sense of $ L^2(J; L^2) $ and $ L^\infty(J; L^2) $-norm. Finally, a numerical experiment is presented to test these theoretical results.&lt;/p&gt;&lt;/abstract&gt;

The primary photolysis of aqueous carbonate di-anions.

We study the primary photolysis dynamics of aqueous carbonate, CO32-(aq), and hydrogen carbonate, HCO3-(aq), when they are excited at λ = 200 nm. The photolysis is recorded with sub-picosecond time resolution using UV pump-Vis probe and UV pump-IR probe transient absorption spectroscopy and interpreted with the aid of density functional theory calculations. When CO32- is excited via single photon absorption at λ = 200 nm, Φ(t = 20 ps) = 82 ± 5% of the excited di-anions either detach an electron or dissociate. The electron detachment takes place from the excited state in t < 1 ps and forms ground state CO3˙- and eaq-. Dissociation occurs from both the electronic ground and excited states of CO32-. Dissociation from the CO32- excited state is assisted by water molecules and forms CO2˙-, OH˙ and OH-. The dissociation occurs both directly from the Franck-Condon region in t < 1 ps and indirectly with a time constant of τ = 13.9 ± 0.5 ps as the excited state relaxes. Dissociation of vibrationally excited CO32- molecules in the electronic ground state is also assisted by water molecules and forms CO2 and two OH- anions. The dissociation and subsequent vibrational relaxation of CO2 occur with a time constant of τ = 10.2 ± 0.5 ps. The residual 1 - Φ(t = 20 ps) = 18 ± 5% of the excited CO32- di-anions return by internal conversion to the equilibrated CO32- ground state with a time constant of τ = 4.0 ± 0.4 ps. The extinction coefficient of aqueous hydrogen carbonate HCO3-(aq) at λ = 200 nm is an order of magnitude smaller than that of carbonate, so even though the hydrogen carbonate anions dominate the carbonate di-anions in the hydrogen carbonate solution, the primary photolysis of hydrogen carbonate is obscured by the photo-products of carbonate. Hence, we are unable to assess the primary photolysis of hydrogen carbonate. However, the weak one-photon absorption facilitates two-photon ionization of water, which forms hydronium, H3O+, cations. The sudden increase in the acidity induced by two-photon ionization protonates the ground state hydrogen carbonate molecules, thus offering a rare spectroscopic glimpse of aqueous carbonic acid.

Application of Heavy and Underestimated Dynamic Models in Adaptive Receding Horizon Control Without Constraints

In the heuristic “Adaptive Receding Horizon Controller” (ARHC) the available dynamic model of the controlled system usually is placed in the role of a constraint under which various cost functions can be minimized over a horizon. A possible secure design can be making calculations for a “heavy dynamic model” that may produce high dynamical burden that is efficiently penalized by the cost functions and instead of the original nominal trajectory results a “deformed” one that can be realized by the controlled system of “less heavy dynamics”. In the lack of accurate system model a fixed point iterationbased adaptive approach is suggested for the precise realization of this deformed trajectory. To reduce the computational burden of the control the usual approach in which the dynamic model is considered as constraint and Lagrange-multipliers are introduced as co-state variables is evaded. The heavy dynamic model is directly built in the cost and the computationally greedy Reduced Gradient Algorithm is replaced by a transition between the simple and fast Newton-Raphson and the slower Gradient Descent algorithms (GDA). In the paper simulation examples are presented for two dynamically coupled van der Pol oscillators as a strongly nonlinear system. The comparative use of simple nondifferentiable and differentiable cost functions is considered, too.

Effect of Na ion on Cu2O:Co nanocrystals and subsequent attributes in structural and optical characteristics

Cuprous oxide-based nanocrystals were synthesized by conventional sol-gel techniques under an ambient environment. Herein, we systematically examined how Na ion influences the optical and structural characteristics of the nanocrystal. The X-ray diffraction exhibits Cu2O and Cu2O:Co nanocrystals have cubic structure (Pn-3m) with average crystallite sizes of 41.70 and 43.00 nm, respectively. Photoluminescence and FT-IR spectra clearly show the existence of CuO phase due to Na, Co doping. HR-TEM micrograph exhibits that the particle size ranges between 0.89–1.11 μm, and that agglomeration diminishes with Na, Co doping concentration. The optical bandgap of Cu2O based nanocrystals were found to be reduced and improves optical absorption in the visible region. Eventually, XPS measurement gives us the existence of several oxidation states of Cu and Co and a single oxidation state of Na.

Rational design of cobalt catalysts embedded in N-Doped carbon for the alcohol dehydrogenation to carboxylic acids

In this work, a series of Co catalysts (namely Co@NC) was designed and developed from the thermal treatment of ZIF-67. Through extensive screening of various parameters, Co@NC700-H2/Ar exhibited the best catalytic performance in the acceptorless alcohol dehydrogenation to carboxylic acids. It was worth noting that Co@NC700-H2/Ar enabled the efficient synthesis of various aromatic and aliphatic carboxylic acids in moderate to high yields. Besides, it was recyclable for 9 consecutive cycles without activity decay. Moreover, the effects of pyrolysis conditions on the catalytic performance of the resulting materials were thoroughly examined, and the rationale for the superiority of Co@NC700-H2/Ar over the other prepared materials was also provided. Specifically, the pyrolysis atmosphere applying 95% of Ar gas and 5% of H2 gas (H2/Ar) resulted in the generation of ideal porosity and carbon nanotubes (CNTs), while utilizing 700 °C as the optimized pyrolysis temperature was ascribed to the comprehensive evaluation of materials crystallinity and defective structures. Furthermore, additional insights were gained by systematically exploring the oxidation states of Co in different catalysts. It was proposed that the Co–N species played a crucial role in this catalysis, while the contribution of other Co species, including Co0 and Co–O, could not be neglected. Finally, a plausible reaction mechanism was proposed based on the above observations and related literature reports.

A free-standing NiMoO4@Mg-Co(OH)F core-shell nanocomposites supported on Ni foam for asymmetric supercapacitor applications

In this paper, a facile hydrothermal treatment method was employed to successfully synthesize the NiMoO4@Mg-Co(OH)F/NF (NM-MCF) nanocomposites by taking NiMoO4 nanorods as cores, and assembling them with the shell of Mg-Co(OH)F nanosheets. The high conductivity of Mg-Co-F was combined with the nanosheet-assembled core-shell structure, and the multivalent states of Ni, Mg, and Co increased the number of active sites and improved the synergy, contributing greatly to excellent performance of NM-MCF. As expected, the optimized NM-MCF nanocomposite electrode exhibits superb capacitive performance, including excellent specific capacitance of 1630 F g −1 at 1 A g −1 and remarkable cycle stability with 86 % after 6000 cycles. Notably, the NM-MCF exhibits a regular rate capability of 56.5 % from 1 to 15 A g−1, superior to that of NiMoO4 (56.4 %) and Mg-Co(OH)F (28.8 %). Employing the as-prepared NM-MCF composites as positive electrode and activated carbon as negative electrode, the hybrid supercapacitor device with spectacular capacity of maximum energy density and power density up to 28.2 Wh kg−1 and 0.7 kW kg−1 and impressive long-term stability of 73.3 % retention after 6000 cycles is assembled.

High-Performance Oxygen Evolution Reaction Electrocatalysts Discovered via High-Throughput Aerogel Synthesis

A combinatorial high-throughput aerogel synthesis method is developed to systematically study the synergistic catalytic effects of multiple non-noble metal elements. A large number of aerogels with different compositions were synthesized and the corresponding ternary phase diagrams of the composition–electrocatalytic performance were constructed. A batch of aerogel catalysts with outstanding catalytic performance and stability for the electrochemical oxygen evolution reaction (OER) were discovered. Amorphous Fe3Co3Ni2-(oxy)hydroxide with excellent electrocatalytic performance (Tafel slope of 49 mV dec–1 and η100 of 421.48 ± 6.41 mV) was found to achieve a mass activity of 611.23 A/gmetal at 1.63 V (vs RHE). X-ray photoemission spectroscopy (XPS) and X-ray absorption spectroscopy (XAS) studies reveal that the incorporation of a third atom (Ni and Ce) into the binary aerogels enlarges the Co–O atomic distance and reduces the average valence state of Co, which in turn speed up the oxygen evolution reaction kinetics. This work puts forward a strategy for high-throughput synthesis and screening of electrocatalysts and provides valuable data sets for future machine learning and prediction of high-performance electrocatalysts.

Stochastic perturbation method for optimal control problem governed by parabolic PDEs with small uncertainties

In this paper, we investigate the first-order and second-order perturbation approximation schemes for an optimal control problem governed by parabolic PDEs with small uncertainties. First, we propose the finite dimensional noise assumption for the random coefficient, and expand the state and co-state variables up to a certain order with respect to a small parameter by using the perturbation technique. Then, we insert all the expansions into the known deterministic parametric optimality system, and obtain the first-order and second-order optimality systems. These two systems are discretized by the finite element method in space and the backward Euler method in time to establish the first-order and second-order approximate schemes. Further, a priori error estimates are derived for the state, co-state and control variables of the first-order and second-order approximation, respectively. Finally, some numerical experiments are presented to verify the theoretical results.

Transformation of Waste Toner Powder into Valuable Fe2O3 Nanoparticles for the Preparation of Recyclable Co(II)-NH2-SiO2@Fe2O3 and Its Applications in the Synthesis of Polyhydroquinoline and Quinazoline Derivatives.

Ecological recycling of waste materials by converting them into valuable nanomaterials can be considered a great opportunity for management and fortification of the environment. This article deals with the environment-friendly synthesis of Fe2O3 nanoparticles (composed of α-Fe2O3 and γ-Fe2O3) using waste toner powder (WTP) via calcination. Fe2O3 nanoparticles were then coated with silica using TEOS, functionalized with silane (APTMS), and immobilized with Co(II) to get the desired biocompatible and cost-effective catalyst, i.e., Co(II)-NH2-SiO2@Fe2O3. The structural features in terms of evaluation of morphology, particle size, presence of functional groups, polycrystallinity, and metal content over the surface were determined by Fourier transform infrared spectroscopy (FTIR), powder X-ray diffraction (P-XRD), field emission gun-scanning electron microscopy (FEG-SEM), energy-dispersive X-ray analysis (EDX), high resolution-transmission electron microscopy (HR-TEM), X-ray photoelectron spectroscopy (XPS), thermogravimetric analysis (TGA), vibrating sample magnetometry (VSM), Brunauer-Emmett-Teller (BET) analysis, and inductively coupled plasma-atomic emission spectroscopy (ICP-AES) studies. XPS confirmed the (II) oxidation state of Co, and ICP-AES and EDX supported the loading of Co(II) over the surface of the support. P-XRD proved the polycrystalline nature of the Fe2O3 core and even after functionalization. In comparison to previously reported methods, Co(II)-NH2-SiO2@Fe2O3 provides an eco-friendly procedure for the synthesis of polyhydroquinoline and quinazoline derivatives with several advantages such as a short reaction time and high yield. Polyhydroquinoline and quinazoline derivatives are important scaffolds in pharmacologically active compounds. Moreover, the developed nanocatalyst was recyclable, and HR-TEM and P-XRD confirmed the agglomeration in the recycled catalyst resulted in a decrease in yield after the fifth run. The present protocol provides a new strategy of recycling e-waste into a heterogeneous nanocatalyst for the synthesis of heterocycles via multicomponent reactions. This made the synthesized catalyst convincingly more superior to other previously reported catalysts for organic transformations.

Structural, electrical, magnetic & optical properties of Nickel, cobalt doped and Co-doped wurtzite GaN: A first-principle investigation

The structural, electronic, magnetic, and optical properties of Co (6.25%) and Ni (6.25%) monodoped and co-doped GaN were calculated by the DFT+U method using a 2 × 2 × 2 supercell of a total of 32 atoms. The formation energy was negative under both Ga-rich and N-rich conditions for all structures. N-rich condition is found to be favorable for GaN growth. The electronic and magnetic properties changed after doping due to the transition of the 3 d states of Ni, Co and hybridization of 2p of N. Hence monodoped GaN is ferromagnetic whereas the co-doped one is ferrimagnetic due to partial cancellation by the opposite spin of Ni and Co. The doped and co-doped absorption peaks are blue-shifted in the ultraviolet (UV) and redshifted in the visible-infrared (VIS-IR) region from the peaks of pure GaN. A pronounced absorption of VIS-IR is observed in monodoped structures. Both dielectric constant and refractive index increased due to co-doping whereas monodoping had enhanced the properties significantly. This theoretical study suggests the potential application of the Co, Ni monodoped in VIS- IR photonic and co-doped GaN in spintronic devices.

Modest modulation on the electronic structure of Co9S8 by vanadium doping for high-performance rechargeable Zn–air batteries

Rechargeable Zn-air batteries (ZABs) have been attracting research interests worldwide owing to their high energy density, as well as the good safety and low cost. However, the lack of highly active bifunctional electrocatalysts for oxygen reduction reaction (ORR) and oxygen evolution reaction (OER) at air cathode seriously hinders its developments. Herein, a vanadium doped Co9S8 (V-Co9S8) nanoparticles encapsulated in the nitrogen-doped porous carbon nanoflowers (NFs) has been prepared by an innovative double-solvent approach (DSA) that can precisely control the level of vanadium doping. The magnetic measurements and the theoretical calculation consistently demonstrate that the spin state of Co can be optimized by tuning the vanadium doping level. The spin-state modulation raises the d-band center of Co and enables the enhanced Co-S covalency in V-Co9S8, which balances the adsorption/desorption of oxygen intermediates and finally accelerates the oxygen redox kinetic. As a result, V-Co9S8 serving as a cost-effectively bifunctional catalyst for the cathode of a rechargeable ZAB, supplies a superb power density of 345 mW cm−2, an energy density of 814 mAh gZn−1 and a super durability with a stable operation of 840 cycles, which outperforms the noble metal catalyst (Pt/C and RuO2) and the most Co based catalysts reported recently.

Discontinuous Galerkin method for a distributed optimal control problem governed by a time fractional diffusion equation

This paper is devoted to the numerical analysis of a control constrained distributed optimal control problem subject to a time fractional diffusion equation with non-smooth initial data. The solutions of state and co-state are decomposed into singular and regular parts, and some growth estimates are obtained for the singular parts. Following the variational discretization concept, a full discretization is applied to the corresponding state and co-state equations by using linear conforming finite element method in space and piecewise constant discontinuous Galerkin method in time. By the growth estimates, error estimates are derived with non-smooth initial data. In particular, graded temporal grids are used to obtain the first-order temporal accuracy. Finally, numerical experiments are performed to verify the theoretical results.

Optimal Control of SARS Disease via Fractional Model

Severe acute respiratory syndrome (SARS) is a very dangerous disease that affects the human respiratory system. In this article, we discuss the optimal control of this disease via a fractional SVEIR epidemic model together with two control variables (treatment and vaccination). For this purpose, we first design a fractional optimal control problem and then apply Pontryagin's minimal principle in a fractional version to find the optimal control. Also, the forward and backward fractional Euler methods (FEM) are used to solve the state and co-state equations, respectively. The results gave a new treatment and vaccine strategy for breaking dawn and preventing the spread of SARS.

An anode fabricated by Co electrodeposition on ZIF-8/CNTs/CF for peroxymonosulfate (PMS) activation

A Co@ZIF-8/CNTs-CF anode for PMS activation was prepared by Co electrodeposition on carbon felt (CF) modified with ZIF-8 and carbon nanotubes (CNTs). The results showed that the fabricated Co@ZIF-8/CNTs-CF anode was an effective peroxymonosulfate (PMS) activator toward tetracycline (TC) removal. Compared with that in reaction system of bare CF anode + PMS, the reaction system of Co@ZIF-8/CNTs-CF anode + PMS exhibited 3.08 times decrease in the activation energy demanded and 4.21 times increase in the reaction rate constant (k), resulting in a kinetic favorable process of PMS activation by the Co@ZIF-8/CNTs-CF anode. The enhanced activation performance of the fabricated anode was ascribed to the high contents of the pyrrolic N and low valence state of Co in the Co@ZIF-8/CNTs-CF anode. Furthermore, the influence factors on the characteristics of transformation among the generated reactive species during the anodic PMS activation process were comprehensively investigated by the quenching experiments and the electron paramagnetic resonance (EPR) tests. The results showed that the SO4•- and reactive oxygen-containing reactive species (O2•- and 1O2) were generated during the activation of PMS by anode and became the major contributors toward TC removal. The production of 1O2 was through the dismutation of O2•-. In addition, the EPR experiments demonstrated that O2•- was generated mainly through the anodic PMS activation but the electrochemically driven molecular oxygen reduction reaction (ORR) process. The fabricated Co@ZIF-8/CNTs-CF anode for PMS activation provided a reference for the wastewater treatment based on the electrochemical advanced oxidation processes (EAOPs).

Tuning the Spin State of Co3+ by Crystal Facet Engineering for Enhancing the Oxygen Evolution Reaction Activity

Tuning the Spin State of Co³⁺ by Crystal Facet Engineering for Enhancing the Oxygen Evolution Reaction Activity

Control and Trajectory Planning of an Autonomous Bicycle Robot

This paper addresses the modeling and the control of an autonomous bicycle robot where the reference point is the center of gravity. The controls are based on the wheel heading’s angular velocity and the steering’s angular velocity. They have been developed to drive the autonomous bicycle robot from a given initial state to a final state, so that the total running cost is minimized. To solve the problem, the following approach was used: after having computed the control system Hamiltonian, Pontryagin’s Minimum Principle was applied to derive the feasible controls and the costate system of ordinary differential equations. The feasible controls, derived as functions of the state and costate variables, were substituted into the combined nonlinear state–costate system of ordinary differential equations and yielded a control-free, state–costate system of ordinary differential equations. Such a system was judiciously vectorized to easily enable the application of any computer program written in Matlab, Octave or Scilab. A Matlab computer program, set as the main program, was developed to call a Runge–Kutta function coded into Matlab to solve the combined control-free, state–costate system of ordinary differential equations coded into a Matlab function. After running the program, the following results were obtained: seven feasible state functions from which the feasible trajectory of the robot is derived, seven feasible costate functions, and two feasible control functions. Computational simulations were developed and provided in order to persuade the readers of the effectiveness and the reliability of the approach.