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.
Invasive tree species pose a grave global threat to biodiversity, particularly impacting vulnerable grassland ecosystems. This paper presents a novel approach to simulate the invasion of ecosystems ...by these aggressive species. Our innovative coupled ODE‐PDE model offers explicit invasion simulations, a significant leap beyond traditional integro‐difference matrix models. With robust theoretical properties, including solution positivity, and drawing upon PDE theory, our model offers deeper insights into invasion behavior. Our model reveals that invasions undergo a prolonged quiescent phase before rapidly transitioning to a consistent rate of advancement after a critical point. This study not only sheds light on invasion dynamics but also charts a course for future exploration and potential model extensions.
Lipids are oxidized in vivo by multiple oxidizing species with different properties, some by regulated manner to produce physiological mediators, while others by random mechanisms to give detrimental ...products. Vitamin E plays an important role as a physiologically essential antioxidant to inhibit unregulated lipid peroxidation by scavenging lipid peroxyl radicals to break chain propagation independent of the type of free radicals which induce chain initiation. Kinetic data suggest that vitamin E does not act as an efficient scavenger of nitrogen dioxide radical, carbonate anion radical, and hypochlorite. The analysis of regio- and stereo-isomer distribution of the lipid oxidation products shows that, apart from lipid oxidation by CYP enzymes, the free radical-mediated lipid peroxidation is the major pathway of lipid oxidation taking place in humans. Compared with healthy subjects, the levels of racemic and trans,trans-hydro (pero)xyoctadecadienoates, specific biomarker of free radical lipid oxidation, are elevated in the plasma of patients including atherosclerosis and non-alcoholic fatty liver diseases. α-Tocopherol acts as a major antioxidant, while γ-tocopherol scavenges nitrogen dioxide radical, which induces lipid peroxidation, nitration of aromatic compounds and unsaturated fatty acids, and isomerization of cis-fatty acids to trans-fatty acids. It is essential to appreciate that the antioxidant effects of vitamin E depend on the nature of both oxidants and substrates being oxidized. Vitamin E, together with other antioxidants such as vitamin C, contributes to the inhibition of detrimental oxidation of biological molecules and thereby to the maintenance of human health and prevention of diseases.
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•Lipids are oxidized in vivo by multiple oxidizing species.•Antioxidant effect of vitamin E depends on oxidizing species and target substrate.•Vitamin E inhibits free radical mediated lipid peroxidation efficiently.•Vitamin E alone cannot cope with detrimental oxidative modification of lipids.•Multiple antioxidants are required to protect biological molecules from oxidation.
We consider the so called combined energy of a deformation between two concentric annuli and minimize it, provided that it keep order of the boundaries. It is an extension of the corresponding result ...of Euclidean energy. It is interesting that the minimizers are certain radial mappings and they exist if and only if the annulus on the image domain is not too thin, provided that the original annulus is fixed. This in turn implies a Nitsche type phenomenon. Next we consider the combined distortion and obtain certain related results which are dual to the results for combined energy, which also involve some Nitsche type phenomenon.
The main part of the paper is concerned with the total combined energy, a certain integral operator, defined as a convex linear combination of the combined energy and combined distortion, of diffeomorphisms between two concentric annuli A(1,r) and B(1,R). First we construct radial minimizers of total combined energy, then we prove that those radial minimizers are absolute minimizers on the class of all mappings between the annuli under certain constraint. This extends the main result obtained by Iwaniec and Onninen in 9.
We present a control design for semilinear and quasilinear 2 × 2 hyperbolic partial differential equations with the control input at one boundary and a nonlinear ordinary differential equation ...coupled to the other. The controller can be designed to asymptotically stabilize the system at an equilibrium or relative to a reference signal. Two related but different controllers for semilinear and general quasilinear systems are presented and the additional challenges in quasilinear systems are discussed. Moreover, we present an observer that estimates the distributed PDE state and the unmeasured ODE state from measurements at the actuated boundary only, which can be used to also solve the output feedback control problem.
This paper develops a fault compensation scheme for a class of Euler-Bernoulli beam-ODE cascade systems to deal with certain boundary input faults, using a model transform-based backstepping control ...design. A model transformation is introduced to convert the Euler-Bernoulli beam-ODE cascade system to a Schrödinger-ODE cascade system. The FTC law is designed to compensate boundary input faults for the Schrödinger-ODE cascade system by constructing a normal intermediate system and a new exponentially stable target system. The purposes of the two-step backstepping are to bring in a state feedback and to improve the system performance. The performance of the target system, intermediate system, and Schrödinger-ODE cascade system is analysed in the Riesz basis frame. The stability analysis of the original Euler-Bernoulli beam-ODE cascade system is further provided. Simulation results illustrate the effectiveness of the proposed fault compensation scheme.
Computer algebra systems play an important role in science as they facilitate the development of new theoretical models. The resulting symbolic equations are often implemented in a compiled ...programming language in order to provide fast and portable codes for practical applications. We describe sympy2c, a new Python package designed to bridge the gap between the symbolic development and the numerical implementation of a theoretical model. sympy2c translates symbolic equations implemented in the SymPy Python package to C/C++ code that is optimized using symbolic transformations. The resulting functions can be conveniently used as an extension module in Python. sympy2c is used within the PyCosmo Python package to solve the Einstein–Boltzmann equations, a large system of ODEs describing the evolution of linear perturbations in the Universe. After reviewing the functionalities and usage of sympy2c, we describe its implementation and optimization strategies. This includes, in particular, a novel approach to generate optimized ODE solvers making use of the sparsity of the symbolic Jacobian matrix. We demonstrate its performance using the Einstein–Boltzmann equations as a test case. sympy2c is general and potentially useful for various areas of computational physics. sympy2c is publicly available at https://cosmology.ethz.ch/research/software-lab/sympy2c.html.
We analyze the problem of controlling parameter-dependent systems. We introduce the notion of averaged control according to which the quantity of interest is the average of the states with respect to ...the parameter.
First we consider the problem of controllability for linear finite-dimensional systems and show that a necessary and sufficient condition for averaged controllability is an averaged rank condition, in the spirit of the classical rank condition for linear control systems, but involving averaged momenta of any order of the matrices generating the dynamics and representing the control action.
We also describe some open problems and directions of possible research, in particular on the average controllability of evolution partial differential equations. In this context we analyze also the averaged version of a classical optimal control problem for a parameter dependent elliptic equation and derive the corresponding optimality system.
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