Journal articles: 'Differential system state equations'

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Tukhtasinov, Muminjon, Gafurjan Ibragimov, Sarvinoz Kuchkarova, and Risman Mat Hasim. "Differential Games for an Infinite 2-Systems of Differential Equations." Mathematics 9, no. 13 (June 23, 2021): 1467. http://dx.doi.org/10.3390/math9131467.

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A pursuit differential game described by an infinite system of 2-systems is studied in Hilbert space l2. Geometric constraints are imposed on control parameters of pursuer and evader. The purpose of pursuer is to bring the state of the system to the origin of the Hilbert space l2 and the evader tries to prevent this. Differential game is completed if the state of the system reaches the origin of l2. The problem is to find a guaranteed pursuit and evasion times. We give an equation for the guaranteed pursuit time and propose an explicit strategy for the pursuer. Additionally, a guaranteed evasion time is found.

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Trigeassou, Jean-Claude, and Nezha Maamri. "Optimal State Control of Fractional Order Differential Systems: The Infinite State Approach." Fractal and Fractional 5, no. 2 (April 5, 2021): 29. http://dx.doi.org/10.3390/fractalfract5020029.

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Optimal control of fractional order systems is a long established domain of fractional calculus. Nevertheless, it relies on equations expressed in terms of pseudo-state variables which raise fundamental questions. So in order remedy these problems, the authors propose in this paper a new and original approach to fractional optimal control based on a frequency distributed representation of fractional differential equations called the infinite state approach, associated with an original formulation of fractional energy, which is intended to really control the internal system state. In the first step, the fractional calculus of variations is revisited to express appropriate Euler Lagrange equations. Then, the quadratic optimal control of fractional linear systems is formulated. Thanks to a frequency discretization technique, the previous theoretical equations are converted into an equivalent large dimension integer order system which permits the implementation of a feasible optimal solution. A numerical example illustrates the validity of this new approach.

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JIANG, YAO-LIN. "STEADY-STATE METHODS OF DIFFERENTIAL-ALGEBRAIC EQUATIONS IN CIRCUIT SIMULATION." Journal of Circuits, Systems and Computers 14, no. 02 (April 2005): 383–93. http://dx.doi.org/10.1142/s0218126605002313.

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In the paper, we study the steady-state methods of large dynamic systems. For a nonlinear system of differential-algebraic equations with a known period T, we decouple it, in function space, into linear subsystems by quasilinearization. The resulting linear dynamic systems can be solved by a waveform Krylov subspace method. For the autonomous case, that is, the period T is unknown, the well-known shooting process can be applied where Newton iterations are computed with pseudo-inverse.

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GREENBAUN, N. N. "THE MARGINAL STABILITY OF AUTONOMOUS DIFFERENTIAL EQUATIONS." International Journal of Bifurcation and Chaos 03, no. 03 (June 1993): 785–88. http://dx.doi.org/10.1142/s0218127493000702.

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Solutions in the vicinity of a steady state solution to a system of autonomous nonlinear differential equations are of interest to modelers. The usual method for determining marginal stability of the steady state is the Routh-Hurwitz criterion. The method offered here is less complicated and more efficient when the number of state variables exceeds three.

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Ibănescu, M., and R. Ibănescu. "New approach in solving derivative causality problems in the bond-graph method." IOP Conference Series: Materials Science and Engineering 1235, no. 1 (March 1, 2022): 012053. http://dx.doi.org/10.1088/1757-899x/1235/1/012053.

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Abstract From the bond-graph model of a dynamic system, the state equations can be obtained. When the system contains only energy storing elements in integral causality, the system of state equations is immediately and directly determined. When the model contains at least one energy storing element in derivative causality, the resulted system of differential-algebraic equations arises some difficulties in finding the final form of the system of state equations. The work presents a new method of deducing the state equations in case of bond-graph models with one, or several energy storing elements in derivative causality. This method is based on the kinetic energy of the system and offers the possibility to avoid a difficult mathematical calculus for the transition from a system of differential algebraic equations to the system of state equations.

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Chen, Xiaopeng, and Jinqiao Duan. "State space decomposition for non-autonomous dynamical systems." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 141, no. 5 (September 26, 2011): 957–74. http://dx.doi.org/10.1017/s0308210510000661.

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The decomposition of state spaces into dynamically different components is helpful for understanding dynamics of complex systems. A Conley-type decomposition theorem is proved for non-autonomous dynamical systems defined on a non-compact but separable state space. Specifically, the state space can be decomposed into a chain-recurrent part and a gradient-like part. This result applies to both non-autonomous ordinary differential equations on a Euclidean space (which is only locally compact), and to non-autonomous partial differential equations on an infinite-dimensional function space (which is not even locally compact). This decomposition result is demonstrated by discussing a few concrete examples, such as the Lorenz system and the Navier–Stokes system, under time-dependent forcing.

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Zawadzki, Andrzej. "Application of local coordinates rectification in linearization of selected parameters of dynamic nonlinear systems." COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering 33, no. 5 (August 26, 2014): 1819–30. http://dx.doi.org/10.1108/compel-11-2013-0358.

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Purpose – The purpose of this paper is to aim to an application of element of the theory of differential geometry for building the state space transformation, linearizing nonlinear dynamic system into a linear form. Design/methodology/approach – It is assumed that the description of nonlinear electric circuits with concentrated parameters or electromechanical systems is given by nonlinear system of differential equations of first order (state equations). The goal is to find transformation which leads nonlinear state equation (written in one coordinate system) to the linear in the other – sought coordinate system. Findings – The necessary conditions fulfilled by nonlinear system undergoing linearization process are presented. Numerical solutions of the nonlinear equations of state together with linearized system obtained from direct transformation of the state space are included (transformation input – the state of the nonlinear system). Originality/value – Application of first order exact differential forms for determining the transformation linearizing the nonlinear state equation. Simple linear models obtained with the use of the linearizing transformation are very useful (mainly because of the known and well-mastered theory of linear systems) in solving of various practical technical problems.

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Zhu, Qingfeng, Yufeng Shi, Jiaqiang Wen, and Hui Zhang. "A Type of Time-Symmetric Stochastic System and Related Games." Symmetry 13, no. 1 (January 12, 2021): 118. http://dx.doi.org/10.3390/sym13010118.

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This paper is concerned with a type of time-symmetric stochastic system, namely the so-called forward–backward doubly stochastic differential equations (FBDSDEs), in which the forward equations are delayed doubly stochastic differential equations (SDEs) and the backward equations are anticipated backward doubly SDEs. Under some monotonicity assumptions, the existence and uniqueness of measurable solutions to FBDSDEs are obtained. The future development of many processes depends on both their current state and historical state, and these processes can usually be represented by stochastic differential systems with time delay. Therefore, a class of nonzero sum differential game for doubly stochastic systems with time delay is studied in this paper. A necessary condition for the open-loop Nash equilibrium point of the Pontriagin-type maximum principle are established, and a sufficient condition for the Nash equilibrium point is obtained. Furthermore, the above results are applied to the study of nonzero sum differential games for linear quadratic backward doubly stochastic systems with delay. Based on the solution of FBDSDEs, an explicit expression of Nash equilibrium points for such game problems is established.

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Klamka, J., A. Babiarz, and M. Niezabitowski. "Banach fixed-point theorem in semilinear controllability problems – a survey." Bulletin of the Polish Academy of Sciences Technical Sciences 64, no. 1 (March 1, 2016): 21–35. http://dx.doi.org/10.1515/bpasts-2016-0004.

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Abstract The main aim of this article is to review the existing state of art concerning the complete controllability of semilinear dynamical systems. The study focus on obtaining the sufficient conditions for the complete controllability for various systems using the Banach fixedpoint theorem. We describe the results for stochastic semilinear functional integro-differential system, stochastic partial differential equations with finite delays, semilinear functional equations, a stochastic semilinear system, a impulsive stochastic integro-differential system, semilinear stochastic impulsive systems, an impulsive neutral functional evolution integro-differential system and a nonlinear stochastic neutral impulsive system. Finally, two examples are presented.

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Glagolev, Mikhail V., Aleksandr F. Sabrekov, and Vladimir M. Goncharov. "Delay differential equations as a tool for mathematical modelling of population dynamic." Environmental Dynamics and Global Climate Change 9, no. 2 (November 27, 2018): 40–63. http://dx.doi.org/10.17816/edgcc10483.

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The manuscript constitutes a lecture from a course “Mathematical modelling of biological processes”, adapted to the format of the journal paper. This course of lectures is held by one of authors in Ugra State University. Delay differential equations are widely used in different ecological and biological problems. It has to do with the fact that delay differential equations are able to take into account that different biological processes depend not only on the state of the system at the moment but on the state of the system in previous moments too. The most popular case of using delay differential equations in biology is modelling in population ecology (including the modelling of several interacting populations dynamic, for example, in predator-prey system). Also delay differential equations are considered in demography, immunology, epidemiology, molecular biology (to provide mathematical description of regulatory mechanisms in a cell functioning and division), physiology as well as for modelling certain important production processes (for example, in agriculture). In the beginning of the paper as introduction some basic concepts of differential difference equation theory (delay differential equations are specific type of differential difference equations) is considered and their classification is presented. Then it is discussed in more detail how such an important equations of population dynamic as Maltus equation and logistic (Verhulst-Pearl) equation are transformed into corresponsive delay differential equations – Goudriaan-Roermund and Hutchinson. Then several discussion questions on using of a delay differential equations in biological models are considered. It is noted that in a certain cases using of a delay differential equations lead to an incorrect behavior from the point of view of common sense. Namely solution of Goudriaan-Roermund equation with harvesting, stopped when all species were harvested, shows that spontaneous generation takes place in the system. This incorrect interpretation has to do with the fact that delay differential equations are used to simplify considered models (that are usually are systems of ordinary differential equations). Using of integro-differential equations could be more appropriate because in these equations background could be taken into account in a more natural way. It is shown that Hutchinson equation can be obtained by simplification of Volterra integral equation with a help of Diraq delta function. Finally, a few questions of analytical and numerical solution of delay differential equations are discussed.

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Bashirov, Agamirza E., and Maher Jneid. "On Partial Complete Controllability of Semilinear Systems." Abstract and Applied Analysis 2013 (2013): 1–8. http://dx.doi.org/10.1155/2013/521052.

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Many control systems can be written as a first-order differential equation if the state space enlarged. Therefore, general conditions on controllability, stated for the first-order differential equations, are too strong for these systems. For such systems partial controllability concepts, which assume the original state space, are more suitable. In this paper, a sufficient condition for the partial complete controllability of semilinear control system is proved. The result is demonstrated through examples.

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Ibănescu, Radu, and Mihaela Ibănescu. "A Method to Find the Differential Equations from the Bond Graph Containing Inertial Elements in Derivative Causality." Bulletin of the Polytechnic Institute of Iași. Machine constructions Section 67, no. 4 (December 1, 2021): 35–44. http://dx.doi.org/10.2478/bipcm-2021-0021.

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Abstract The bond-graph method is used for finding the equations which describe the systems dynamics, by analysing the way of power transmission from the source to the working elements. When the energy storage elements I and C are in integral causality, the number of state equations equals the number of these elements. If there are also energy storage elements in derivative causality, the system contains a number of differential equations equal to the number of energy storage elements in integral causality and a number of algebraic equations equal to the number of energy storage elements in derivative causality. The work presents a new method for finding the system of differential equations for mechanical systems with one degree of freedom, starting from the original system which contains both algebraic and differential equations.

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FILIPPOVA, TATIANA. "SET-VALUED DYNAMICS IN PROBLEMS OF MATHEMATICAL THEORY OF CONTROL PROCESSES." International Journal of Modern Physics B 26, no. 25 (September 10, 2012): 1246010. http://dx.doi.org/10.1142/s0217979212460101.

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The dynamics and properties of set-valued states of differential control systems with uncertainties in initial data are studied. It is assumed that the dynamical system has a special structure, in which nonlinear terms in the right-hand sides of related differential equations are quadratic in state coordinates. We construct external and internal ellipsoidal estimates of reachable sets of nonlinear control system and find differential equations of proposed ellipsoidal estimates of reachable sets of nonlinear control system. The results obtained for quadratic system nonlinearities are extended to other types of control systems under uncertainty.

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BERTOTTI, MARIA LETIZIA, and MARCELLO DELITALA. "FROM DISCRETE KINETIC AND STOCHASTIC GAME THEORY TO MODELLING COMPLEX SYSTEMS IN APPLIED SCIENCES." Mathematical Models and Methods in Applied Sciences 14, no. 07 (July 2004): 1061–84. http://dx.doi.org/10.1142/s0218202504003544.

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This paper deals with some methodological aspects related to the discretization of a class of integro-differential equations modelling the evolution of the probability distribution over the microscopic state of a large system of interacting individuals. The microscopic state includes both mechanical and socio-biological variables. The discretization of the microscopic state generates a class of dynamical systems defining the evolution of the densities of the discretized state. In general, this yields a system of partial differential equations replacing the continuous integro-differential equation. As an example, a specific application is discussed, which refers to modelling in the field of social dynamics. The derivation of the evolution equation needs the development of a stochastic game theory.

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Yang, Z., and J. P. Sadler. "A Numerically Efficient Algorithm for Steady-State Response of Flexible Mechanism Systems." Journal of Mechanical Design 115, no. 4 (December 1, 1993): 848–55. http://dx.doi.org/10.1115/1.2919278.

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Presented in this work is a numerically efficient algorithm for treating the periodic steady-state response of flexible mechanisms as the solution to separated two-point boundary value problems. The finite element method is applied to discretize continuous elastic mechanisms systems and a set of second-order ordinary differential equations is obtained with periodically time-varying coefficient matrices and forcing vectors. Modal analysis techniques are employed to decouple these equations into a number of single scalar ordinary differential equations in modal basis. The periodic time-boundary conditions at both ends of a fundamental period equal to a cycle of input motion are mathematically separated by introducing auxiliary variables, thus resulting in a so-called almost-block-diagonal matrix for linear algebraic systems of equations. Solving such a system is computationally less expensive than solving a general linear algebraic system. Examples are included to illustrate the procedures applied to a four-bar linkage through which computing time is compared with other approaches.

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Asl, Farshid Maghami, and A. Galip Ulsoy. "Analysis of a System of Linear Delay Differential Equations." Journal of Dynamic Systems, Measurement, and Control 125, no. 2 (June 1, 2003): 215–23. http://dx.doi.org/10.1115/1.1568121.

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A new analytic approach to obtain the complete solution for systems of delay differential equations (DDE) based on the concept of Lambert functions is presented. The similarity with the concept of the state transition matrix in linear ordinary differential equations enables the approach to be used for general classes of linear delay differential equations using the matrix form of DDEs. The solution is in the form of an infinite series of modes written in terms of Lambert functions. Stability criteria for the individual modes, free response, and forced response for delay equations in different examples are studied, and the results are presented. The new approach is applied to obtain the stability regions for the individual modes of the linearized chatter problem in turning. The results present a necessary condition to the stability in chatter for the whole system, since it only enables the study of the individual modes, and there are an infinite number of them that contribute to the stability of the system.

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Rigatos, G., and P. Siano. "Feedback control of the multi-asset Black–Scholes PDE using differential flatness theory." International Journal of Financial Engineering 03, no. 02 (June 2016): 1650008. http://dx.doi.org/10.1142/s2424786316500080.

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A method for feedback control of the multi-asset Black–Scholes PDE is developed. By applying semi-discretization and a finite differences scheme the multi-asset Black–Scholes PDE is transformed into a state-space model consisting of ordinary nonlinear differential equations. For this set of differential equations it is shown that differential flatness properties hold. This enables to solve the associated control problem and to succeed stabilization of the options’ dynamics. It is shown that the previous procedure results into a set of nonlinear ordinary differential equations (ODEs) and to an associated state equations model. For the local subsystems, into which a Black–Scholes PDE is decomposed, it becomes possible to apply boundary-based feedback control. The controller design proceeds by showing that the state-space model of the Black–Scholes PDE stands for a differentially flat system. Next, for each subsystem which is related to a nonlinear ODE, a virtual control input is computed, that can invert the subsystem’s dynamics and can eliminate the subsystem’s tracking error. From the last row of the state-space description, the control input (boundary condition) that is actually applied to the multi-asset Black–Scholes PDE system is found. This control input contains recursively all virtual control inputs which were computed for the individual ODE subsystems associated with the previous rows of the state-space equation. Thus, by tracing the rows of the state-space model backwards, at each iteration of the control algorithm, one can finally obtain the control input that should be applied to the Black–Scholes PDE system so as to assure that all its state variables will converge to the desirable setpoints.

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Mann, B. P., and B. R. Patel. "Stability of Delay Equations Written as State Space Models." Journal of Vibration and Control 16, no. 7-8 (June 2010): 1067–85. http://dx.doi.org/10.1177/1077546309341111.

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In this paper we describe a new approach to examine the stability of delay differential equations that builds upon prior work using temporal finite element analysis. In contrast to previous analyses, which could only be applied to second-order delay differential equations, the present manuscript develops an approach which can be applied to a broader class of systems: systems that may be written in the form of a state space model. A primary outcome from this work is a generalized framework to investigate the asymptotic stability of autonomous delay differential equations with a single time delay. Furthermore, this approach is shown to be applicable to time-periodic delay differential equations and equations that are piecewise continuous.

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Kashchenko, Ilia. "Normalization of a System with Two Large Delays." International Journal of Bifurcation and Chaos 24, no. 08 (August 2014): 1440021. http://dx.doi.org/10.1142/s0218127414400215.

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In this paper, the local dynamics of differential equations is studied with two asymptotically large proportional delays. Depending on the parameters, critical cases in the problem of the stability of the equilibrium state are identified. In all critical cases, special evolutionary equations (quasinormal forms) are built. Their nonlocal dynamics determine the local behavior of solutions of the original equations.

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SRI NAMACHCHIVAYA, N., and VOLKER WIHSTUTZ. "ALMOST SURE ASYMPTOTIC STABILITY OF SCALAR STOCHASTIC DELAY EQUATIONS: FINITE STATE MARKOV PROCESS." Stochastics and Dynamics 12, no. 01 (March 2012): 1150010. http://dx.doi.org/10.1142/s0219493712003560.

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In this paper, we study the almost-sure asymptotic stability of scalar delay differential equations with random parametric fluctuations which are modeled by a Markov process with finitely many states. The techniques developed for the determination of almost-sure asymptotic stability of finite dimensional stochastic differential equations will be extended to delay differential equations with random parametric fluctuations. For small intensity noise, we construct an asymptotic expansion for the exponential growth rate (the maximal Lyapunov exponent), which determines the almost-sure stability of the stochastic system.

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Kalis, H. "THE EXACT FINITE-DIFFERENCE SCHEME FOR VECTOR BOUNDARY‐VALUE PROBLEMS WITH PIECE‐WISE CONSTANT COEFFICIENTS." Mathematical Modelling and Analysis 3, no. 1 (December 15, 1998): 114–23. http://dx.doi.org/10.3846/13926292.1998.9637094.

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We will consider the exact finite‐difference scheme for solving the system of differential equations of second order with piece‐wise constant coefficients. It is well‐known, that the presence of large parameters at first order derivatives or small parameters at second order derivatives in the system of hydrodynamics and magnetohydrodynamics (MHD) equations (large Reynolds, Hartmann and others numbers) causes additional difficulties for the applications of general classical numerical methods. Thus, important to work out special methods of solution, the so‐called uniform converging computational methods. This gives a basis for the development of special monotone finite vector‐difference schemes with perturbation coefficient of function‐matrix for solving the system of differential equations. Special finite‐difference approximations are constructed for a steady‐state boundary‐value problem, systems of parabolic type partial differential equations, a system of two MHD equations, 2‐D flows and MHD‐flows equations in curvilinear orthogonal coordinates.

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Bulanov, Denis Mikhailovich, and Victor Vasil’yevich Sazonov. "Steady-state rotational motion of the Photon M-2 satellite." Keldysh Institute Preprints, no. 63 (2021): 1–36. http://dx.doi.org/10.20948/prepr-2021-63.

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At the end of the flight, the attitude motion of the Photon M-2 satellite (it was in orbit 2005.05.31-2005.06.16) can be described by a generalized conservative system of differential equations. The secular change in the own kinetic moment of this satellite is described by the so-called evolutionary equations of Beletsky, which also form a generalized conservative system. The preprint examines the relationship between these systems. The satellite motion equations are reduced to equations of the 4th order describing the motion of the satellite axis of symmetry. Beletsky's equations are of the second order and describe the secular motion of the ort of the satellite's own kinetic moment. The solutions of these systems of equations corresponding to the real movements of the satellite are, respectively, conditionally periodic and periodic. The solutions of the 4th-order system are dominated by two frequencies – high and low ones. The spectral analysis showed that the low frequency coincides with the frequency of solutions of Beletsky’s equations. And the solutions of these equations coincide with the low-frequency component in the solution of the 4th-order system with respect to the variables that determine the direction of the axis of symmetry of the satellite.

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Arguchintsev, Alexander, and Vasilisa Poplevko. "An Optimal Control Problem by a Hybrid System of Hyperbolic and Ordinary Differential Equations." Games 12, no. 1 (March 3, 2021): 23. http://dx.doi.org/10.3390/g12010023.

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This paper deals with an optimal control problem for a linear system of first-order hyperbolic equations with a function on the right-hand side determined from controlled bilinear ordinary differential equations. These ordinary differential equations are linear with respect to state functions with controlled coefficients. Such problems arise in the simulation of some processes of chemical technology and population dynamics. Normally, general optimal control methods are used for these problems because of bilinear ordinary differential equations. In this paper, the problem is reduced to an optimal control problem for a system of ordinary differential equations. The reduction is based on non-classic exact increment formulas for the cost-functional. This treatment allows to use a number of efficient optimal control methods for the problem. An example illustrates the approach.

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Draouil, Olfa, and Bernt Øksendal. "Optimal insider control of stochastic partial differential equations." Stochastics and Dynamics 18, no. 01 (November 6, 2017): 1850014. http://dx.doi.org/10.1142/s0219493718500144.

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We study the problem of optimal insider control of an SPDE (a stochastic evolution equation) driven by a Brownian motion and a Poisson random measure. Our optimal control problem is new in two ways: (i) The controller has access to inside information, i.e. access to information about a future state of the system. (ii) The integro-differential operator of the SPDE might depend on the control. In the first part of the paper, we formulate a sufficient and necessary maximum principle for this type of control problem, in two cases: The control is allowed to depend both on time [Formula: see text] and on the space variable [Formula: see text]. The control is not allowed to depend on [Formula: see text]. In the second part of the paper, we apply the results above to the problem of optimal control of an SDE system when the inside controller has only noisy observations of the state of the system. Using results from non-linear filtering, we transform this noisy observation SDE inside control problem into a full observation SPDE insider control problem. The results are illustrated by explicit examples.

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Lisowski, Józef. "The Sensitivity of State Differential Game Vessel Traffic Model." Polish Maritime Research 23, no. 2 (April 1, 2016): 14–18. http://dx.doi.org/10.1515/pomr-2016-0015.

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Abstract The paper presents the application of the theory of deterministic sensitivity control systems for sensitivity analysis implemented to game control systems of moving objects, such as ships, airplanes and cars. The sensitivity of parametric model of game ship control process in collision situations have been presented. First-order and k-th order sensitivity functions of parametric model of process control are described. The structure of the game ship control system in collision situations and the mathematical model of game control process in the form of state equations, are given. Characteristics of sensitivity functions of the game ship control process model on the basis of computer simulation in Matlab/Simulink software have been presented. In the end, have been given proposals regarding the use of sensitivity analysis to practical synthesis of computer-aided system navigator in potential collision situations.

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Hernandez-Gonzalez, Miguel, and Michael V. Basin. "State estimation for stochastic polynomial systems with switching in the state equation." Transactions of the Institute of Measurement and Control 40, no. 9 (November 21, 2017): 2732–39. http://dx.doi.org/10.1177/0142331217737134.

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The problem of designing a mean-square filter has been studied for stochastic polynomial systems, where the state equation switches between two different nonlinear functions, over linear observations. A switching signal depends on a random variable modelled as a Bernoulli distributed sequence that takes the quantities of zero and one. The differential equations for the state estimate and the error covariance matrix are obtained in a closed form by expressing the conditional expectation of polynomial terms as functions of the estimate and covariance matrix. Finite-dimensional filtering equations are obtained for a particular case of a third-degree polynomial system. Numerical simulations are carried out in two cases of switching between different linear and second degree polynomial functions.

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VALENCIANO, J., and M. A. J. CHAPLAIN. "AN EXPLICIT SUBPARAMETRIC SPECTRAL ELEMENT METHOD OF LINES APPLIED TO A TUMOUR ANGIOGENESIS SYSTEM OF PARTIAL DIFFERENTIAL EQUATIONS." Mathematical Models and Methods in Applied Sciences 14, no. 02 (February 2004): 165–87. http://dx.doi.org/10.1142/s0218202504003155.

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In this paper we consider a numerical solution to Anderson and Chaplain's tumour angiogenesis model1 over two-dimensional complex geometry. The numerical solution of the governing system of non-linear evolutionary partial differential equations is obtained using the method of lines: after a spatial semi-discretisation based on the subparametric Legendre spectral element method is performed, the original system of partial differential equations is replaced by an augmented system of stiff ordinary differential equations in autonomous form, which is then advanced forward in time using an explicit time integrator based on the fourth-order Chebyshev polynomial. Numerical simulations show the convergence of the steady state numerical solution towards the linearly stable steady state analytical solution.

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Lynch, A. G., and M. J. Vanderploeg. "A Symbolic Formulation for Linearization of Multibody Equations of Motion." Journal of Mechanical Design 117, no. 3 (September 1, 1995): 441–45. http://dx.doi.org/10.1115/1.2826698.

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This paper presents a method for obtaining linearized state space representations of open or closed loop multibody dynamic systems. The paper develops a symbolic formulation for multibody dynamic systems which result in an explicit set of symbolic equations of motion. The symbolic equations are then used to perform symbolic linearizations. The resulting symbolic, linear equations are in terms of the system parameters and the equilibrium point, and are valid for any equilibrium point. Finally, a method is developed for reducing a linearized, constrained multibody system consisting of a mixed set of algebraic-differential equations to a reduced set of differential equations in terms of an independent coordinate set. An example is used to demonstrate the technique.

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FJÄLLBORG, MIKAEL, J. MARK HEINZLE, and CLAES UGGLA. "Self-gravitating stationary spherically symmetric systems in relativistic galactic dynamics." Mathematical Proceedings of the Cambridge Philosophical Society 143, no. 3 (November 2007): 731–52. http://dx.doi.org/10.1017/s0305004107000540.

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AbstractWe study equilibrium states in relativistic galactic dynamics which are described by stationary solutions of the Einstein–Vlasov system for collisionless matter. We recast the equations into a regular three-dimensional system of autonomous first order ordinary differential equations on a bounded state space. Based on a dynamical systems analysis we derive new theorems that guarantee that the steady state solutions have finite radii and masses.

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Valliammal, N., C. Ravichandran, Zakia Hammouch, and Haci Mehmet Baskonus. "A New Investigation on Fractional-Ordered Neutral Differential Systems with State-Dependent Delay." International Journal of Nonlinear Sciences and Numerical Simulation 20, no. 7-8 (November 18, 2019): 803–9. http://dx.doi.org/10.1515/ijnsns-2018-0362.

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AbstractFractional differential equations with delay behaviors occur in fields like physical and biological ones with state-dependent delay or nonconstant delay and has drawn the attention of researchers. The main goal of the present work is to study the existence of mild solutions of neutral differential system along state-dependent delay in Banach space. By employing the fractional theory, noncompact measure and Mönch’s theorem, we investigate the existence results for neutral differential equations of fractional order with state-dependent delay. An illustration of derived results is offered.

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Chen, Li, Zhen Wu, and Zhiyong Yu. "Delayed Stochastic Linear-Quadratic Control Problem and Related Applications." Journal of Applied Mathematics 2012 (2012): 1–22. http://dx.doi.org/10.1155/2012/835319.

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We discuss a quadratic criterion optimal control problem for stochastic linear system with delay in both state and control variables. This problem will lead to a kind of generalized forward-backward stochastic differential equations (FBSDEs) with Itô’s stochastic delay equations as forward equations and anticipated backward stochastic differential equations as backward equations. Especially, we present the optimal feedback regulator for the time delay system via a new type of Riccati equations and also apply to a population optimal control problem.

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MICHIELS, W., and D. ROOSE. "LIMITATIONS OF DELAYED STATE FEEDBACK: A NUMERICAL STUDY." International Journal of Bifurcation and Chaos 12, no. 06 (June 2002): 1309–20. http://dx.doi.org/10.1142/s0218127402005091.

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Stabilization of a class of linear time-delay systems can be achieved by a numerical procedure, called the continuous pole placement method [Michiels et al., 2000]. This method can be seen as an extension of the classical pole placement algorithm for ordinary differential equations to a class of delay differential equations. In [Michiels et al., 2000] it was applied to the stabilization of a linear time-invariant system with an input delay using static state feedback. In this paper we study the limitations of such delayed state feedback laws. More precisely we completely characterize the class of stabilizable plants in the 2D-case. For that purpose we make use of numerical continuation techniques. The use of delayed state feedback in various control applications and the effect of its limitations are briefly discussed.

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Sendrescu, Dorin. "Distribution-Based Identification of Yield Coefficients in a Baker’s Yeast Bioprocess." Mathematical Problems in Engineering 2012 (2012): 1–21. http://dx.doi.org/10.1155/2012/789156.

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A distribution-based identification procedure for estimation of yield coefficients in a baker’s yeast bioprocess is proposed. This procedure transforms a system of differential equations to a system of algebraic equations with respect to unknown parameters. The relation between the state variables is represented by functionals using techniques from distribution theory. A hierarchical structure of identification is used, which allows obtaining a linear algebraic system of equations in the unknown parameters. The coefficients of this algebraic system are functionals depending on the input and state variables evaluated through some test functions from distribution theory. First, only some state equations are evaluated throughout test functions to obtain a set of linear equations in parameters. The results of this first stage of identification are used to express other parameters by linear equations. The process is repeated until all parameters are identified. The performances of the method are analyzed by numerical simulations.

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BOJOWALD, MARTIN, and AURELIANO SKIRZEWSKI. "EFFECTIVE EQUATIONS OF MOTION FOR QUANTUM SYSTEMS." Reviews in Mathematical Physics 18, no. 07 (August 2006): 713–45. http://dx.doi.org/10.1142/s0129055x06002772.

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In many situations, one can approximate the behavior of a quantum system, i.e. a wave function subject to a partial differential equation, by effective classical equations which are ordinary differential equations. A general method and geometrical picture are developed and shown to agree with effective action results, commonly derived through path integration, for perturbations around a harmonic oscillator ground state. The same methods are used to describe dynamical coherent states, which in turn provide means to compute quantum corrections to the symplectic structure of an effective system.

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Kapustyan, Vladimir, and Vyacheslav Maksimov. "On attaining the prescribed quality of a controlled fourth order system." International Journal of Applied Mathematics and Computer Science 24, no. 1 (March 1, 2014): 75–85. http://dx.doi.org/10.2478/amcs-2014-0006.

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Abstract In this paper, we discuss a method of auxiliary controlled models and its application to solving some robust control problems for a system described by differential equations. As an illustration, a system of nonlinear differential equations of the fourth order is used. A solution algorithm, which is stable with respect to informational noise and computational errors, is presented. The algorithm is based on a combination of online state/input reconstruction and feedback control methods.

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Bharath, S., B. C. Nakra, and K. N. Gupta. "Mathematical Model of a Railway Pneumatic Brake System With Varying Cylinder Capacity Effects." Journal of Dynamic Systems, Measurement, and Control 112, no. 3 (September 1, 1990): 456–62. http://dx.doi.org/10.1115/1.2896164.

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Governing equations for the analysis of pressure transient are derived from the principle of conservation of mass and momentum for a pneumatic brake system, which consists of a train pipe connected to a number of linear actuators (brake cylinders with piston displacement). The governing one-dimensional non-linear partial differential equations for the train pipe, non-linear ordinary differential equations for the brake cylinders, and second-order differential equation of motion for piston displacement are solved to determine the pressure transients in the brake system for a step change in pressure at the inlet. The governing equations are nondimensionalized and reduced to a set of ordinary nonlinear differential difference equations and integrated by standard numerical methods. The flow is considered isothermal, and the friction effects for turbulent and laminar flow are evaluated by quasi-steady state approximation. The auxiliary reservoir volume effect is also included. The results are compared with the experimental data obtained on a brake test rig.

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ALFIFI, H. Y. "SEMI-ANALYTICAL SOLUTIONS FOR THE BRUSSELATOR REACTION–DIFFUSION MODEL." ANZIAM Journal 59, no. 2 (October 2017): 167–82. http://dx.doi.org/10.1017/s1446181117000311.

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Semi-analytical solutions are derived for the Brusselator system in one- and two-dimensional domains. The Galerkin method is processed to approximate the governing partial differential equations via a system of ordinary differential equations. Both steady-state concentrations and transient solutions are obtained. Semi-analytical results for the stability of the model are presented for the identified critical parameter value at which a Hopf bifurcation occurs. The impact of the diffusion coefficients on the system is also considered. The results show that diffusion acts to stabilize the systems better than the equivalent nondiffusive systems with the increasing critical value of the Hopf bifurcation. Comparison between the semi-analytical and numerical solutions shows an excellent agreement with the steady-state transient solutions and the parameter values at which the Hopf bifurcations occur. Examples of stable and unstable limit cycles are given, and Hopf bifurcation points are shown to confirm the results previously calculated in the Hopf bifurcation map. The usefulness and accuracy of the semi-analytical results are confirmed by comparison with the numerical solutions of partial differential equations.

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Sarangapani, P., T. Thiessen, and W. Mathis. "Differential Algebraic Equations of MOS Circuits and Jump Behavior." Advances in Radio Science 10 (October 2, 2012): 327–32. http://dx.doi.org/10.5194/ars-10-327-2012.

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Abstract. Many nonlinear electronic circuits showing fast switching behavior exhibit jump effects which occurs when the state space of the electronic system contains a fold. This leads to difficulties during the simulation of these systems with standard circuit simulators. A method to overcome these problems is by regularization, where parasitic inductors and capacitors are added at the suitable locations. However, the transient solution will not be reliable if this regularization is not done in accordance with Tikhonov's Theorem. A geometric approach is taken to overcome these problems by explicitly computing the state space and jump points of the circuit. Until now, work has been done in analyzing example circuits exhibiting this behavior for BJT transistors. In this work we apply these methods to MOS circuits (Schmitt trigger, flip flop and multivibrator) and present the numerical results. To analyze the circuits we use the EKV drain current model as equivalent circuit model for the MOS transistors.

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Popov, V. G. "System of cracks under the impact of plane elastic waves." Journal of Physics: Conference Series 2231, no. 1 (April 1, 2022): 012004. http://dx.doi.org/10.1088/1742-6596/2231/1/012004.

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Abstract The problem of determining the two-dimensional diffraction field and the stress state arising from the interaction of elastic waves with a system of arbitrarily located cracks is solved. Cracks are located in an unbounded elastic medium which is in a plane deformation state. The method of solution is based on the use of discontinuous solutions of the equations of motion of elastic media and the reduction of the original problem to a system of singular integro-differential equations. The obtained systems are solved numerically the mechanical quadrature method.

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Ma, Junxia, Qiuling Fei, Fan Guo, and Weili Xiong. "Variational Bayesian Iterative Estimation Algorithm for Linear Difference Equation Systems." Mathematics 7, no. 12 (November 22, 2019): 1143. http://dx.doi.org/10.3390/math7121143.

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Many basic laws of physics or chemistry can be written in the form of differential equations. With the development of digital signals and computer technology, the research on discrete models has received more and more attention. The estimates of the unknown coefficients in the discretized difference equation can be obtained by optimizing certain criterion functions. In modern control theory, the state-space model transforms high-order differential equations into first-order differential equations by introducing intermediate state variables. In this paper, the parameter estimation problem for linear difference equation systems with uncertain noise is developed. By transforming system equations into state-space models and on the basis of the considered priors of the noise and parameters, a variational Bayesian iterative estimation algorithm is derived from the observation data to obtain the parameter estimates. The unknown states involved in the variational Bayesian algorithm are updated by the Kalman filter. A numerical simulation example is given to validate the effectiveness of the proposed algorithm.

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Gordon, Brandon W., and Sheng Liu. "A Singular Perturbation Approach for Modeling Differential-Algebraic Systems." Journal of Dynamic Systems, Measurement, and Control 120, no. 4 (December 1, 1998): 541–45. http://dx.doi.org/10.1115/1.2801500.

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Dynamic systems described by an implicit mixed set of Differential and Algebraic Equations (DAEs) are often encountered in control system modeling and analysis due to inherent constraints in the system. A key difficulty in control and simulation of DAE systems is that they are not expressed in an explicit state space representation. This paper describes a general approach based on singular perturbation analysis for adding fast dynamics to a system of DAEs so that they can be expressed in an explicit state space form. Conditions for asymptotic convergence and approximation methods are investigated. The approach is illustrated for a model of a two-phase flow heat exchanger.

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Han, Xiaoshuang, Mingyan Teng, and Ming Fang. "Well-posedness and Stability of the Repairable System with Three Units and Vacation." Journal of Systems Science and Information 2, no. 1 (February 25, 2014): 54–76. http://dx.doi.org/10.1515/jssi-2014-0054.

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AbstractThe stability of the repairable system with three units and vacation was investigated by two different methods in this note. The repairable system is described by a set of ordinary differential equation coupled with partial differential equations with initial values and integral boundaries. To apply the theory of positive operator semigroups to discuss the repairable system, the system equations were transformed into an abstract Cauchy problem on some Banach lattice. The system equations have a unique non-negative dynamic solution and positive steady-state solution and dynamic solution strongly converges to steady-state solution were shown on the basis of the detailed spectral analysis of the system operator. Furthermore, the Cesáro mean ergodicity of the semigroupT(t) generated by the system operator was also shown through the irreducibility of the semigroup.

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Crank, Keith N. "A method for approximating the probability functions of a Markov chain." Journal of Applied Probability 25, no. 4 (December 1988): 808–14. http://dx.doi.org/10.2307/3214302.

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This paper presents a method of approximating the state probabilities for a continuous-time Markov chain. This is done by constructing a right-shift process and then solving the Kolmogorov system of differential equations recursively. By solving a finite number of the differential equations, it is possible to obtain the state probabilities to any degree of accuracy over any finite time interval.

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Crank, Keith N. "A method for approximating the probability functions of a Markov chain." Journal of Applied Probability 25, no. 04 (December 1988): 808–14. http://dx.doi.org/10.1017/s0021900200041607.

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This paper presents a method of approximating the state probabilities for a continuous-time Markov chain. This is done by constructing a right-shift process and then solving the Kolmogorov system of differential equations recursively. By solving a finite number of the differential equations, it is possible to obtain the state probabilities to any degree of accuracy over any finite time interval.

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Farhang, K., and A. Midha. "Steady-State Response of Periodically Time-Varying Linear Systems, With Application to an Elastic Mechanism." Journal of Mechanical Design 117, no. 4 (December 1, 1995): 633–39. http://dx.doi.org/10.1115/1.2826732.

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This paper presents the development of an efficient and direct method for evaluating the steady-state response of periodically time-varying linear systems. The method is general, and its efficacy is demonstrated in its application to a high-speed elastic mechanism. The dynamics of a mechanism comprised of elastic members may be described by a system of coupled, inhomogeneous, nonlinear, second-order partial differential equations with periodically time-varying coefficients. More often than not, these governing equations may be linearized and, facilitated by separation of time and space variables, reduced to a system of linear ordinary differential equations with variable coefficients. Closed-form, numerical expressions for response are derived by dividing the fundamental time period of solution into subintervals, and establishing an equal number of continuity constraints at the intermediate time nodes, and a single periodicity constraint at the end time nodes of the period. The symbolic solution of these constraint equations yields the closed-form numerical expression for the response. The method is exemplified by its application to problems involving a slider-crank mechanism with an elastic coupler link.

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Tanwani, Aneel, and Stephan Trenn. "Determinability and state estimation for switched differential–algebraic equations." Automatica 76 (February 2017): 17–31. http://dx.doi.org/10.1016/j.automatica.2016.10.024.

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Fiagbedzi, Y. A., and A. E. Pearson. "A state observer for systems described by functional differential equations." Automatica 26, no. 2 (March 1990): 321–31. http://dx.doi.org/10.1016/0005-1098(90)90126-3.

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MORALES-RODRIGO, CRISTIAN, and J. IGNACIO TELLO. "GLOBAL EXISTENCE AND ASYMPTOTIC BEHAVIOR OF A TUMOR ANGIOGENESIS MODEL WITH CHEMOTAXIS AND HAPTOTAXIS." Mathematical Models and Methods in Applied Sciences 24, no. 03 (December 29, 2013): 427–64. http://dx.doi.org/10.1142/s0218202513500553.

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We consider a system of differential equations modeling tumor angiogenesis. The system consists of three equations: two parabolic equations with chemotactic terms to model endothelial cells and tumor angiogenesis factors coupled to an ordinary differential equation which describes the evolution of the fibronectin concentration. We study global existence of solutions and, under extra assumption on the initial data of the fibronectin concentration we obtain that the homogeneous steady state is asymptotically stable.

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Verdugo, Anael. "Linear Analysis of an Integro-Differential Delay Equation Model." International Journal of Differential Equations 2018 (2018): 1–6. http://dx.doi.org/10.1155/2018/5035402.

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This paper presents a computational study of the stability of the steady state solutions of a biological model with negative feedback and time delay. The motivation behind the construction of our system comes from biological gene networks and the model takes the form of an integro-delay differential equation (IDDE) coupled to a partial differential equation. Linear analysis shows the existence of a critical delay where the stable steady state becomes unstable. Closed form expressions for the critical delay and associated frequency are found and confirmed by approximating the IDDE model with a system of N delay differential equations (DDEs) coupled to N ordinary differential equations. An example is then given that shows how the critical delay for the DDE system approaches the results for the IDDE model as N becomes large.

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Librovich, B. V., A. F. Nowakowski, F. C. G. A. Nicolleau, and T. M. Michelitsch. "Non-Equilibrium Evaporation/Condensation Model." International Journal of Applied Mechanics 09, no. 08 (December 2017): 1750111. http://dx.doi.org/10.1142/s1758825117501113.

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A new mathematical model for non-equilibrium evaporation/condensation including boiling effect is proposed. A simplified differential-algebraic system of equations is obtained. A code to solve numerically this differential-algebraic system has been developed. It is designed to solve both systems of equations with and without the boiling effect. Numerical calculations of ammonia–water systems with various initial conditions, which correspond to evaporation and/or condensation of both components, have been performed. It is shown that, although the system evolves quickly towards a quasi-equilibrium state, it is necessary to use a non-equilibrium evaporation model to calculate accurately the evaporation/condensation rates, and consequently all the other dependent variables.

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Journal articles on the topic 'Differential system state equations'

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