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Noether symmetries and conserved quantities for Birkhoffian systems with time delay
Abstract
The Noether symmetries and conserved quantities for Birkhoffian systems with time delay are proposed and studied. First, the Pfaff–Birkhoff principle with time delay is proposed, and Birkhoff’s equations with time delay are obtained. Second, based on the invariance of the Pfaff action with time delay under a group of infinitesimal transformations, the Noether symmetric transformations and the Noether quasisymmetric transformations of the system are defined, and the criteria of the Noether symmetries are established. Finally, the relationship between the symmetries and the conserved quantities are studied, and the Noether theorems for Birkhoffian systems with time delay are established. Some examples are given to illustrate the application of the results.
... Frederico and Torres [27] preliminarily introduced the classical Noether's theory to the time-delay calculus of variations. Indeed, Noether's theory has been applied to various problems involving time-delay, such as non-smooth extremals of variational problems [28], isoperimetric variational problems [29], high-order variational problems [30], non-conservative systems [31], nonholonomic systems [32], Hamiltonian systems [33], Birkhoffian systems [34], generalized Herglotz variational problems [35][36][37], and dynamical systems in fractional [38,39] and time-scale frameworks [6,40]. ...
... Equations (26), (29), and (32) can each be used as the criterion equation for the three symmetries above, respectively. Indeed, Equations (33) and (34) are often referred to as the Noether identity when r = 1. ...
... In the intervals [t 1 − τ 1 , t 1 − τ 2 ) and [t 1 − τ 2 , t 1 ), when τ 1 = τ 2 , the first two formulas of Equations (33) and (34) are not obtained in Ref. [31] because of the calculation problems with respect to the non-isochronous variation. ...
The study of multi-time-delay dynamical systems has highlighted many challenges, especially regarding the solution and analysis of multi-time-delay equations. The symmetry and conserved quantity are two important and effective essential properties for understanding complex dynamical behavior. In this study, a multi-time-delay non-conservative mechanical system is investigated. Firstly, the multi-time-delay Hamilton principle is proposed. Then, multi-time-delay non-conservative dynamical equations are deduced. Secondly, depending on the infinitesimal group transformations, the invariance of the multi-time-delay Hamilton action is studied, and Noether symmetry, Noether quasi-symmetry, and generalized Noether quasi-symmetry are discussed. Finally, Noether-type conserved quantities for a multi-time-delay Lagrangian system and a multi-time-delay non-conservative mechanical system are obtained. Two examples in terms of a multi-time-delay non-conservative mechanical system and a multi-time-delay Lagrangian system are given.
... It is a natural development of the Hamiltonian mechanics and has valuable applications in hadronic physics, spatial mechanics, statistical mechanics, biophysics, and engineering [23][24][25][26]. Over the past two decades, important achievements, including the symmetries and conserved quantities of Birkhoffian systems, the Birkhoffian dynamic inverse problems, the stability of motion of Birkhoff's equations, the Birkhoffian systems with time delay, and the fractional Birkhoffian systems, have been made [27][28][29][30][31][32][33][34][35][36][37][38]. ...
... where (36) then the Mei symmetry of the system can lead to the new conserved quantity ...
The time-scale dynamic equations play an important role in modeling complex dynamical processes. In this paper, the Mei symmetry and new conserved quantities of time-scale Birkhoff’s equations are studied. The definition and criterion of the Mei symmetry of the Birkhoffian system on time scales are given. The conditions and forms of new conserved quantities which are found from the Mei symmetry of the system are derived. As a special case, the Mei symmetry of time-scale Hamilton canonical equations is discussed and new conserved quantities for the Hamiltonian system on time scales are derived. Two examples are given to illustrate the application of results.
... The influence of time delay becomes a very important factor in scientific research to achieve more accurate and more objective results. Since TÈl'sgol'c'sT work [1] in 1964, the dynamical equations in the framework of difference and differential have been investigated with delayed arguments extensively and the results proved to be effective in reflecting a better essence of things and development law [2][3][4][5][6][7]. Nonetheless, in reality, discrepancies still remain, sometimes even essential differences. ...
... The classical Hamilton canonical equations turn to be a kind of general dynamic equation in the sense of a non-canonical transformation, that is, Birkhoff's equation which is richer in content than Hamilton canonical equations. Thus, it's desirable to discuss the delayed Birkhoffian system [6,7] on time scales. ...
The theory of time scales which unifies differential and difference analysis provides a new perspective for scientific research. In this paper, we derive the canonical equations of a delayed Hamiltonian system in a time scales version and prove the Noether theorem by using the method of reparameterization with time. The results extend not only the continuous version of the Noether theorem with delayed arguments but also the discrete one. As an application of the results, we find a Noether-type conserved quantity of a delayed Emden-Fowler equation on time scales.
... Noether's theorem explains all conservation laws of classical mechanics; for example, the conservation of energy comes from the invariance of the system under time translations, the conservation of momentum comes from the invariance of the system under spatial translations, and the conservation of the moment of momentum comes from the invariance of the system under spatial rotations. Nowadays, the celebrated Noether's theorem is a well-known tool in constrained mechanical systems, such as holonomic systems [2][3][4][5], non-holonomic systems [2,6-9], Birkhoffian systems [10-13], dynamical systems with time delay [14][15][16][17], fractional calculus of variations [18][19][20][21][22], and variational problems of Herglotz type [23,24]. However, non-standard Lagrangians and non-standard Hamiltonians may make the description easier in some cases, for example, when dealing with nonlinear dynamics. ...
... and the former of Eq. (7), condition (16) can be expressed as ...
This paper focuses on studying Noether’s theorems for dynamical systems with two kinds of non-standard Hamiltonians, respectively, namely exponential Hamiltonian and power-law Hamiltonian. Firstly, the differential equations of motion for dynamical systems with exponential Hamiltonian and power-law Hamiltonian are established. Secondly, according to the invariance of the action under the infinitesimal transformations, the definitions and criteria of Noether symmetric transformations and Noether quasi-symmetric transformations are given. Then, Noether’s theorems for dynamical systems with exponential Hamiltonian and power-law Hamiltonian are obtained, respectively. Finally, two examples are given to illustrate the applications of the results.
... Mei [32] established the basic theoretical framework of the Birkhoffian dynamics. In recent years, the study of the Birkhoffian dynamics gained significant headway [35][36][37][38][39][40][41][42][43][44][45][46][47]. However, the symmetry and the conserved quantity for Birkhoffian systems based on the variational problems of Herglotz type have not been investigated yet. ...
... , r ) . (42) Theorem 3 can be called Noether's theorem of the Hamiltonian system for a variational problem of Herglotz type. By this theorem we can get a corresponding conserved quantity from a given Noether symmetric transformation of the system. ...
Herglotz proposed a generalized variational principle through his work on contact transformations and their connections with Hamiltonian systems and Poisson brackets, which provides an effective method to study the dynamics of nonconservative systems. In this paper, the variational problem of Herglotz type for a Birkhoffian system is presented and the differential equations of motion for the system are established. The invariance of the Pfaff–Herglotz action under a group of infinitesimal transformations and its connection with the conserved quantities of the system are studied, and Noether’s theorem and its inverse for the Herglotz variational problem are derived. The variational problem of Herglotz type for a Birkhoffian system reduces to the classical Pfaff–Birkhoff variational problem under classical conditions. Thus, it contains Noether’s theorem for the classical Birkhoffian system as a special case. And since Birkhoffian mechanics is a natural generalization of Hamiltonian mechanics, the results we obtained contain Noether’s theorem of Herglotz variational problems for Hamiltonian systems and Lagrangian systems as special cases. In the end of the paper, we give two examples to illustrate the application of the results.
... In addition, a new way of interpreting the famous Noether theorem has emerged, which is to introduce the delayed parameters into the Noether theorem. 19 This result is pregnant because timedelay problems play a crucial role in the application of various variational problems, such as Lagrangian frameworks, 20 Hamiltonian frameworks, 21 Pfaff-Birkhoff principle, 22 generalized Herglotz variational problems, 23,24 ordinary differential equation problems, 25,26 high-order version variational problems, 27 nonsmooth extension, 28 isoperimetric problems, 29 fractional frameworks, 30,31 and dynamical systems in time scale. 32,33 However, research on multiple time delays is not merely an extension of the original argument, incorporating a different delayed argument. ...
The Birkhoffian system, an important mechanical system in analytical mechanics, is studied in the background of multiple time delays. Both the multi-delay Birkhoffian system and the Hamiltonian system are investigated. First, the more general framework is discussed. The Pfaff–Birkhoff action is presented with multiple time delays. The multi-delay differential equations of motion are derived for both Birkhoffian and Hamiltonian models. Second, the invariance and two different kinds of Noether symmetries of the multi-delay Pfaff action are studied. Third, due to the effects of different time delays, the corresponding Noether-type theorems of multi-delay Birkhoffian systems are obtained. Two examples are presented to demonstrate the applications of the results.
... The constrained Birkhoff s equations with multiplier form on time scales can be obtained as [22] ¶R l ( ) ...
Time scale is a new and powerful tool for dealing with complex dynamics problems. The main result of this study is the exact invariants and adiabatic invariants of the generalized Birkhoffian system and the constrained Birkhoffian system on time scales. Firstly, we establish the differential equations of motion for the above two systems and give the corresponding Noether symmetries and exact invariants. Then, the perturbation to the Noether symmetries and the adiabatic invariants for the systems mentioned above under the action of slight disturbance are investigated, respectively. Finally, two examples are provided to show the practicality of the findings.
... One can find a conserved quantity from a Noether symmetry by using the intrinsic relation between the conserved quantity and the symmetry. Afterward, the theories of symmetry and conserved quantity are extended and applied to various kinds of the dynamical systems, such as nonconservative systems [2, 3], nonholonomic systems [4-6], fractional systems [7][8][9][10][11], time delay systems [12][13][14][15], Herglotz systems [16,17], time scales systems [18][19][20][21][22] and approximate systems [23,24]. But all these results are limited to the systems based on standard Lagrangians, and the Chaplygin systems with nonstandard Lagrangians have not been involved yet. ...
In this paper, the Noether theorems and their inverse theorems for generalized Chaplygin systems with two types of nonstandard Lagrangians, related to exponential and power-law Lagrangian, are explored and presented. The variational principles for the Chaplygin systems with nonstandard Lagrangian are derived, and the generalized Chaplygin equations for the corresponding systems are established, the Noether transformations are considered, from which the corresponding conserved quantities are deduced. And their inverse theorems for nonstandard generalized Chaplygin systems are given. Two examples show the validity of the results.
... Hamilton principle is a special case of the Pfaff-Birkhoff principle; Hamilton canonical equations are a special case of Birkhoff's equations. Hamilton canonical equations remain the same under a canonical transformation and become Birkhoff's equations under a general noncanonical transformation [31][32][33][34][35]. ...
In this paper, the approximate Noether theorems for approximate Birkhoffian systems are presented and discussed. The approximate Birkhoff equations for the systems are established. The Noether identities for approximate Birkhoffian systems are given, which based upon the Noether symmetry and quasi-symmetry, and the relationship between the approximate Noether symmetries and approximate conservation laws for the systems are established, and the approximate Noether theorems are obtained. The results show that the results under the approximate Hamiltonian systems are a special cases of the approximate Birkhoffian systems, while the results under the approximate Lagrangian systems is equivalent to that under the approximate Hamiltonian systems. Finally, two examples are given to illustrate the application of the results.
... Bartosiewicz and his co-workers [24] studied the time-scale second Euler-Lagrange equation, and derived Noether s conserved quantities on time scales by using this equation. Subsequently, many scholars [25][26][27][28][29][30][31][32][33][34] began to carry out a series of studies on time-scale the variational principle for constrained mechanical systems and Noether s theorem of different dynamic systems. However, on a time scale, there are few studies on Lie symmetry in mechanical systems. ...
The Lie theorem for Birkhoffian systems with time-scale nonshifted variational problems are studied, including free Birkhoffian system, generalized Birkhoffian system and constrained Birkhoffian system. First, the time-scale nonshifted generalized Pfaff-Birkhoff principle is established, and the dynamical equations for three Birkhoffian systems under nonshifted variational problems are deduced. Afterwards, in the time-scale nonshifted variational problems, by introducing the infinitesimal transformations, Lie symmetry for free Birkhoffian system, generalized Birkhoffian system and constrained Birkhoffian system are defined respectively. Then Lie symmetry theorems for three kinds of Birkhoffian systems are deduced and proved. In the end, three examples are given to explain the applications for the results.
... Ref. [44] gave a numerical scheme to solve the fractional Birkhoff equations using variational integrators. Other existing literature mostly deals with the Noether symmetries and conserved quantities for fractional Birkhoffian systems [45][46][47]. Ref. [48] studied the optimal control of a Birkhoffian system based on variational discretization. For a wider range of theoretical and practical needs, this work will focus on the FOCPs in the Birkhoffian sense whose dynamic constraints are described by the fractional forced Birkhoff equations. ...
This paper gives a general numerical scheme for the optimal control problem of fractional Birkhoffian systems. The fractional forced Birkhoff equations within Riemann–Liouville fractional derivatives are derived from the fractional Pfaff–Birkhoff–d’Alembert principle which includes the control as an external force term. Following the strategy of variational integrators, the fractional Pfaff–Birkhoff–d’Alembert principle is directly discretized to develop the equivalent discrete fractional forced Birkhoff equations that served as the equality constraints of the optimization problem. Together with the initial and final state constraints on the configuration space, the original optimal control problem is converted into a nonlinear optimization problem subjected to a system of algebraic constraints, which can be solved by existing algorithms. An illustrative example is given to show the efficiency and simplicity of the proposed method.
... Based on the second Euler-Lagrange equations, they proposed another method to find the Noether conserved quantity. Afterwards, according to these two methods, many scholars have obtained some results have been obtained in the study of variational principle, dynamical equations, and Noether symmetries for the different mechanical systems, such as references [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31]. ...
The time-scale version of Noether symmetry and conservation laws for three Birkhoffian mechanics, namely, nonshifted Birkhoffian systems, nonshifted generalized Birkhoffian systems, and nonshitfed constrained Birkhoffian systems, are studied. Firstly, on the basis of the nonshifted Pfaff-Birkhoff principle on time scales, Birkhoff’s equations for nonshifted variables are deduced; then, Noether’s quasi-symmetry for the nonshifted Birkhoffian system is proved and time-scale conserved quantity is presented. Secondly, the nonshifted generalized Pfaff-Birkhoff principle on time scales is proposed, the generalized Birkhoff’s equations for nonshifted variables are derived, and Noether’s symmetry for the nonshifted generalized Birkhoffian system is established. Finally, for the nonshifted constrained Birkhoffian system, Noether’s symmetry and time-scale conserved quantity are proposed and proved. The validity of the result is proved by examples.
... Ref. [44] gave a numerical scheme to solve the fractional Birkhoff equations using variational integrators. Other existing literatures mostly deal with the Noether symmetries and conserved quantities for fractional Birkhoffian systems [45][46][47]. Ref. [48] studied the optimal control of a Birkhoffian system based on variational discretization. For a wider range of theoretical and practical needs, this work will focus on the FOCPs in the Birkhoffian sense whose dynamic constraints are described by the fractional forced Birkhoff equations. ...
This paper deals with the optimal control of fractional Birkhof-fian systems based on the numerical method of variational integrators. Firstly, the fractional forced Birkhoff equations within Riemann–Liouville fractional derivatives are derived from the fractional Pfaff–Birkhoff–d'Alembert principle. Secondly, by directly discretizing the fractional Pfaff–Birkhoff–d'Alembert principle, we develop the equivalent discrete fractional forced Birkhoff equations, which are served as the equality constraints of the optimization problem. Together with the initial and final state constraints on the configuration space, the original optimal control problem is converted into a nonlinear optimization problem subjected to a system of algebraic constraints, which can be solved by the existing methods such as sequential quadratic programming. Finally, an example is given to show the efficiency and simplicity of the proposed method.
... In 2013, in reference [8], we extended the results of [7] in three aspects: from Lagrange system to general non-conservative system; from a group of point transformations corresponding to generalized coordinates and time to a group of transformations that depend on generalized velocities; from Noether symmetry to Noether quasi-symmetry. In recent years, Noether's theorems with time delay have been extended to high-order variational problems [9], fractional systems [10], Hamilton systems [11], nonholonomic systems [12], Birkhoff systems [13,14], and dynamics on time scales [15,16], etc. Although some important results have been obtained in the dynamics modeling of time-delay systems and its Noether's theorems, in general, the research in this field is still in the preliminary stage and is still an open topic. ...
Because Herglotz’s variational problem achieves the variational representation of non-conservative dynamic processes, its research has attracted wide attention. The aim of this paper is to explore Herglotz’s variational problem for a non-conservative system with delayed arguments under Lagrangian framework and its Noether’s theorem. Firstly, we derive the non-isochronous variation formulas of Hamilton–Herglotz action containing delayed arguments. Secondly, for the Hamilton–Herglotz action case, we define the Noether symmetry and give the criterion of symmetry. Thirdly, we prove Herglotz type Noether’s theorem for non-conservative system with delayed arguments. As a generalization, Birkhoff’s version and Hamilton’s version for Herglotz type Noether’s theorems are presented. To illustrate the application of our Noether’s theorems, we give two examples of damped oscillators.
... Noether's theorem reveals the inherent relation between the Noether symmetry and the conserved quantity [16,17]. Recently, a series of important advances have been made in the research of Noether symmetry and conserved quantity [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36]. In this section, we establish Noether's theorem for the time-delayed Hamiltonian system of Herglotz type. ...
The variational problem of Herglotz type and Noether's theorem for a time-delayed Hamiltonian system are studied. Firstly, the variational problem of Herglotz type with time delay in phase space is proposed, and the Hamilton canonical equations with time delay based on the Herglotz variational problem are derived. Secondly, by using the relationship between the non-isochronal variation and the isochronal variation, two basic formulae of variation of the Hamilton–Herglotz action with time delay in phase space are derived. Thirdly, the definition and criterion of the Noether symmetry for the time-delayed Hamiltonian system are established and the corresponding Noether's theorem is presented and proved. The theorem we obtained contains Noether's theorem of a time-delayed Hamiltonian system based on the classical variational problem and Noether's theorem of a Hamiltonian system based on the variational problem of Herglotz type as its special cases. At the end of the paper, an example is given to illustrate the application of the results.
... The well-known Noether theorem reveals a connection between symmetries and conserved quantities (Noether, 1918). New directions of the applications for Noether theorems such as the fractional dynamical equations (Atanackovic´et al., 2009;Frederico and Torres, 2007;Zhai and Zhang, 2016), the dynamical equations with time delay (Frederico and Torres, 2012;Jin and Zhang, 2015;Zhai and Zhang, 2014), and the equations derived by the variational problems of Herglotz type (Santos et al., 2015;Zhang, 2016a) can be found. And the method of Noether symmetry which presents the invariance of action under the infinitesimal transformations has been used successfully to find the conserved quantities for dynamical systems on time scales. ...
The theory of time scales that can unify and extend continuous and discrete analysis has proved to be more accurate in modeling the dynamic process. The Lie symmetry approach, which is an effective way to deal with different kinds of dynamical equations in a variety of areas of applied science, is to be analyzed on time scales. We begin with the Lie group of point infinitesimal transformations on time scales and its corresponding extensions. And the invariance of dynamical equations on time scales under the infinitesimal transformations is discussed. More specifically, the Lie symmetries for dynamical equations of mechanical systems on time scales including Lagrangian systems on time scales, Hamiltonian systems on time scales, and Birkhoffian systems on time scales are investigated as applications. Thus, the corresponding conserved quantities for mechanical systems on time scales are derived by using the Lie symmetries. Examples are given to illustrate the application of the results.
... Hence, it is of great significance to study. Up to now many results about Noether theory have been obtained, such as Noether theory based on fractional models, see [7,35,80,81]; Noether theory with time delay, see [46,77]; Noether theory for nonlinear dynamical systems, see [36,82]; as well as Noether theory for fractional systems of variable order, see [62,76]. Recently, Noether theory was extended to time scales, see for instance, [18,29,50,51,55,63,79,83]. ...
... Then the Noether symmetries and conserved quantities for fractional Birkhoffian systems were discussed in Refs. [53][54][55][56]. Fractional Birkhoff equations in terms of Riesz fractional derivatives were obtained in Ref. [57]. ...
In this paper, we generalize the Pfaff–Birkhoff principle to the case of containing fractional derivatives and obtain the so-called fractional Pfaff–Birkhoff–d’Alembert principle. The fractional Birkhoff equations in the sense of Riemann–Liouville fractional derivative are derived. Under the framework of variational integrators, we develop the discrete fractional Birkhoff equations by approximating the Riemann–Liouville fractional derivative with the shifted Grünwald–Letnikov fractional derivative. The resulting algebraic equations can be served as an algorithm to numerically solve the fractional Birkhoff equations. A numerical example is demonstrated to show the validity and applicability of the presented methodology.
... One can find a conserved quantity from a Noether symmetry by using the intrinsic relation between the conserved quantity and the symmetry, which broke through the traditional approaches for finding the conserved quantities by the law of the conservation of energy of the system, the law of the conservation of momentum, and the law of the conservation of angular momentum. Afterward, the theory of symmetry and conserved quantity is extended and applied to various kinds of constrained mechanical systems, such as holonomic nonconservative systems [2,3], nonholonomic systems [3][4][5][6], fractional dynamical systems [7][8][9][10][11], and dynamics with time delay [12][13][14][15]. But all these results are limited to the systems based on standard Lagrangians, and the systems with nonstandard Lagrangians have not been involved yet. ...
The Noether theorem and its inverse theorem for the nonlinear dynamical systems with nonstandard Lagrangians are studied. In this paper, two kinds of nonstandard Lagrangians, namely exponential Lagrangians and power-law Lagrangians, are discussed. For each case, the Hamilton principle based on the action with nonstandard Lagrangians is established, the differential equations of motion for the dynamical systems with nonstandard Lagrangians are obtained, and two basic formulae for the variation in Hamilton action with nonstandard Lagrangians are derived. The definitions and the criteria of the Noether symmetric transformations and the Noether quasi-symmetric transformations are given. The Noether theorem and its inverse theorem are established, which reveal the intrinsic relation between the symmetry and the conserved quantity for the systems with nonstandard Lagrangians. Two examples are given to illustrate the application of the results.
... The research on symmetries and conserved quantities of mechanical systems possesses important theoretical and practical significance. The well-known Noether symmetry has broad applications in mathematics, dynamics, and physics [3][4][5][6][7][8], and it always can lead to conserved quantities. In fact, it is also named variational symmetry [4,9]. ...
By Kirchhoff dynamic analogy, the thin elastic rod static equals to rotation of rigid body dynamic. The analytical mechanics methods reflect their advantages in the study of the modeling and equilibrium and stability of elastic rod static, especially for the constrained problems. The Lagrangian structure of the equation of motion for elastic rod is deduced from the integral variational principle. The definition of conformal invariance of Mei symmetry of elastic rod in Lagrangian form is given. The determining equation of conformal invariance of Mei symmetry is obtained based on the Lie point transformation group. The relation between conformal invariance of Mei symmetry and Mei symmetry is discussed. The structure equation and conserved quantity by using the Lagrangian structure along arc coordinate deduced from conformal invariance of Mei symmetry of elastic rod are constructed. Take rod with circular cross section as example to illustrate the application of the results get in this paper. These conserved quantities will be helpful in the study of exact solutions and stability, as well as the numerical simulation of the thin elastic rod nonlinear mechanics.
... In 2012, the Noether symmetry theorems for variational and optimal control problems with time delay were first discussed by Frederico and Torres [24]. In 2013, the Noether symmetries for dynamics of nonconservative Lagrange system, nonconservative Hamilton system, and Birkhoffian system with time delay were studied by Zhang and his coworkers [25][26][27], and the relationship between the Noether symmetries and the conserved quantities was established. Up to now, more and more dynamical problems with time delay and its relevant mathematical theories are presented. ...
The paper focuses on studying the Noether theorem for nonholonomic systems with time delay. Firstly, the differential equations of motion for nonholonomic systems with time delay are established, which is based on the Hamilton principle with time delay and the Lagrange multiplier rules. Secondly, based upon the generalized quasi-symmetric transformations for nonconservative systems with time delay, the Noether theorem for corresponding holonomic systems is given. Finally, we obtain the Noether theorem for the nonholonomic nonconservative systems with time delay. At the end of the paper, an example is given to illustrate the application of the results.
This paper deals with the Noether symmetry and conserved quantity for Birkhoffian system with delayed arguments in version of time scales. Firstly, the Pfaff–Birkhoff principle with delayed arguments in which the variables are defined on time scales is established, through which the corresponding time-delayed Birkhoff’s equations on time scales are derived. Secondly, the invariance of the Pfaff action with delayed arguments on time scales under the infinitesimal transformations of one-parameter group is analyzed. Further, the Noether theorem for the system which reveals the relationship between the symmetry and conserved quantity is proved by applying a technique of time reparameterization and performing the linear change of the variables with delayed arguments. Finally, the results incorporate the classical Noether-type conserved quantity, the discrete Noether-type conserved quantity and the quantum Noether-type conserved quantity. Also, an example illustrates the application of the results.
The generalized variational principle of Herglotz type provides an effective way to study the problems of conservative and non-conservative systems in a unified way. According to the differential variational principle of Herglotz type, we study the adiabatic invariants for a disturbed Birkhoffian system in this paper. Firstly, the differential equations of motion of the Birkhoffian system based upon this variational principle are given, and the exact invariant of Herglotz type of the system is introduced. Secondly, a new type of adiabatic invariants for the system under the action of small perturbation is obtained. Thirdly, the inverse theorem of adiabatic invariant for the disturbed Birkhoffian system of Herglotz type is obtained. Finally, an example is given.
In this paper, the Noether theorem for generalized Chaplygin system on time scales is proposed and studied. The generalized Chaplygin formula for nonholonomic system on time scales is derived. The Noether theorems for generalized Chaplygin system on time scales are established, and two special cases of the Noether theorems for continuous and discrete generalized Chaplygin systems are given. Finally, two examples are given to illustrate the applications of the results.
This paper focuses on studying the Noether symmetries and the conserved quantities for a time-delayed Birkhoffian system of Herglotz type. The generalized Herglotz variational principle is extended to a time-delayed Birkhoffian system, and Birkhoff's equations of Herglotz type with time delay are established. The definition and the criterion of Noether symmetric transformation for the time-delayed Birkhoffian system of Herglotz type are given, which are based upon the formulae of non-isochronal variation of the Pfaff–Herglotz action deduced in this paper. Noether's theorem for the time-delayed Birkhoffian system of Herglotz type is proved. The theorem reveals the inherent relation between the Noether symmetries and the conserved quantities of the system, and contains Noether's theorem for the time-delayed Birkhoffian system based on the classical variational problem and Noether's theorem for the time-delayed Hamiltonian system of Herglotz type as its specials. At the end of the paper, an example is given to illustrate the application of the results.
In general, optimal control problems rely on numerically rather than analytically solving methods, due to their nonlinearities. The direct method, one of the numerically solving methods, is mainly to transform the optimal control problem into a nonlinear optimization problem with finite dimensions, via discretizing the objective functional and the forced dynamical equations directly. However, in the procedure of the direct method, the classical discretizations of the forced equations will reduce or affect the accuracy of the resulting optimization problem as well as the discrete optimal control. In view of this fact, more accurate and efficient numerical algorithms should be employed to approximate the forced dynamical equations. As verified, the discrete variational difference schemes for forced Birkhoffian systems exhibit excellent numerical behaviors in terms of high accuracy, long-time stability and precise energy prediction. Thus, the forced dynamical equations in optimal control problems, after being represented as forced Birkhoffian equations, can be discretized according to the discrete variational difference schemes for forced Birkhoffian systems. Compared with the method of employing traditional difference schemes to discretize the forced dynamical equations, this way yields faithful nonlinear optimization problems and consequently gives accurate and efficient discrete optimal control. Subsequently, in the paper we are to apply the proposed method of numerically solving optimal control problems to the rendezvous and docking problem of spacecrafts. First, we make a reasonable simplification, i.e., the rendezvous and docking process of two spacecrafts is reduced to the problem of optimally transferring the chaser spacecraft with a continuously acting force from one circular orbit around the Earth to another one. During this transfer, the goal is to minimize the control effort. Second, the dynamical equations of the chaser spacecraft are represented as the form of the forced Birkhoffian equation. Then in this case, the discrete variational difference scheme for forced Birkhoffian system can be employed to discretize the chaser spacecraft's equations of motion. With further discretizing the control effort and the boundary conditions, the resulting nonlinear optimization problem is obtained. Finally, the optimization problem is solved directly by the nonlinear programming method and then the discrete optimal control is achieved. The obtained optimal control is efficient enough to realize the rendezvous and docking process, even though it is only an approximation of the continuous one. Simulation results fully verify the efficiency of the proposed method for numerically solving optimal control problems, if the fact that the time step is chosen to be very large to limit the dimension of the optimization problem is noted.
Conserved quantities for Hamiltonian systems on time scales with nabla derivatives and delta derivatives are presented. First, Hamilton principle on time scales with nabla derivatives is established and Hamilton canonical equation with nabla derivatives is obtained. Second, Noether identity and Noether theorem for Hamiltonian systems with nabla derivatives are achieved. Third, Hamilton canonical equation with delta derivatives, Noether identity and Noether theorem for Hamiltonian systems with delta derivatives are gotten through duality principle on the basis of the corresponding results with nabla derivatives. Fourth, some special cases of Noether identity and Noether theorem are given. And finally, two examples are devoted to illustrate the methods and results.
This paper deals with the Noether theorem for non-conservative systems with time delay on time scales. First, the Hamilton principle for the non-conservative system with time delay on time scales is proposed, and Lagrange equations with time delay on time scales are obtained. Second, based on the invariance of the Hamilton action for the non-conservative system with time delay under a group of infinitesimal transformations, the Noether symmetric transformations, the Noether quasi-symmetric transformations and the generalized Noether quasi-symmetric transformations of the system are defined on time scales, respectively, and the criteria of the Noether symmetries are established. Finally, the relationship between the symmetries and the conserved quantities are studied, and the Noether theorems for non-conservative systems with time delay on time scales are established. Some examples are given to illustrate the application of the results.
Based on Riemann-Liouville fractional derivatives, conserved quantities and adiabatic invariants for fractional generalized Birkhoffian systems are investigated. Firstly, fractional generalized Birkhoff equations are obtained by studying fractional generalized Pfaff-Birkhoff principle. Secondly, the definition of fractional generalized quasi-symmetry is given, the criteria of fractional generalized quasi-symmetry and the corresponding conserved quantity are achieved for fractional generalized Birkhoffian systems. Thirdly, perturbation to symmetry and adiabatic invariants for disturbed fractional generalized Birkhoffian systems are presented. Finally, an example is given to illustrate the results.
Scalar-field cosmology can be regarded as one of the significant fields of research in recent years. This paper is dedicated to a thorough investigation of the symmetries and conservation laws of the geodesic equations associated to a specific exact cosmological solution of a scalar-field potential which was originally motivated by six-dimensional Einstein-Maxwell theory. The mentioned string inspired Friedmann-Robertson-Lama tre-Walker (FRLW) solution is one of the noteworthy solutions of Einstein field equations. For this purpose, first of all the Christoffel symbols and the corresponding system of geodesic equations are computed and then the associated Lie symmetries are totally analyzed. Moreover, the algebraic structure of the Lie algebra of local symmetries is briefly investigated and a complete classification of the symmetry subalgebras is presented. Besides by applying the resulted symmetry operators the invariant solutions of the system of geodesic equations are discussed. In addition, the Noether symmetries and the Killing vector fields of the geodesic Lagrangian are determined and the corresponding optimal system of one-dimensional subalgebras is constructed. Mainly, an entire set of local conservation laws is computed for our analyzed scalar-field cosmological solution. For this purpose, two distinct procedures are applied: the celebrated Noether’s theorem and the direct method which is fundamentally based on a systematic application of Euler differential operators which annihilate any divergence expression identically.
Birkhoff equations on time scales and Noether theorem for Birkhoffian system on time scales are studied. First, some necessary knowledge of calculus on time scales are reviewed. Second, Birkhoff equations on time scales are obtained. Third, the conditions for invariance of Pfaff action and conserved quantities are presented under the special infinitesimal transformations and general infinitesimal transformations, respectively. Fourth, some special cases are given. And finally, an example is given to illustrate the method and results.
In this paper, we focus on discussing the fractional Noether symmetries and fractional conserved quantities for non-conservative Hamilton system with time delay. Firstly, the fractional Hamilton canonical equations of non-conservative system with time delay are established; secondly, based upon the invariance of the fractional Hamilton action with time delay under the infinitesimal transformations of group, we obtain the definitions and criterion of fractional Noether symmetric transformations, fractional Noether quasi-symmetric transformations and fractional generalized Noether quasi-symmetric transformations in phase space; finally, the relationship between the fractional Noether symmetries and fractional conserved quantities with time delay in phase space is established. At the end of the paper, some examples are given to illustrate the application of the results.
This paper presents the variational problems for fractional Birkhoffian systems, and its corresponding symmetries and conserved quantities are further studied. First, the fractional Pfaff variational problems in terms of Riemann-Liouville fractional derivatives are proposed, and the fractional Pfaff-Birkhoff-d’Alembert principle is established. Thus, the fractional Birkhoff’s equations are derived. Second, we derive two basic formulae for the variation of Pfaff action and give the definitions and criteria of fractional Noether symmetries and quasi-symmetries. Third, we establish the Noether theorems of fractional Birkhoffian systems which reveal the relationship between fractional symmetries and fractional conserved quantities. Some special cases are discussed, showing the universality of our results. Two examples are analyzed to illustrate the applications of the results in detail.
In this work, fractional integral calculus is applied in order to derive Lagrangian mechanics of nonconservative systems. In the proposed method, fractional time in-tegral introduces only one parameter, α, while in other models an arbitrary number of fractional parameters (orders of derivatives) appears. Some results on Hamilto-nian part of mechanics, namely Hamilton equations, are obtained and discussed in detail.
In this paper, the Noether symmetries and the conserved quantities for a Hamilton system with time delay are discussed. Firstly, the variational principles with time delay for the Hamilton system are given, and the Hamilton canonical equations with time delay are established. Secondly, according to the invariance of the function under the infinitesimal transformations of the group, the basic formulas for the variational of the Hamilton action with time delay are discussed, the definitions and the criteria of the Noether symmetric transformations and quasi-symmetric transformations with time delay are obtained, and the relationship between the Noether symmetry and the conserved quantity with time delay is studied. In addition, examples are given to illustrate the application of the results.
For a weakly nonholonomic system, the Lie symmetry and approximate Hojman conserved quantity of Appell equations are studied. Based on the Appell equations for a weakly nonholonomic system under special infinitesimal transformations of a group in which the time is invariable, the definition of the Lie symmetry of the weakly nonholonomic system and its first-degree approximate holonomic system are given. With the aid of the structure equation that the gauge function satisfies, the exact and approximate Hojman conserved quantities deduced directly from the Lie symmetry are derived. Finally, an example is given to study the exact and approximate Hojman conserved quantity of the system.
A special Lie symmetry and Hojman conserved quantity of the Appell equations for a Chetaev nonholonomic system are studied. The differential equations of motion and Appell equations of the Chetaev nonholonomic system are established. Under the special Lie symmetry group transformations in which the time is invariable, the determining equation of the special Lie symmetry of the Appell equations for a Chetaev nonholonomic system is given, and the expression of the Hojman conserved quantity is deduced directly from the Lie symmetry. Finally, an example is given to illustrate the application of the results.
In a recent article (see [8]), we derived necessary and sufficient conditions for minima for the fixed-endpoint problem in
the calculus of variations involving a constant delay in the phase coordinates. These conditions are expressed, explicitly,
in terms of the first and second variations. The vanishing of the first variation is characterized in terms of an extended
Euler's equation, just as for delay-free problems, but the characterization of the conditions on the second variation remained
unsolved. In this paper we convert, through the ‘method of steps’, the delay problem into one without delay. Although this
problem will not have fixed-endpoint constraints, it allows us to introduce, in a natural way, the concept of ‘conjugate sequence’;
this solves the main difficulty for delay problems, namely, to give initial conditions for existence and uniqueness of solutions
of the Hamiltonian system (which is a difference–differential system with both advanced and retarded arguments). The conditions
on the second variation are then characterized by an extra condition which is based exclusively on a solution of a given matrix
Riccati equation.
Fractional actionlike variational problems have recently gained importance in studying dynamics of nonconservative systems. In this note we address multidimensional fractional actionlike problems of the calculus of variations.
In this paper, we consider an optimal control problem with time-delay. The state and the control variables contain various
constant time-delays. This allows us to represent the necessary conditions in an explicit form. Solution of this problem with
infinite terminal time is also given.
The generalized Pfaff-Birkhoff-dAlembert principle and the form invariance of the generalized Birkhoff s equations under the infinitesimal transformation of group are studied. The criterion of the form invariance is given and the Noether conserved quantity and a new conserved quantity are obtained by the invariance.
This paper surveys recent advances in the study on the stability and bifurcation of time-delay systems. The review focuses mainly on the achievements of authors' team in the fundamental studies of stability, Hopf bifurcation and utilization of time delays to improve the system stability, as well as the delay effects on the stability of some typical systems or structures, including an aircraft wing, an inverted double-pendulum and networks. Finally, the paper addresses some open problems for future investigations.
The generalized Pfaff-Birkhoff-d'Alembert principle and the form invariance of the generalized Birkhoff's equations under the infinitesimal transformation of group are studied. The criterion of the form invariance is given and the Noether conserved quantity and a new conserved quantity are obtained by the invariance.
The Noether symmetries and the conserved quantities of dynamics for non-conservative systems with time delay are proposed and studied. Firstly, the Hamilton principle for non-conservative systems with time delay is established, and the Lagrange equations with time delay are obtained. Secondly, based upon the invariance of the Hamilton action with time delay under a group of infinitesimal transformations which depends on the generalized velocities, the generalized coordinates and the time, the Noether symmetric transformations and the Noether quasi-symmetric transformations of the system are defined and the criteria of the Noether symmetries are established. Finally, the relationship between the symmetries and the conserved quantities are studied, and the Noether theory of non-conservative systems with time delay is established At the end of the paper, some examples are given to illustrate the application of the results.
The effects of constraints on Noether symmetries and conserved quantities in a Birkhoffian system are studied. Firstly, the differential equations of the Birkhoffian system are established. Secondly, the criteria of Noether symmetries of the system are given. Thirdly, when constraints are inserted in the system, the variations that occur in the Noether symmetries are discussed, and the conditions under which the Noether symmetries and the conserved quantities of the system will remain unchanged are given. Finally, the results are applied to an illustrative example.
Noether’s theorem and its inverse of nonconservative systems subject to first-order nonlinear nonholonomic constraints are obtained by introducing the generalized quasi-symmetry of the infinitesimal transformation of the group of transformations G r for the given dynamical systems.
The fractional Pfaffian variational problems and the fractional Noether theory are studied under a fractional model presented by El-Nabulsi. Firstly, the fractional action-like Pfaffian variational problem is presented, the El-Nabulsi–Pfaff–Birkhoff–d’Alembert fractional principle is established, then the El-Nabulsi–Birkhoff fractional equations are derived; secondly, the definitions and criteria of the fractional Noether symmetric transformations are given, which are based on the invariance of El-Nabulsi–Pfaffian action under the infinitesimal transformations of group, then the inner relationship between a fractional Noether symmetry and a fractional conserved quantity is established; finally, two examples are given to illustrate the application of the results.
For a Birkhoffian system, a new Lie symmetrical method to find a conserved quantity is given. Based on the invariance of the equations of motion for the system under a general infinitesimal transformation of Lie groups, the Lie symmetrical determining equations are obtained. Then, several important relationships which reveal the interior properties of the Birkhoffian system are given. By using these relationships, a new Lie symmetrical conservation law for the Birkhoffian system is presented. The new conserved quantity is constructed in terms of infinitesimal generators of the Lie symmetry and the system itself without solving the structural equation which may be very difficult to solve. Furthermore, several deductions are given in the special infinitesimal transformations and the results are reduced to a Hamiltonian system. Finally, one example is given to illustrate the method and results of the application.
The weakly nonholonomic system is a nonholonomic system whose constraint equations contain a small parameter. The special Mei symmetry and approximate conserved quantity of Appell equations for a weakly nonholonomic system are studied. Appell equations for a weakly nonholonomic system are established and the definition and the criterion of the special Mei symmetry of the system are given. The structure equation of the special Mei symmetry for a weakly nonholonomic system and approximate conserved quantity deduced from the special Mei symmetry of the system are obtained. Finally, special approximate conserved quantity issues of Appell equations for a two freedom degrees weakly nonholonomic system are investigated using the results of this paper.
Perturbation to Noether symmetries and adiabatic invariants of discrete nonholonomic nonconservative mechanical systems on an uniform lattice are investigated. Firstly, we review Noether symmetry and conservation laws of a nonholonomic nonconservative system. Secondly, we study continuous Noether symmetry of a discrete nonholonomic system, give the Noether symmetry criterion and theorem of discrete corresponding holonomic system and nonholonomic system. Thirdly, we study perturbation to Noether symmetry of the discrete nonholonomic nonconservative system, give the criterion of perturbation to Noether symmetry for this system, and based on the definition of adiabatic invariants, we construct the theorem under which can lead to Noether adiabatic invariants for this system, and the forms of discrete Noether adiabatic invariants are given. Finally, we give an example to illustrate our results.
This paper studies a type of integral and reduction of the generalized Birkhoffian system. An existent condition and the form of the integral are obtained. By using the integral, the dimension of the system can be reduced two degrees. An example is given to illustrate the application of the results.
This paper proposes firstly the Pfaff-Birkhoff-D'Alembert's principle and obtains the equations of motion in terms of the
independent variables for a constrained Birkhoff's system from the principle. Secondly, it establishes the equations of perturbation
of the system. Finally, it obtains the stability criteria by using the Liapunov direct method and the firstly approximate
method.
In 1983, the Birkhoffian mechanics was proposed. Supplementary studies for this new mechanics are given. The studies include the Birkhoffian mechanics of holonomic and non-holonomic systems, the integration theory of Birkhoffian mechanics, the inverse problem of Birkhoffian mechanics and the stability of motion of Birkhoffian mechanics.
This paper focuses on studying the Poisson theory and the integration method of a Birkhoffian system in the event space. The Birkhoff's equations in the event space are given. The Poisson theory of the Birkhoffian system in the event space is established. The definition of the Jacobi last multiplier of the system is given, and the relation between the Jacobi last multiplier and the first integrals of the system is discussed. The researches show that for a Birkhoffian system in the event space, whose configuration is determined by (2n + 1) Birkhoff's variables, the solution of the system can be found by the Jacobi last multiplier if 2n first integrals are known. An example is given to illustrate the application of the results.
It is shown that conserved quantities for lagrangian systems can be associated with symmetry groups which do not leave the action integral invariant. An analogy with a recently described invariant for hamiltonian systems is pointed out.
A weakly nonholonomic system is a nonholonomic system whose constraint equations contain a small parameter. The form invariance and the approximate conserved quantity of the Appell equations for a weakly nonholonomic system are studied. The Appell equations for the weakly nonholonomic system are established, and the definition and the criterion of form invariance of the system are given. The structural equation of form invariance for the weakly nonholonomic system and the approximate conserved quantity deduced from the form invariance of the system are obtained. Finally, an example is given to illustrate the application of the results.
The existence of non-Noether invariants is related to the transformation properties of the Euler equations under symmetries which do not preserve the action. A simpler derivation of the invariants is provided.
We consider optimal harvesting of systems described by stochastic differential equations with delay. We focus on those situations where the value function of the harvesting problem depends on the initial path of the process in a simple way, namely through its value at 0 and through some weighted averages
A verification theorem of variational inequality type is proved. This is applied to solve explicitly some classes of optimal harvesting delay problems
The invariance properties of second order dynamical systems under velocity dependent transformations of the coordinates and time are studied. For Lagrangian systems the connection between Noether conserved quantities and dynamical symmetries is not too direct; however, the author shows that for general systems dynamical symmetries always possess associated conserved quantities, which are invariants of the symmetry group itself. In the special case of point symmetries this yields the result that the associated conserved quantity is an invariant of the first extended group.
The variational integrators of autonomous Birkhoff systems are obtained by the discrete variational principle. The geometric structure of the discrete autonomous Birkhoff system is formulated. The discretization of mathematical pendulum shows that the discrete variational method is as effective as symplectic scheme for the autonomous Birkhoff systems.
For a generalized Birkhoffian system with the action of small disturbance, the Lie symmetrical perturbation and a new type
of non-Noether adiabatic invariants are presented. On the basis of the invariance of disturbed generalized Birkhoffian system
under general infinitesimal transformation of group, the determining equation of Lie symmetrical perturbation of the system
is constructed. Based on the definition of higher-order adiabatic invariants of a mechanical system, a new type of non-Noether
adiabatic invariants of a disturbed generalized Birkhoffian system is obtained by investigating the Lie symmetrical perturbation.
Then, a new type of exact invariants of non-Noether type is given, furthermore adiabatic invariants and exact invariants of
non-Noether type are obtained under the special infinitesimal transformation of group. Finally, an example is given to illustrate
the application of the method and results.
KeywordsGeneralized Birkhoffian system–Lie symmetry–Symmetrical perturbation–Non-Noether exact invariant–Non-Noether adiabatic invariant
An optimal control problem with a prescribed performance index for parabolic systems with time delays is investigated. A necessary condition for optimality is formulated and proved in the form of a maximum principle. Under additional conditions, the maximum principle gives sufficient conditions for optimality. It is also shown that the optimal control is unique. As an illustration of the theoretical consideration, an analytic solution is obtained for a time-delayed diffusion system.
Various first-order and second-order sufficient conditions of optimality for calculus of variations problems with delayed argument are formulated. The cost functionals are not required to be convex. A second-order sufficient condition is shown to be related to the existence of solutions of a Riccati-type matrix differential inequality.
A variational problem with delayed argument is investigated. The existing necessary conditions are first reviewed. New results, two conjugate-point conditions, are then derived for this problem. The method of proof is similar to that used by Bliss for the classical problem. An example shows that the two conditions are not equivalent, and that the first-order necessary conditions, the strengthened Legendre conditions, and the conjugate-point conditions do not in general constitute a set of sufficient conditions for the delay problem. It is shown that a special case, referred to as the separated-integrand problem, leads to considerable simplification of the results for the general problem.
In this paper we consider a rather general optimal control problem involving ordinary differential equations with delayed arguments and a set of equality and inequality restrictions on state- and control variables. For this problem a maximum principle is given in pointwise form, using variational techniques. From this maximum principle necessary conditions are derived, as well as a Lagrange-like multiplier rule. Details may be found in ref. [2], together with extensions to the Hamilton-Jacobi equation and free end point problems.
This paper deals with variational and optimal control problems with delayed argument and presents analogs of the classical
necessary conditions for optimality for problems in (n + 1)-space. It is mainly concerned with the functional
J(y) = òab f[t,y(t\user2-t),y(t),[(y)\dot](t\user2-t),[(y)\dot](t)] dt J(y) = \int_a^b {f[t,y(t\user2{--}\tau ),y(t),\dot y(t\user2{--}\tau ),\dot y(t)] dt}
There are no side conditions; τ is a positive real number; andy is a continuous piecewise smooth vector function havingn components. The fundamental lemma of the calculus of variations is used in deriving an analog of the Euler equations. The
usual construction is utilized in obtaining analogs of the Weierstrass and Legendre conditions. Also found is a fourth necessary
condition involving the least proper value associated with a boundary value problem related to the second variation. A sufficient
condition is obtained by the use of a simple expansion method. The last station of the paper outlines an extension of a maximal
principle obtained by Hestenes to control problems which involve delays in both the state variable and the control variable.
Noether's theorem and Noether's inverse theorem for mechanical systems with nonconservative forces are established. The existence of first integrals depends on the existence of solutions of the generalized Noether-Bessel-Hagen equation or, which is the same, on the existence of solutions of the Killing system of partial differential equations. The theory is based on the idea that the transformations of time and generalized coordinates together with dissipative forces determine the transformations of generalized velocities, as it is the case with variations in a variational principle of Hamilton's type for purely nonconservative mechanics [17], [18]. Using the theory a few new first integrals for nonconservative problems are obtained.Der Noethersche Satz und seine Umkehrung werden fr mechanische Systeme mit nichtkonservativen Krften aufgestellt. Die Existenz von Erstintegralen hngt von der Existenz von Lsungen der verallgemeinerten Noether-Bessel-Hagen-Gleichung oder, gleichbedeutend, von der von Lsungen des Killingschen Systems partieller Differentialgleichungen ab. Die Theorie fut auf der Idee, da Transformationen von Zeit, verallgemeinerten Koordinaten und dissipativen Krften die Transformation der verallgemeinerten Geschwindigkeiten bestimmen; wie im Fall von Variationen in einem Variationsprinzip von Hamiltonscher Art fr rein nichtkonservative Systeme [17], [18]. Unter Verwendung dieser Theorie werden einige neue Erstintegrale nichtkonservativer Probleme erhalten.
Various first-order and second-order sufficient conditions of optimality for nonlinear optimal control problems with delayed argument are formulated. The functions involved are not required to be convex. Second-order sufficient conditions are shown to be related to the existence of solutions of a Riccati-type matrix differential inequality. Their relation with the second variation is discussed.
The sufficient condition of stability of linear system with delay in the form of frequency equality is received. Efficiency
of the received criterion is shown on examples.
Noether's theory of dynamical systems with unilateral constraints by introducing the generalized quasi-symmetry of the infinitesimal
transformation for the transformation group G, is presented and two examples to illustrate the application of the result are
given.
The optimal control of a system whose state is governed by a nonlinear autonomous Volterra integrodifferential equation with unbounded time interval is considered. Specifically, it is assumed that the delay occurs only in the state variable. We are concerned with the existence of an overtaking optimal trajectory over an infinite horizon. The existence result that we obtain extends the result of Carlson (Ref. 1) to a situation where the trajectories are not necessary bounded. Also, we study the structure of approximate solutions for the problem on a finite interval.
For a nonlinear nonholonomic constrained mechanical system with the action of small forces of perturbation, Lie symmetries,
symmetrical perturbation, and a new type of non-Noether adiabatic invariants are presented in general infinitesimal transformation
of Lie groups. Based on the invariance of the equations of motion for the system under general infinitesimal transformation
of Lie groups, the Lie symmetrical determining equations, constraints restriction equations, additional restriction equations,
and exact invariants of the system are given. Then, under the action of small forces of perturbation, the determining equations,
constraints restriction equations, and additional restriction equations of the Lie symmetrical perturbation are obtained,
and adiabatic invariants of the Lie symmetrical perturbation, the weakly Lie symmetrical perturbation, and the strongly Lie
symmetrical perturbation for the disturbed nonholonomic system are obtained, respectively. Furthermore, a set of non-Noether
exact invariants and adiabatic invariants are given in the special infinitesimal transformations. Finally, one example is
given to illustrate the application of the method and results.
KeywordsDisturbed nonlinear nonholonomic system–Lie group–Lie symmetrical perturbation–Exact invariant–Adiabatic invariant