Abstract: The theory and applications of dynamic derivatives on time scales have recently received considerable attention. The primary purpose of this paper is to give basic properties of diamond-alpha derivatives which are a linear combination of delta and nabla dynamic derivatives on time scales. We prove a generalized version of Jensen’s inequality on time scales via the diamond-alpha integral and present some corollaries, including Holder’s and Minkowski’s diamond-alpha integral inequalities. Copyright (c) 2008 Moulay Rchid Sidi Ammi et al.
Abstract: 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. (C) 2008 American Institute of Physics.
Abstract: 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. (C) 2008 American Institute of Physics.
Abstract: The theory and applications of dynamic derivatives on time scales have recently received considerable attention. The primary purpose of this paper is to give basic properties of diamond-alpha derivatives which are a linear combination of delta and nabla dynamic derivatives on time scales. We prove a generalized version of Jensen's inequality on time scales via the diamond-alpha integral and present some corollaries, including Holder's and Minkowski's diamond-alpha integral inequalities. Copyright (c) 2008 Moulay Rchid Sidi Ammi et al.
Abstract: Using the recent formulation of Noether's theorem for the problems of the calculus of variations with fractional derivatives, the Lagrange multiplier technique, and the fractional Euler-Lagrange equations, we prove a Noether-like theorem to the more general context of the fractional optimal control. As a corollary, it follows that in the fractional case the autonomous Hamiltonian does not define anymore a conservation law. Instead, it is proved that the fractional conservation law adds to the Hamiltonian a new term which depends on the fractional-order of differentiation, the generalized momentum and the fractional derivative of the state variable.
Abstract: We show that for any variational symmetry of the problem of the calculus of variations on time scales there exists a conserved quantity along the respective Euler-Lagrange extremals. (c) 2008 Elsevier Inc. All rights reserved.
Abstract: In this note, we contrast two transformation-based methods to deduce absolute extrema and the corresponding extremizers. Unlike variation-based methods, the transformation-based methods of Carlson and Leitmann and the recent one of Silva and Torres are direct in that they permit obtaining solutions by inspection.
Abstract: We show that for any variational symmetry of the problem of the calculus of variations on time scales there exists a conserved quantity along the respective Euler-Lagrange extremals. (c) 2008 Elsevier Inc. All rights reserved.
Abstract: We analyze the spatially semidiscrete piecewise linear finite element method for a nonlocal parabolic equation resulting from thermistor problem. Our approach is based on the properties of the elliptic projection defined by the bilinear form associated with the variational formulation of the finite element method. We assume minimal regularity of the exact solution that yields optimal order error estimate. The full discrete backward Euler method and the Crank-Nicolson-Galerkin scheme are also considered. Finally, a simple algorithm for solving the fully discrete problem is proposed. (c) 2007 IMACS. Published by Elsevier B.V. All rights reserved.
Abstract: In this note, we contrast two transformation-based methods to deduce absolute extrema and the corresponding extremizers. Unlike variation-based methods, the transformation-based methods of Carlson and Leitmann and the recent one of Silva and Torres are direct in that they permit obtaining solutions by inspection.
Abstract: Using the recent formulation of Noether’s theorem for the problems of the calculus of variations with fractional derivatives, the Lagrange multiplier technique, and the fractional Euler-Lagrange equations, we prove a Noether-like theorem to the more general context of the fractional optimal control. As a corollary, it follows that in the fractional case the autonomous Hamiltonian does not define anymore a conservation law. Instead, it is proved that the fractional conservation law adds to the Hamiltonian a new term which depends on the fractional-order of differentiation, the generalized momentum and the fractional derivative of the state variable.
Abstract: We study a system of nonlinear partial differential equations resulting from the traditional modelling of oil engineering within the framework of the mechanics of a continuous medium. Recent results on the problem provide existence, uniqueness and regularity of the optimal solution. Here we obtain the first necessary optimality conditions.
Abstract: We address the problem of obtaining well-defined criteria for multiple criteria optimal control problems. Necessary and sufficient conditions for ail objective functional to be nonessential are proved. The results provide effective tools for determining nonessential objectives in multiobjective optimal control problems.
Abstract: The study of problems of the calculus of variations with compositions is a quite recent subject with origin in dynamical systems governed by chaotic maps. Available results are reduced to a generalized Euler-Lagrange equation that contains a new term involving inverse images of the minimizing trajectories. In this work, we prove a generalization of the necessary optimality condition of DuBois-Reymond for variational problems with compositions. With the help of the new obtained condition, a Noether-type theorem is proved. An application of our main result is given to a problem appearing in the chaotic setting when one consider maps that are ergodic.
Abstract: Evolution strategies are inspired in biology and form part of a larger research field known as evolutionary algorithms. Those strategies perform a random search in the space of admissible functions, aiming to optimize some given objective function. We show that simple evolution strategies are a useful tool in optimal control, permitting one to obtain, in an efficient way, good approximations to the solutions of some recent and challenging optimal control problems.
Abstract: Fractional (or non-integer) differentiation is an important concept both from theoretical and applicational points of view. The study of problems of the calculus of variations with fractional derivatives is a rather recent subject, the main result being the fractional necessary optimality condition of Euler-Lagrange obtained in 2002. Here we use the notion of Euler-Lagrange fractional extremal to prove a Noether-type theorem. For that we propose a generalization of the classical concept of conservation law, introducing an appropriate fractional operator. (c) 2007 Elsevier Inc. All rights reserved.
Abstract: The study of problems of the calculus of variations with compositions is a quite recent subject with origin in dynamical systems governed by chaotic maps. Available results are reduced to a generalized Euler-Lagrange equation that contains a new term involving inverse images of the minimizing trajectories. In this work, we prove a generalization of the necessary optimality condition of DuBois-Reymond for variational problems with compositions. With the help of the new obtained condition, a Noether-type theorem is proved. An application of our main result is given to a problem appearing in the chaotic setting when one consider maps that are ergodic.
Abstract: Evolution strategies are inspired in biology and form part of a larger research field known as evolutionary algorithms. Those strategies perform a random search in the space of admissible functions, aiming to optimize some given objective function. We show that simple evolution strategies are a useful tool in optimal control, permitting one to obtain, in an efficient way, good approximations to the solutions of some recent and challenging optimal control problems.
Abstract: We address the problem of obtaining well-defined criteria for multiple criteria optimal control problems. Necessary and sufficient conditions for ail objective functional to be nonessential are proved. The results provide effective tools for determining nonessential objectives in multiobjective optimal control problems.
Abstract: We study a system of nonlinear partial differential equations resulting from the traditional modelling of oil engineering within the framework of the mechanics of a continuous medium. Recent results on the problem provide existence, uniqueness and regularity of the optimal solution. Here we obtain the first necessary optimality conditions.
Abstract: We derive Euler-Lagrange-type equations for fractional action-like integrals of the calculus of variations which depend on the Riemann-Liouville derivatives of order (alpha, beta), alpha > 0, beta > 0, recently introduced by Cresson. Some interesting consequences are obtained and discussed. Copyright (c) 2007 John Wiley & Sons, Ltd.
Abstract: Fractional (or non-integer) differentiation is an important concept both from theoretical and applicational points of view. The study of problems of the calculus of variations with fractional derivatives is a rather recent subject, the main result being the fractional necessary optimality condition of Euler-Lagrange obtained in 2002. Here we use the notion of Euler-Lagrange fractional extremal to prove a Noether-type theorem. For that we propose a generalization of the classical concept of conservation law, introducing an appropriate fractional operator. (c) 2007 Elsevier Inc. All rights reserved.
Abstract: We derive Euler-Lagrange-type equations for fractional action-like integrals of the calculus of variations which depend on the Riemann-Liouville derivatives of order (alpha, beta), alpha > 0, beta > 0, recently introduced by Cresson. Some interesting consequences are obtained and discussed. Copyright (c) 2007 John Wiley & Sons, Ltd.
Abstract: We use a computer algebra system to compute, in an efficient way, optimal control variational symmetries up to a gauge term. The symmetries are then used to obtain families of Noether’s first integrals, possibly in the presence of nonconservative external forces. As an application, we obtain eight independent first integrals for a sub-Riemannian nilpotent problem (2,3,5,8).
Abstract: Newton’s problem of minimal resistance is one of the first problems of optimal control: it was proposed, and its solution given, by Isaac Newton in his masterful Principia Mathematica, in 1686. The problem consists of determining, in dimension three, the shape of an axis-symmetric body, with assigned radius and height, which offers minimum resistance when it is moving in a resistant medium. The problem has a very rich history and is well documented in the literature. Of course, at a first glance, one suspects that the two dimensional case should be well known. Nevertheless, we have looked into numerous references and asked at least as many experts on the problem, and we have not been able to identify a single source. Solution was always plausible to everyone who thought about the problem, and writing it down was always thought not to be worthwhile. Here we show that this is not the case: the two-dimensional problem is richer than the classical one, being, in some sense, more interesting. Novelties include: (i) while in the classical three-dimensional problem only the restricted case makes sense (without restriction on the monotonicity of admissible functions the problem does not admit a local minimum), we prove that in dimension two the unrestricted problem is also well-posed when the ratio of height versus radius of base is greater than a given quantity; (ii) while in three dimensions the (restricted) problem has a unique solution, we show that in the restricted two-dimensional problem the minimizer is not always unique - when the height of the body is less or equal than its base radius, there exists infinitely many minimizing functions.
Abstract: We obtain a method to compute effective first integrals by combining Noether’s principle with the Kozlov-Kolesnikov integrability theorem. A sufficient condition for the integrability by quadratures of optimal control problems with controls taking values on open sets is obtained. We illustrate our approach on some problems taken from the literature. An alternative proof of the integrability of the sub-Riemannian nilpotent Lie group of type (2,3,5) is also given.
Abstract: We use a computer algebra system to compute, in an efficient way, optimal control variational symmetries up to a gauge term. The symmetries are then used to obtain families of Noether's first integrals, possibly in the presence of nonconservative external forces. As an application, we obtain eight independent first integrals for a sub-Riemannian nilpotent problem (2,3,5,8).
Abstract: Newton's problem of minimal resistance is one of the first problems of optimal control: it was proposed, and its solution given, by Isaac Newton in his masterful Principia Mathematica, in 1686. The problem consists of determining, in dimension three, the shape of an axis-symmetric body, with assigned radius and height, which offers minimum resistance when it is moving in a resistant medium. The problem has a very rich history and is well documented in the literature. Of course, at a first glance, one suspects that the two dimensional case should be well known. Nevertheless, we have looked into numerous references and asked at least as many experts on the problem, and we have not been able to identify a single source. Solution was always plausible to everyone who thought about the problem, and writing it down was always thought not to be worthwhile. Here we show that this is not the case: the two-dimensional problem is richer than the classical one, being, in some sense, more interesting. Novelties include: (i) while in the classical three-dimensional problem only the restricted case makes sense (without restriction on the monotonicity of admissible functions the problem does not admit a local minimum), we prove that in dimension two the unrestricted problem is also well-posed when the ratio of height versus radius of base is greater than a given quantity; (ii) while in three dimensions the (restricted) problem has a unique solution, we show that in the restricted two-dimensional problem the minimizer is not always unique - when the height of the body is less or equal than its base radius, there exists infinitely many minimizing functions.
Abstract: We obtain a method to compute effective first integrals by combining Noether's principle with the Kozlov-Kolesnikov integrability theorem. A sufficient condition for the integrability by quadratures of optimal control problems with controls taking values on open sets is obtained. We illustrate our approach on some problems taken from the literature. An alternative proof of the integrability of the sub-Riemannian nilpotent Lie group of type (2,3,5) is also given.
Abstract: The mathematical model of a real flexible elastic system with distributed and discrete parameters is considered. It is a partial differential equation with non-classical boundary conditions. Complexity of the boundary conditions makes it impossible to find exact analytical solutions. To address the problem, we use the asymptotical method of small parameters together with the numerical method of normal fundamental systems of solutions. These methods allow us to investigate vibrations, and a technique for determination of complex eigenvalues of the considered boundary value problem is developed. The conditions, at which vibration processes of different characteristics take place, are defined. The dependence of the vibration frequencies on the physical parameters of the hybrid system is studied. We show that introduction of different feedbacks into the system allows one to control the frequency spectrum, in which excitation of vibrations is possible. (c) 2005 Elsevier Ltd. All rights reserved.
Abstract: The problem of minimum resistance is studied for a body moving with constant velocity in a rarefied medium of chaotically moving point particles in the Euclidean space R-d. The distribution of the velocities of the particles is assumed to be radially symmetric. Under additional assumptions on the distribution function a complete classification of the bodies of least resistance is carried out. In the case of dimension three or more there exist two kinds of solution: a body similar to the solution of the classical Newton problem and a union of two such bodies 'glued together' along the rear parts of their surfaces. In the two-dimensional case there exist solutions of five distinct types: (a) a trapezium; (b) an isosceles triangle; (c) the union of an isosceles triangle and a trapezium with a common base; (d) the union of two isosceles triangles with a common base; (e) the union of two triangles and a trapezium. Cases (a)-(d) are realized for an arbitrary velocity distribution of the particles, while case (e) is realized only for some distributions. Two limit cases are considered: when the average velocity of the particles is large and when it is small in comparison with the velocity of the body. Finally, the analytic results so obtained are used for the numerical study of a particular case: the problem of the motion of a body in a rarefied homogeneous monatornic ideal gas of positive temperature in R-2 and in R-3.
Abstract: The mathematical model of a real flexible elastic system with distributed and discrete parameters is considered. It is a partial differential equation with non-classical boundary conditions. Complexity of the boundary conditions makes it impossible to find exact analytical solutions. To address the problem, we use the asymptotical method of small parameters together with the numerical method of normal fundamental systems of solutions. These methods allow us to investigate vibrations, and a technique for determination of complex eigenvalues of the considered boundary value problem is developed. The conditions, at which vibration processes of different characteristics take place, are defined. The dependence of the vibration frequencies on the physical parameters of the hybrid system is studied. We show that introduction of different feedbacks into the system allows one to control the frequency spectrum, in which excitation of vibrations is possible. (c) 2005 Elsevier Ltd. All rights reserved.
Abstract: The problem of minimum resistance is studied for a body moving with constant velocity in a rarefied medium of chaotically moving point particles in the Euclidean space R-d. The distribution of the velocities of the particles is assumed to be radially symmetric. Under additional assumptions on the distribution function a complete classification of the bodies of least resistance is carried out. In the case of dimension three or more there exist two kinds of solution: a body similar to the solution of the classical Newton problem and a union of two such bodies ‘glued together’ along the rear parts of their surfaces. In the two-dimensional case there exist solutions of five distinct types: (a) a trapezium; (b) an isosceles triangle; (c) the union of an isosceles triangle and a trapezium with a common base; (d) the union of two isosceles triangles with a common base; (e) the union of two triangles and a trapezium. Cases (a)-(d) are realized for an arbitrary velocity distribution of the particles, while case (e) is realized only for some distributions. Two limit cases are considered: when the average velocity of the particles is large and when it is small in comparison with the velocity of the body. Finally, the analytic results so obtained are used for the numerical study of a particular case: the problem of the motion of a body in a rarefied homogeneous monatornic ideal gas of positive temperature in R-2 and in R-3.
Abstract: For nonsmooth Euler-Lagrange extremals, Noether's conservation laws cease to be valid. We show that Emmy Noether's theorem of the calculus of variations is still valid in the wider class of Lipschitz functions, as long as one restrict the Euler-Lagrange extremals to those which satisfy the DuBois-Reymond necessary condition. In the smooth case all Euler-Lagrange extremals are DuBois-Reymond extremals, and the result gives a proper extension of the classical Noether's theorem. This is in contrast with the recent developments of Noether's symmetry theorems to the optimal control setting, which give rise to non-proper extensions when specified for the problems of the calculus of variations. Results are also obtained for variational problems with higher-order derivatives.
Abstract: For nonsmooth Euler-Lagrange extremals, Noether’s conservation laws cease to be valid. We show that Emmy Noether’s theorem of the calculus of variations is still valid in the wider class of Lipschitz functions, as long as one restrict the Euler-Lagrange extremals to those which satisfy the DuBois-Reymond necessary condition. In the smooth case all Euler-Lagrange extremals are DuBois-Reymond extremals, and the result gives a proper extension of the classical Noether’s theorem. This is in contrast with the recent developments of Noether’s symmetry theorems to the optimal control setting, which give rise to non-proper extensions when specified for the problems of the calculus of variations. Results are also obtained for variational problems with higher-order derivatives.
Abstract: We consider the Lagrange problem of optimal control with unrestricted controls and address the question: under what conditions can we assure optimal controls are bounded? This question is related to one of Lipschitzian regularity of optimal trajectories, and the answer to it is crucial in closing the gap between the conditions arising in existence theory and necessary optimality conditions. Rewriting the Lagrange problem in a parametric form, we obtain a relation between the applicability conditions of the Pontryagin maximum principle to the latter problem and the Lipschitzian regularity conditions for the original problem. Under the standard hypotheses of coercivity of the existence theory, the conditions imply that the optimal controls are essentially bounded, assuring the applicability of the classical necessary optimality conditions like the Pontryagin maximum principle. The result extends previous Lipschitzian regularity results to cover optimal control problems with general nonlinear dynamics.
Abstract: For nonsmooth Euler-Lagrange extremals, Noether's conservation laws cease to be valid. We show that Emmy Noether's theorem of the calculus of variations is still valid in the wider class of Lipschitz functions, as long as one restrict the Euler-Lagrange extremals to those which satisfy the DuBois-Reymond necessary condition. In the smooth case all Euler-Lagrange extremals are DuBois-Reymond extremals, and the result gives a proper extension of the classical Noether's theorem. This is in contrast with the recent developments of Noether's symmetry theorems to the optimal control setting, which give rise to non-proper extensions when specified for the problems of the calculus of variations. Copyright (C) 2003 IFAC.
Abstract: We consider the Lagrange problem of optimal control with unrestricted controls and address the question: under what conditions can we assure optimal controls are bounded? This question is related to one of Lipschitzian regularity of optimal trajectories, and the answer to it is crucial in closing the gap between the conditions arising in existence theory and necessary optimality conditions. Rewriting the Lagrange problem in a parametric form, we obtain a relation between the applicability conditions of the Pontryagin maximum principle to the latter problem and the Lipschitzian regularity conditions for the original problem. Under the standard hypotheses of coercivity of the existence theory, the conditions imply that the optimal controls are essentially bounded, assuring the applicability of the classical necessary optimality conditions like the Pontryagin maximum principle. The result extends previous Lipschitzian regularity results to cover optimal control problems with general nonlinear dynamics.
Abstract: We obtain a discrete time analog of E. Noether's theorem in Optimal Control, asserting that integrals of motion associated to the discrete time Pontryagin Maximum Principle can be computed from the quasi-invariance properties of the discrete time Lagrangian and discrete time control system. As corollaries, conservation laws for discrete problems of the calculus of variations are obtained. Copyright (C) 2003 IFAC.
Abstract: Conservation laws, i.e. conserved quantities along Euler-Lagrange extremals, which are obtained on the basis of Noether's theorem, play a prominent role in mathematical analysis and physical applications. In this paper we present a general and constructive method to obtain conserved quantities along the Pontryagin extremals of optimal control problems, which axe invariant under a family of transformations that explicitly change all (time, state, control) variables.
Abstract: We obtain a generalization of E. Noether's theorem for the optimal control problems. The generalization involves a one-parameter family of smooth maps which may depend also on the control and a Lagrangian which is invariant up to an addition of an exact differential.
Abstract: Conservation laws, i.e. conserved quantities along Euler-Lagrange extremals, which are obtained on the basis of Noether’s theorem, play a prominent role in mathematical analysis and physical applications. In this paper we present a general and constructive method to obtain conserved quantities along the Pontryagin extremals of optimal control problems, which axe invariant under a family of transformations that explicitly change all (time, state, control) variables.
Abstract: We obtain a generalization of E. Noether’s theorem for the optimal control problems. The generalization involves a one-parameter family of smooth maps which may depend also on the control and a Lagrangian which is invariant up to an addition of an exact differential.
Abstract: We survey some conditions for Lipschitzian regularity of minimizers in various problems of the calculus of variations and optimal control theory. Some recent results obtained by the authors are presented as well.
Abstract: We study the Lagrange Problem of Optimal Control with a functional integral(a)(b) L (t, x (t), u (t)) dt and control-affine dynamics (x) over dot = f (t, x) + g (t, x)u and (a priori) unconstrained control u is an element of R-m. We obtain conditions under which the minimizing controls of the problem are bounded-a fact which is crucial for the applicability of many necessary optimality conditions, like, for example, the Pontryagin Maximum Principle. As a corollary we obtain conditions for the Lipschitzian regularity of minimizers of the Basic Problem of the Calculus of Variations and of the Problem of the Calculus of Variations with higher-order derivatives.
Abstract: We study the Lagrange Problem of Optimal Control with a functional integral(a)(b) L (t, x (t), u (t)) dt and control-affine dynamics (x) over dot = f (t, x) + g (t, x)u and (a priori) unconstrained control u is an element of R-m. We obtain conditions under which the minimizing controls of the problem are bounded-a fact which is crucial for the applicability of many necessary optimality conditions, like, for example, the Pontryagin Maximum Principle. As a corollary we obtain conditions for the Lipschitzian regularity of minimizers of the Basic Problem of the Calculus of Variations and of the Problem of the Calculus of Variations with higher-order derivatives.
