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The problem of reduction of multisymplectic manifolds by the action of Lie groups is stated and discussed, as a previous step to give a fully covariant scheme of reduction for classical field theories with symmetries., Peer Reviewed, Postprint (author's final draft)

We present a unified geometric framework for describing both the Lagrangian and Hamiltonian formalisms of regular and non-regular time-dependent mechanical systems, which is based on the approach of Skinner and Rusk (1983). The dynamical equations of motion and their compatibility and consistency are carefully studied, making clear that all the characteristics of the Lagrangian and the Hamiltonian formalisms are recovered in this formulation. As an example, it is studied a semidiscretization of the nonlinear wave equation proving the applicability of the proposed formalism., Peer Reviewed

We present a unified geometric framework for describing both the Lagrangian and Hamiltonian formalisms of regular and non-regular time-dependent mechanical systems, which is based on the approach of Skinner and Rusk (1983). The dynamical equations of motion and their compatibility and consistency are carefully studied, making clear that all the characteristics of the Lagrangian and the Hamiltonian formalisms are recovered in this formulation. As an example, it is studied a semidiscretization of the nonlinear wave equation proving the applicability of the proposed formalism.

A geometric approach to time-dependent optimal control problems is proposed. This formulation is based on the Skinner and Rusk formalism for Lagrangian and Hamiltonian systems. The corresponding unified formalism developed for optimal control systems allows us to formulate geometrically the necessary conditions given by Pontryagin’s Maximum Principle, providing that the differentiability with respect to controls is assumed and the space of controls is open. Furthermore, our method is also valid for implicit optimal control systems and, in particular, for the so-called descriptor systems (optimal control problems including both differential and algebraic equations).

We consider Hamiltonian systems in first-order multisymplectic field theories. We review the properties of Hamiltonian systems in the so-called restricted multimomentum bundle, including the variational principle which leads to the Hamiltonian field equations. In an analogous way to how these systems are defined in the so-called extended (symplectic) formulation of nonautonomous mechanics, we introduce Hamiltonian systems in the extended multimomentum bundle. The geometric properties of these systems are studied, the Hamiltonian equations are analyzed using integrable multivector fields, the corresponding variational principle is also stated, and the relation between the extended and the restricted Hamiltonian systems is established. All these properties are also adapted to certain kinds of submanifolds of the multimomentum bundles in order to cover the case of almost-regular field theories.

A geometric approach to time-dependent optimal control problems is proposed. This formulation is based on the Skinner and Rusk formalism for Lagrangian and Hamiltonian systems. The corresponding unified formalism developed for optimal control systems allows us to formulate geometrically the necessary conditions given by Pontryagin's Maximum Principle, providing that the differentiability with respect to controls is assumed and the space of controls is open. Furthermore, our method is also valid for implicit optimal control systems and, in particular, for the so-called descriptor systems (optimal control problems including both differential and algebraic equations).

In 1983, the dynamics of a mechanical system was represented by a first-order system on a suitable phase space by R. Skinner and R. Rusk. The corresponding unified formalism developed for optimal control systems allows us to formulate geometrically the necessary conditions given by Pontryagin's Maximum Principle, as long as the differentiability with respect to controls is assumed.

We consider Hamiltonian systems in first-order multisymplectic field theories. First we review the construction and properties of Hamiltonian systems in the so-called restricted multimomentum bundle using Hamiltonian sections, including the variational principle which leads to the Hamiltonian field equations. Then, we introduce Hamiltonian systems in the extended multimomentum bundle, in an analogous way to how these systems are defined in non-autonomous (symplectic) mechanics or in the so-called extended (symplectic) formulation of autonomous mechanics. The corresponding variational principle is also stated for these extended Hamiltonian systems and, after studying the geometric properties of these systems, we establish the relation between the extended and the restricted ones. These definitions and properties are also adapted to submanifolds of the multimomentum bundles in order to cover the case of almost-regular field theories.

We consider Hamiltonian systems in first-order multisymplectic field theories. First we review the construction and properties of Hamiltonian systems in the so-called restricted multimomentum bundle using Hamiltonian sections, including the variational principle which leads to the Hamiltonian field equations. Then, we introduce Hamiltonian systems in the extended multimomentum bundle, in an analogous way to how these systems are defined in non-autonomous (symplectic) mechanics or in the so-called extended (symplectic) formulation of autonomous mechanics. The corresponding variational principle is also stated for these extended Hamiltonian systems and, after studying the geometric properties of these systems, we establish the relation between the extended and the restricted ones. These definitions and properties are also adapted to submanifolds of the multimomentum bundles in order to cover the case of almost-regular field theories.