64 research outputs found
On some aspects of the geometry of non integrable distributions and applications
We consider a regular distribution D in a Riemannian manifold (M, g). The LeviCivita connection on (M, g) together with the orthogonal projection allow to endow the space of sections of D with a natural covariant derivative, the intrinsic connection. Hence we have two different covariant derivatives for sections of D, one directly with the connection in (M, g) and the other one with this intrinsic connection. Their difference is the second fundamental form of D and we prove it is a significant tool to characterize the involutive and the totally geodesic distributions and to give a natural formulation of the equation of motion for mechanical systems with constraints. The two connections also give two different notions of curvature, curvature tensors and sectional curvatures, which are compared in this paper with the use of the second fundamental form.Peer ReviewedPostprint (author's final draft
Strict abnormal extremals in nonholonomic and kinematic control systems
In optimal control problems, there exist different kinds of
extremals, that is, curves candidates to be solution:
abnormal, normal and strictly abnormal. The key point for
this classification is how those extremals depend on the
cost function. We focus on control systems such as
nonholonomic control mechanical systems and the associated
kinematic systems as long as they are equivalent.
With all this in mind, first we study conditions to relate
an optimal control problem for the mechanical system with
another one for the associated kinematic system. Then,
Pontryagin's Maximum Principle will be used to connect the
abnormal extremals of both optimal control problems.
An example is given to glimpse what the abnormal solutions
for kinematic systems become when they are considered as
extremals to the optimal control problem for the
corresponding nonholonomic mechanical systems
Constraint algorithm for extremals in optimal control problems
A characterization of different kinds of extremals of optimal control problems
is given if we take an open control set. A well known constraint algorithm for
implicit differential equations is adapted to the study of such problems. Some
necessary conditions of Pontryagin’s Maximum Principle determine the primary
constraint submanifold for the algorithm. Some examples in the control literature,
such as subRiemannian geometry and control-affine systems, are revisited
to give, in a clear geometric way, a subset where the abnormal, normal and strict
abnormal extremals stand
Optimal control problems for affine connection control systems: characterization of extremals
Pontryagin’s Maximum Principle [8] is considered as an outstanding achievement of
optimal control theory. This Principle does not give sufficient conditions to compute an optimal
trajectory; it only provides necessary conditions. Thus only candidates to be optimal trajectories,
called extremals, are found. Maximum Principle gives rise to different kinds of them and, particularly,
the so-called abnormal extremals have been studied because they can be optimal, as Liu and
Sussmann, and Montgomery proved in subRiemannian geometry [5, 7].
We build up a presymplectic algorithm, similar to those defined in [2, 3, 4, 6], to determine
where the different kinds of extremals of an optimal control problem can be. After describing such
an algorithm, we apply it to the study of extremals, specially the abnormal ones, in optimal control
problems for affine connection control systems [1]. These systems model the motion of different
types of mechanical systems such as rigid bodies, nonholonomic systems and robotic arms [1].Peer Reviewe
Optimal control, contact dynamics and Herglotz variational problem
In this paper, we combine two main topics in mechanics and optimal control theory: contact Hamiltonian systems and Pontryagin maximum principle. As an important result, among others, we develop a contact Pontryagin maximum principle that permits to deal with optimal control problems with dissipation. We also consider the Herglotz optimal control problem, which is simultaneously a generalization of the Herglotz variational principle and an optimal control problem. An application to the study of a thermodynamic system is provided.M. de León and M. Lainz acknowledge the partial finantial support from MINECO Grants MTM2016-76-072-P and PID2019-106715GB-C21 and the ICMAT Severo Ochoa projects SEV2015-0554 and CEX2019-000904-S. M. Lainz wishes to thank MICINN and ICMAT for a FPI-Severo Ochoa predoctoral contract PRE2018-083203. M.C. Muñoz-Lecanda acknowledges the financial support from the Spanish Ministerio de Ciencia, Innovación y Universidades project PGC2018-098265-B-C33 and the Secretary of University and Research of the Ministry of Business and Knowledge of the Catalan Government project 2017–SGR–932. We also thank Maria Barbero-Liñan for the fruitful conversations we had with her about the PMP. Finally, we would like to thank the referees for their careful read and their constructive inputs.Peer ReviewedPostprint (published version
Structural aspects of Hamilton–Jacobi theory
The final publication is available at Springer via http://dx.doi.org/10.1142/S0219887816500171In our previous papers [11, 13] we showed that the Hamilton–Jacobi problem can be regarded as a way to describe a given dynamics on a phase space manifold in terms of a family of dynamics on a lower-dimensional manifold. We also showed how constants of the motion help to solve the Hamilton–Jacobi equation. Here we want to delve into this interpretation by considering the most general case: a dynamical system on a manifold that is described in terms of a family of dynamics (‘slicing vector fields’) on lower-dimensional manifolds. We identify the relevant geometric structures that lead from this decomposition of the dynamics to the classical Hamilton– Jacobi theory, by considering special cases like fibred manifolds and Hamiltonian dynamics, in the symplectic framework and the Poisson one. We also show how a set of functions on a tangent bundle can determine a second-order dynamics for which they are constants of the motion.Peer ReviewedPostprint (author's final draft
Skinner-Rusk unified formalism for optimal control systems and applications
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)
Unified formalism for non-autonomous mechanical systems
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
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