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We give a geometric description of variational principles in mechanics, with special attention to constrained systems. For the general case of nonholonomic constraints, a unified variational approach is given, and the equations of motion of…
We consider a nonlinear system, affine with respect to an unbounded control $u$ which is allowed to range in a closed cone. To this system we associate a Bolza type minimum problem, with a Lagrangian having sublinear growth with respect to…
Recently, continuous-time dynamical systems have proved useful in providing conceptual and quantitative insights into gradient-based optimization, widely used in modern machine learning and statistics. An important question that arises in…
We present a Pontryagin maximum principle for discrete time optimal control problems with (a) pointwise constraints on the control actions and the states, (b) frequency constraints on the control and the state trajectories, and (c)…
In many problems in optimal control, one seeks to minimise an objective function subject to constraints on the velocity of the system. Imposing these constraints directly -- the ``hard-constrained'' approach -- is often analytically and…
We constructed a symplectic realization of the dynamic structure of two interacting spin-two fields in three dimensions. A significant simplification refers to the treatment of constraints: instead of performing a Hamiltonian analysis…
This paper studies optimal trajectory-tracking for driftless, x-flat nonlinear systems with three states and two inputs. The tracking problem is formulated in Bolza form with a quadratic cost of the tracking error and its derivative.…
Trajectory optimization is a fundamental problem in robotics. While optimization of continuous control trajectories is well developed, many applications require both discrete and continuous, i.e., hybrid, controls. Finding an optimal…
This paper is a review of results which have been recently obtained by applying mathematical concepts drawn, in particular, from differential geometry and topology, to the physics of Hamiltonian dynamical systems with many degrees of…
The first part of the paper studies a class of optimal control problems in Bolza form, where the dynamics is linear w.r.t.~the control function. A necessary condition is derived, for the optimality of a trajectory which starts at a…
This paper establishes a general framework for describing hybrid dynamical systems which is particularly suitable for numerical simulation. In this context, the data structures used to describe the sets and functions which comprise the…
Hybrid systems are dynamical systems with continuous-time and discrete-time components in their dynamics. When hybrid systems are defined on a principal bundle we are able to define two classes of impacts for the discrete-time transition of…
The dynamic equation of mass point in rotating coordinates is governed by Coriolis and centrifugal force, besides a corotating potential relative to frame. Such a system is no longer a canonical Hamiltonian system so that the construction…
This paper develops numerical methods for optimal control of mechanical systems in the Lagrangian setting. It extends the theory of discrete mechanics to enable the solutions of optimal control problems through the discretization of…
In this note, we develop the first-order theory of optimal control problems with box constraints on the control. We emphasize the precise modification of Pontryagin's maximum principle when the admissible control set is compact, the…
Pointwise minimum norm control laws for hybrid dynamical systems are proposed. Hybrid systems are given by differential equations capturing the continuous dynamics or flows, and by difference equations capturing the discrete dynamics or…
This paper investigates a family of dynamical systems arising from an evolutionary re-interpretation of certain optimal control and optimization problems. We focus particularly on the application in image registration of the theory of…
We extend the convergence analysis for methods solving PDE-constrained optimal control problems containing both discrete and continuous control decisions based on relaxation and rounding strategies to the class of first order semilinear…
A complete geometric classification of symmetries of autonomous Hamiltonian mechanical systems is established; explaining how to obtain their associated conserved quantities in all cases. In particular, first we review well-known results…
In this note we describe how some objects from generalized geometry appear in the qualitative analysis and numerical simulation of mechanical systems. In particular we discuss double vector bundles and Dirac structures. It turns out that…