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We consider control problems for the variational inequality describing a single degree of freedom elasto-plastic oscillator. We are particularly interested in finding the "critical excitation", i.e., the lowest energy input excitation that drives the system between the prescribed initial and final states within a given time span. This is a control problem for a state evolution described by a variational inequality. We obtain Pontryagin’s necessary condition of optimality. An essential difficulty lies with the non continuity of adjoint variables.
We consider the kinematics and control of a sphere rolling of a curved surface and analyze its rotation by mapping the system to the precession of a spin 1/2 in a magnetic field of variable magnitude and direction. This mapping is useful in understanding the role of the geometrical phase and generalizes the kinematic control problem of a ball rolling on a plane.
The objective of this article is to present geometric and numerical techniques developed to study the orbit transfer between Keplerian elliptic orbits in the two-body problem or between quasi-Keplerian orbits in the Earth-Moon transfer when low propulsion is used. We concentrate our study on the energy minimization problem. From Pontryagin’s maximum principle, the optimal solution can be found solving the shooting equation for smooth Hamiltonian dynamics. A first step in the analysis is to find in the Kepler case an analytical solution for the averaged Hamiltonian, which corresponds to a Riemannian metric. This will allow to compute the solution for the original Kepler problem, using a numerical continuation method where the smoothness of the path is related to the conjugate point condition. Similarly, the solution of the Earth-Moon transfer is computed using geometric and numerical continuation techniques.
We present here a general theory, and give a specific example, showing that there exist time invariant Markov decision problems, with no time variation in the model which, when optimized over an infinite interval, have optimal closed loop control laws that are time varying. Although similar behavior was observed much earlier for specific problems arising in chemical and aeronautical engineering, this work is not applicable to Markov decision problems because of the specific form of the constraints involving the action of the semigroup of stochastic matrices on the standard simplex and the bilinear structure that goes along with rate control for Markov processes. The results given here are especially interesting insofar as they are analogous to the optimal solutions of stochastic control problems associated with Carnot cycles. As in some earlier work, the conditions under which time varying controls are optimal are characterized in terms of the the second variation about a singular solution. In this case the second variation is expressible in terms of a kernel function and conditions under which the second variation is positive definite can be checked by determining if the transform of this kernel is positive real or not.
This paper presents several classical mechanical systems with nonholonomic constraints from the point of view of sub-Riemannian geometry. For those systems that satisfy the bracket generating condition the system can move continuously between any two given states. However, the paper provides a counterexample to show that the bracket generating condition is not also a sufficient condition for connectivity. All possible motions of the system correspond to curves tangent to the distribution defined by the nonholonomic constraints. Among the connecting curves we distinguish an optimal one which minimizes a certain energy induced by a natural sub-Riemannian metric on the non-integrable distribution. The paper discusses several classical problems such as the knife edge, the skater, the rolling disk and the nonholonomic bicycle.