Existence Of Optimal Trajectories Research Articles

Time minimization problem on the group of motions of a plane with admissible control in a half-disc

Abstract The time minimization problem with admissible control in a half-disc is considered on the group of motions of a plane. The control system under study provides a model of a car on the plane that can move forwards or rotate in place. Optimal trajectories of such a system are used to detect salient curves in image analysis. In particular, in medical image analysis such trajectories are used for tracking vessels in retinal images. The problem is of independent interest in geometric control theory: it provides a model example when the set of values of the control parameters contains zero at the boundary. The problem of controllability and existence of optimal trajectories is studied. By analysing the Hamiltonian system of the Pontryagin maximum principle the explicit form of extremal controls and trajectories is found. Optimality of the extremals is partially investigated. The structure of the optimal synthesis is described. Bibliography: 33 titles.

Averaged time-optimal control problem in the space of positive Borel measures

We introduce a time-optimal control theory in the space ℳ+ (ℝd) of positive and finite Borel measures. We prove some natural results, such as a dynamic programming principle, the existence of optimal trajectories, regularity results and an HJB equation for the value function in this infinite-dimensional setting. The main tool used is the superposition principle (by Ambrosio–Gigli–Savaré) which allows to represent the trajectory in the space of measures as weighted superposition of classical characteristic curves in ℝd.

Accumulation of tangent points with the boundary and Lagrangian manifolds in problems with phase constraints

The phenomenon of the existence of optimal trajectories having an infinite number of tangent points with the boundary of the phase constraint is studied. The optimal synthesis and the structure of the corresponding Lagrangian manifolds are obtained. The main tool is the blowing up procedure for the resolution of the singularities of the Poincare mapping of the constraint boundary on itself. A singularity appears at the point of accumulation of the tangent points with the boundary.

Approximate Euler-Lagrange inclusion, approximate transversality condition, and sensitivity analysis of convex parametric problems of calculus of variations

We study the first-order behaviour of the optimal value function associated to a convex parametric problem of calculus of variations. An important feature of this paper is that we do not assume the existence of optimal trajectories for the unperturbed problem. The concepts of approximate Euler-Lagrange inclusion and approximate transversality condition are key ingredients in the writing of our sensitivity results.

Optimal control on an infinite time horizon: The turnpike approach

The optimal control problem with infinite horizon is defined. Under assumptions motivated by the theory of optimal economic growth the controllability of the system, the turnpike property, the asymptotic behavior and the existence of optimal trajectories are fully explored.