Regular path-constrained time-optimal control problems in three-dimensional flow fields

Abstract

This article concerns a class of time-optimal state constrained control problems with dynamics defined by an ordinary differential equation involving a three-dimensional steady flow vector field. The problem is solved via an indirect method based on the maximum principle in Gamkrelidze's form. The proposed computational method essentially uses a certain regularity condition imposed on the data of the problem. The property of regularity guarantees the continuity of the measure multiplier associated with the state constraint, and ensures the appropriate behavior of the corresponding numerical procedure which, in general, consists in computing the entire field of extremals for the problem in question. Several examples of vector fields are considered to illustrate the computational approach.

Necessary Optimality Conditions for a Class of Control Problems with State Constraint

“…In the interior of every boundary interval, the function χ is of class C 1 with the derivative identically zero. The continuity of χ at entry points readily follows from (13) and (9). Let now t ex be an exit point of u.…”

“…Of course, the other approaches clearly prevail in a wider perspective, when problems of greater complexity are also taken into account. They then produce optimality conditions, which can be effectively used in optimal control computations (see [11][12][13][14]). Consider a control system described by a state equation .…”

A Maximum Principle for a Time-Optimal Bilevel Sweeping Control Problem

“…Moreover, as shown in [4] and further exploited in [27], additional controllability, and regularity assumptions may be imposed on the optimal control problem data ensuring that the measure multipliers of the Maximum Principle are continuous. This fact is very important to ensure the efficiency of numerical computational algorithms based on indirect methods using necessary conditions of optimality and was exploited in [12], and [30].…”

Fundamentals of Synthesized Optimal Control

“…There are two main directions for solving the problem of optimal control: direct and indirect approaches. The indirect approach based on the Pontryagin's maximum principle [2][3][4] solves optimal control by formulating it as a boundary-value problem, in which it is necessary to find the initial conditions for a system of differential equations for conjugate variables. Its optimal solution is highly accurate, however, very sensitive to the formulation of additional conditions that the control must satisfy, along with ensuring the maximum of the Hamiltonian, which are generally very difficult to set in practice for problems with complex phase constraints.…”