Locally deforming continuation method based on a shooting method for a class of optimal control problems

Abstract

This paper proposes a new continuation method for solving optimal control problems. The proposed method is based on a shooting method. In the proposed method, a cost function of an optimal control problem is locally deformed to find the solution of the problem in a stable way. This paper also analyses a relationship between the variation of the continuation parameter and the proximity of the solutions before and after a deformation in the proposed method.The obtained relation provides guidance on how to deform the continuation parameter. The effectiveness of this method is confirmed through numerical examples.

Introduction to Compressed Sensing with Python

“…We take the parameters α = 0.8 and λ = 50 in (38). First, we compute the matrix U and y in (40). The Python code is given as follows: g = St( gamma *tlambda ,g + gamma * tU .T @ (ty -tU @ g))…”

Sparse control for continuous‐time systems

“…⊤ with the same partition as Ψ in (63). S 𝛾 is the soft-thresholding function 75 defined by where [⋅] k denotes the kth element of the argument vector, and u k is the kth element of vector u.…”

“…60,61 Efficient algorithms to numerically obtain the maximum hands-off control have been proposed in recent papers. [62][63][64] Discrete-time hands-off control is also important in digital control systems, on which a recent line of research has focused. [65][66][67][68] Finally, practical applications have been proposed for electrically tunable lens, 69 spacecraft maneuvering, 70 and thermally activated building systems, 71 to name a few.…”

“…The maximum hands‐off control has also been extended to time‐optimal control, 51 control in the presence of denial‐of‐service attacks, 52 distributed control, 53,54 continuous control, 55 infinite‐dimensional systems, 56 optimal multiplexing, 57 stochastic control, 58,59 and time‐space sparse control 60,61 . Efficient algorithms to numerically obtain the maximum hands‐off control have been proposed in recent papers 62‐64 . Discrete‐time hands‐off control is also important in digital control systems, on which a recent line of research has focused 65‐68 .…”

A survey on compressed sensing approach to systems and control

“…Fundamental properties of maximum hands-off control, such as the value function [49] and necessary conditions [50] have been investigated. Efficient numerical algorithms for maximum hands-off control have been proposed in recent papers [51][52][53]. Finally, practical applications of maximum hands-off control have been reported for electrically tunable lens [54], spacecraft maneuvering [55], and thermally activated building systems [56].…”