Numerical Solution for a Controlled Nonconvex Sweeping Process

Abstract

The numerical method developed in [30] for optimal control problems involving sweeping processes with smooth sweeping set C is generalized to the case where C is nonsmooth, namely, C is the intersection of a finite number of sublevel sets of smooth functions. The novelty of this extension resides in producing for the general setting a different approach, since the one used for the smooth sweeping sets is not applicable here.

Optimal Control of Several Motion Models

“…[1,3,6,7,8,9,10,11,12,13,16,17,18,19,30,42] and their extensive bibliographies therein. Recently, the authors in [26,27,34,35,42] have introduced and developed an innovative exponential penalization technique (also known as a continuous approximation approach as opposed to the method of discrete approximations) to obtain the existence of solution and derive a set of nonsmooth necessary optimality conditions in the form of Pontryagin maximum principle involving a controlled nonconvex sweeping process governed by a sublevel-sweeping set. This exponential penalization technique allows them to approximate the controlled sweeping differential inclusions by the sequence of standard smooth control systems and hence has successfully demonstrated to be an appropriate technique for developing a numerical algorithm to efficiently compute an approximate solution for certain forms of controlled sweeping processes with smooth data; see [26,27,34,35,42] for details.…”

A Micro-Perspective on Governance in Informal Tented Settlements. (c2022)

“…In [1], Nour and Zeidan propose a numerical algorithm to solve the problem (P ), which improves in several directions the numerical method established in [2]. Indeed, in [2], the problem (P ) is considered with initial condition in the interior of the convex sweeping set C, which is not the case in [1]. More precisely, in [1], no convexity is assumed on the sweeping set C, and the initial state x 0 can be any point in C including the boundary.…”

“…Indeed, in [2], the problem (P ) is considered with initial condition in the interior of the convex sweeping set C, which is not the case in [1]. More precisely, in [1], no convexity is assumed on the sweeping set C, and the initial state x 0 can be any point in C including the boundary. Note that the numerical algorithm proposed in [1] and [2] is second of its kind for (P ) after the work done by Adam and Outrata in [8] where a different techniques are used.…”

“…The method used in [1] and [2] consists of approximating (D) by a standard control system (D γ k ), obtained by replacing in (D) the set N C (x(t)) by the penalty term:…”

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In [1], Nour and Zeidan proposed a numerical algorithm to solve optimal control problems involving sweeping processes. In order to apply this algorithm to real life problems, two numerical methods need to be developed: A numerical method to solve nonlinear differential equations, and a numerical method to find the minimum of an objective function (given only numerically) in finite dimensional spaces.

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