Numerical computation of singular control problems with application to optimal heating and cooling by solar energy

Abstract

Abstract. The method presented here is an extension of the multiple shooting algorithm in order to handle multipoint boundary-value problems and problems of optimal control in the special situation of singular controls or constraints on the state variables. This generalization allows a direct treatment of (nonlinear) conditions at switching points. As an example a model of optimal heating and cooling by solar energy is considered. The model is given in the form of an optimal control problem with three control functions appearing linearly and a first order constraint on the state variables. Numerical solutions of this problem by multiple shooting techniques are presented.

The non‐linear beam via optimal control with bounded state variables

“…On a boundary arc with x(t) = 2 0.05 it follows from u = -As, (12) and (15) that Xe = 0 and X, = 0. Then (14) and the sign condition (47) imply the desired…”

Singular solutions in optimal control: second order conditions and a shooting algorithm

“…Let (x, ū, v) ∈ C be a critical direction. Define ( ξ, ȳ) by the transformation (47) and set h := ȳT . Note that ( 14)-( 15) yield Dη j (x 0 , xT )( ξ0 , ξT + F v,T h) = 0, for j = 1, .…”

“…Remark 4.2. Note that P consists of the directions obtained by transformating the elements of C via (47).…”

“…Throughout this proof, whenever we put ρ i we refer to a positive constant depending on A ∞ , F v ∞ , E ∞ , and/or B ∞ . Let (x, ū, v) ∈ H 2 and ( ξ, ȳ) be defined by Goh's Transformation (47). Thus ( ξ, ū, ȳ) is solution of (48).…”

An Improved Computational Algorithm for Singular Control Problems in Chemical Reaction Engineering

“…These conditions can be utilized to obtain an analytic solution for simple problems described by a few dynamic equations (1-3). However, for problems described by a large number of dynamic equations [3,4] Stutts (23) Thomas (18] Jacobson [24] Edgar and Lapidus [25,26] Maurer [27] Oberele (28) Aly (29) Aly and Chan [30] Aly and Megeed (31) Soliman and Ray (32) Kumar [33) Jacobson [34] Cuthrell and Biegler [29] Downloaded by [University of California Santa Barbara] at 04: 20 17 June 2016 SINGULAR CONTROL PROBLEMS 167 (say 4 or more) numerical solutions are inevitable. There have been a number of numerical techniques proposed for solving singular control problems, which are summarized in Table II.…”