Optimization of Initial Functions for Linear Retarded Systems in ...

Funkcialaj Ekvacioj, 35 (1992) 179-198

Optimization of Initial Functions for Linear Retarded Systems in Banach Spaces By

Shin-ichi NAKAGIRI and Shigeru HARUKI (Kobe University and Okayama University of Science, Japan) Dedicated to Professor Hiroki Tanabe on his 60th birthday

1. Introduction Optimal control theory of retarded systems is further studied in many literatures for both finite and infinite dimensional spaces (see Banks and Manitius [2, 3], Colonius [5, 6], Delfour [7], Delfour and Mitter [11], Gibson [12], Oguztoreli [20] for finite dimensional space and Nababan and Teo [15, 16], Nakagiri [17], Teo [22] for infinite dimensional space). However, in the above literatures the optimization of initial functions appearing as the system control variables has not been considered in spite of its technological importance. One of the reasons may depend on the difficulty for constructing the transposed system which describes the system optimality for initial functions. In this paper we solve the optimization problem by introducing the socalled structural operator (cf. Delfour and Manitius [10], Manitius [14], Nakagiri [18] , the transposed system in Nakagiri [17] and the time reversing plays central role as in the standard operation. The structural operator optimal control problems investigated by Delfour [8], Delfour, Lee and Manitius [9] and Vinter and Kwong [23]. We emphasize here that the structural operator behaves like a controller operator as in [8, 9, 23]. We shall explain the results obtained in this paper. Let $=[0, T]$ , $T>0$ . We consider the following retarded system on a Banach space $X$ involving control variables of initial data: $F$

$)$

$F$

$F$

$I$

$g$

(1.1)

$¥frac{d¥mathrm{x}(t)}{dt}=A_{0}x(t)+¥int_{-h}^{0}d¥eta(s)x(t+s)+u(t)$

(1.2)

$¥mathrm{x}(0)=g^{0}$

(1.3)

$g$

where Let

,

$x(s)=g^{1}(s)$

$=(g^{0}, g^{1})¥in V_{ad}$

$¥mathrm{a}.¥mathrm{e}$

.

$s¥in[-h,$


$0)$

.

$t$

$¥in I$

,

,

is an admissible set. $J=J(g, x)$ be the integral convex cost with respect to the state

$V_{ad}$

$x(t)$

180

Shin-ichi NAKAGIRI and Shigeru HARUKI

and the initial data tion problem :

$g$

$¥in V_{ad}$

.

For the cost

$J$

we consider the following optimiza-

$¥mathrm{P}$

Find an initial data and optimality conditions for which the is minimized subject to the constraint (1.1), (1.2), (1.3).

P. cost

$g$

$J$

$¥in V_{ad}$

We present results on the existence of optimal initial data, necessary optimality conditions, maximal principle and bang-bang principle for the problem P. After the necessary system description is given in Section 2, two existence theorems of optimal initial data are proved in Section 3; one is for bounded admissible set and the other is for unbounded admissible set. In Section 4, we establish two types of necessary optimality conditions which are described by the integral inequalities in terms of the transposed state and the structural operator $F$ . As an application of the main results we give a feedback control law for the regulator problem containing initial data control variables . Section 5 is devoted to studying the ’pointwise’ maximum principle and ’conditional’ bang-bang principle. The maximum principle is derived from the optimality conditions in Section 4 by applying variational technique. Some examples of maximum principle as well as bang-bang principle for technologically important costs are also given in Section 5. $g$

2. Linear retarded system and structural operator Let $X$ be a reflexive Banach space over $C$ or $R$, with norm . Let $h>0$ be fixed and $I_{h}=[-h, 0]$ . We consider the following system (S) described by a linear retarded functional differential equation in $X$ : $|¥cdot|$

(2.1)

$¥frac{dx(t)}{dt}=A_{0}x(t)+¥int_{-h}^{0}d¥eta(s)x(t+s)+u(t)$


. $t>0$

with an initial data

(2.2)

$x(0)=g^{0}$

$x(s)=g^{1}(s)$


.

where $g=(g^{0}, g^{1})¥in M_{p}¥equiv X¥times L_{p}([-h, 0];X)$, -semigroup $¥{T(t);t¥geq 0¥}$ on generates a and is a Stieltjes measure given by

$s¥in[-h,$

$0)$

, ,

$u¥in L_{p}^{1¥mathrm{o}¥mathrm{c}}([0, ¥infty);X)$

$A_{0}$

(2.3)

$¥mathrm{C}_{0}$

$X$

$¥eta(s)=-¥sum_{r=1}^{m}¥chi_{(^{¥_}¥infty,-h_{r}¥mathrm{j}}(s)A_{r}-¥int_{s}^{0}A_{I}(¥xi)d¥xi$

,

$p¥in(1, ¥infty)$

(cf. Tanabe [21]) and $s¥in I_{h}$

,

$¥eta$

,

where denotes the characteristic function of the interval . In (2.3) it is assumed that $0