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26
где Л - полный эллиптический интеграл первого рода для модуля Л, определяемого условием Л/Л' = 2п.К'/К, а бп(г, к), сп(г,к), ¿п(г,к) - эллиптические функции Якоби.
Теорема 2. При всех £ € Т метод
ДО » ¿оdj(aj(f)cosj(> + Ьj(f)smjZ),
¿=1
где
2 2т — 1 .2т — 1
dj =---— V -1Г+1 <*п —-------К сое .7—---тг, ^ =0,...,п- 1,
7 паДсг) " 2п 2п
•» ' ' т=1
является оптимальным на классе Н0
БЛАГОДАРНОСТИ: Работа выполнена при финансовой поддержке Российского Фонда Фундаментальных Исследований (гранты Л» 99-01-01181 и № 00-15-96109)
ON THE OPTIMAL CONTROL OF IMPULSIVE DIFFERENTIAL INCLUSIONS © F.L. Pereira (Portugal), G.N. Silva (Brasil)
Dynamic optimization problems arising in a variety of application areas such as finance, mechanics, resources management, and space navigation, whose solutions might involve discontinuous trajectories have been, over the years, motivating a significant research effort in this class of problems.
In this article, we address the class of impulsive control problems for which the dynamics are defined by a differential inclusion with a vector valued control measure under the weakest assumptions known to date. In particular, we do not require the vector fields associated with the singular term of the dynamics to satisfy the so-called Frobenius condition. We will state the optimal control problem as follows:
(P) Minimize /i(ar(0),a;(l)) (1)
subject to dx(t) e F(t,x(t))dt + G(t:x(t))^i(dt) t e [0,1] (2)
(x(0),x(l)) G C
» e K. (3)
Here, h : lRn x lRn -> 1R is the cost functional, F : [0,1] x lRn <—» V(]Rn), and G : [0,1] x lRn <—> V(lRnxq) are given set-valued functions, C is a closed set in lRn x ]Rn, K is a positive convex pointed cone in lRq.
By e K it is meant that p € C*([0,1]; X), i.e., n{A) € K for any Borel set A C [0,1]. C*([0,1]; K), denotes the set in the dual space of continuous functions from [0,1] to lRq with values in K.
In this new context, we introduce a concept of proper solution which, besides providing a meaning to the dynamic optimization problem, is also endowed with a robustness property allowing to derive necessary conditions of optimality. More specifically, a trajectory x G BV+([0,1]; lRn) is a proper solution to (2) relative to the objective functional (1) if, for all t e (0,1],
x{t) = x(0) + Jo f{T)dr + /(M g(T)fl{dT)
for some £-integrable function /, f(t) e F(t,x(t)), C-a.e., and a /¿-integrable function g, g(t) € G(f, x(t~)\ l¿({t})) ¡1-a.e., where /2(d£) := ¿¿=1 /¿»(^0 and the multifunction G : [0,1] x lRn x K <-» V{JRn) is given by
G(£, q) := <
r {G(i, 2)u>(0} if |a| = 0
. ¿(s) 6 G{t,£{s))7(a) r?(0-a.e., £(rK*-)) = 2
and 7(r/(i)) - 'yi.vi.t )) = «} if M > 0,
with |q| = Yli=iaii u'(‘) *s t^ie Radon-Nicodym derivative of // w.r.t. /2, (£,7) are in AC([0,1];JR" x Mf), !,(•) given by ,(t) := being i?(i) := | \$\v{t)] £${$ > ° }.
Mi(t) := Jj0 fii(ds), Vi > 0 with Mi(0) := 0, and the pair (0,7) is a h-graph completion with 0(a) = 0 on fj(t).
Given a feasible control measure /z and any pair of measurable selections (/, G) of (F, G), a h-graph completion related to (¡i,f,G) is a graph completion (0,7) that minimizes h(y(0),y(l)) over all y(-) over the family of parametrized trajectories
:= {*/(•) : y(s) = /(0(a), y(a))0(a) + G(0(a),j/(a))7(a), 7(a) € tf, (0(a),7(a)) € fi, [0, l]-a.e.}
where Q := {u’ € Rq+l : Y?i=owi = 1 + £([°> 1])}> 7(0) = 0 and 7(*7(0) = /*([°>*]) > Vi G [0.1].
A graph completion associated with a given set-valued measure /i G C*(0,1; A') is any pair of functions (0,7) : [0,1] —> 2R+ x if satisfies 0 is the “inverse” of fj (i.e., 0(a) = £, Va € 77(0) and 7(a), is given by
t \ if #({*}) = 0 • \ M(t~) + r(or)d(T if /!({£}) > 0,
where v{-) e V1 := {t; : r](t) -> #?9|v(a) € tf, £j=1 t’i(a) = 1 + /z([0,1]) Va G rj{t), /-(t) v(a)da = //({*})}•
Now, we state our main result.
Theorem Let (x*,p,*) be a solution to problem (P) whose data satisfies the assumptions stated above. Then there are nonegative number A and a function p € £?K+([0, l];iRn) satisfying A -I- ||p|| ^ 0 and
(-dp(t),dx*(t)) e dHfr(t, x*(t),p(t))dt + dHG(t, x*(t),p(t))p.*(dt) n* and L-a.e.(4)
(p(0),-p(l)) € \Ldh(x*(0),x*(l)) -1- LdNc{x*(0),x*(l)) (5)
0 ^ crK(HG{t,x*(t),p(t))) Vi €[0,1] (6)
0 < aK{HG(t,x*{t),p(t))). p*-a.e. (7)
Furthermore, (-at(s),€t(s)) € dHG(t,^t{s),at(s)) • v*(s) fj(t)-a.e. (8)
0 ^ aK(HG(t,Zt{s),at(s))) f}{t)-a.e. (9)
(x*{t~),p{t~)) = {Zt{v(t~)),(*t(v(t~))) (10)
(®*(0>p(0) = (&(f?(0W(*?(0)) (u)
where the additional “control” extension v* satisfies aK{HG(t,^t{s),at{s))) = HG(t, £t(a), Qt(s)) • v*(s) and f-^v*(s)ds = fi*({t}). Here, (4) has to be interpreted in the sense explained above, <tk-(/c) is the support of the set K at point k, and Hir(t.x*(t),p(t)) := ap(t,x*(t))(p(0) and HG is a “path-valued” multifunction defined by:
{M0} with hG( 0 • UJ*(0 := SUP weK. G(t,x*(t))€G(t,x*(i)){pr( t)G{t,X*{t)) • w} if fi({t}) = 0
{hsG(t) : a G 77} with hsG{t)-v*(s) := supveVttG{tMa))eG(t^t{s)){a'[{s)G{t,^t{s)) - v{s)} if /2({i}) > 0,
where w* is the Radon-Nycodim derivative of /z* w.r.t. //*.
The adopted approach, described in detail in [1], allows us to deal with vector valued control measures without requiring the commutative assumption on the singular vector fields dynamics which can be regarded as a significant extension of the one in [2]. Besides an inherently natural interpretation of the state trajectory which takes into account the interaction between the evolving state variable and the impulsive integrating control measure at times when the trajectory is discontinuous, this approach is amenable to the derivation of necessary conditions of optimality under hypotheses on the data which allow measurable time dependence and a time varying control constraint set.
From the engineering point of view, our approach is suitable to address important classes of problems involving the coordinated control of multiple dynamic systems. A simple example consists in the problem where the goal is to control a set of dynamical systems with several viable configurations in order to maximize a given performance criterion in the execution of the given task. By configuration, it is meant a set of specific constraints or evolution laws which might be associated with a certain region of the state space. Let us assume that the transition between any two configurations is not instantaneous but that takes place through a evolution which depends on some specific laws. Let us assume also that the mentioned task can only be executed when the system is “operational”, i.e., anyone of the configurations is well established. Observe that the solution of the problem involves control optimization not only in the operational mode but also in the management of the transitions between configurations. On the other hand, it is of interest that time flows only when the task of interest is being executed. This means that, for this time frame, the transition between configurations (i.e., singular evolution) would be instantaneous, although, for the purpose of the system performance evaluation, the evolution of the singular component would have to be taken into account.
REFERENCES
1. Pereira F. and Silva GNecessary conditions of optimality for vector-valued impulsive control problems 11 Systems and Control Letters. 2000.
2. Pereira F., Silva G. and Vinter R, Necessary conditions of optimality for impulsive differential inclusions 11 Proc. Conf. Decision and Control’98, Tampa, FL, December, 16-18, 1998.
ЛИНЕЙНАЯ ОЦЕНКА ПРОГНОЗИРОВАНИЯ НА ОСНОВЕ МЕТОДА
ОБЕЛЯЮЩЕГО ФИЛЬТРА
© В.В. Савченко, A.B. Баринов, A.A. Шкулев (Нижний Новгород)
Задача прогнозирования случайного временного ряда Х(п) относится к числу центральных задач статистического анализа информации. При этом наибольший интерес вызывают оценки прогнозирования линейного вида
М
х(п) = '52aix(n-i), (1)
t=i
