CC BY
4DIFFERENTIAL EQUATIONS AND
CONTROL PROCESSES N4, 2005 Electronic Journal, reg. N P23275 at 07.03.97
http://www.neva.ru/journMl e-mail: diff@osipenko.stu.neva.ru
Optimal control
Time-Optimal Control Problem For Infinite Order Parabolic Equations With Control Constraints
G. M. Bahaa
Mathematics Department, Faculty of Science, Beni-Suef University, Beni- Suef, Egypt e-mail: Bahaa_gmhotmail.com
35K20, 49J20, 49K20, 93C20
Abstract
A time optimal control problem for infinite order parabolic equations is considered. A time optimal control problem is replaced by an equivalent one with a performance index in the form of integral form. Constraints on controls are assumed. To obtain the optimality conditions for the Neumann problem, the generalization of the Dubovitskii-Milyutin Theorem given by Walczak in
Refs.[41,42] was applied.
1 Introduction
In recent years, significant emphasis has been given to the study of optimal control for systems governed by parabolic and hyperbolic partial differential equations with first boundary conditions or with Cauchy conditions. In these studies, the differential equations are either in general form or in divergence form. Questions concerning necessary conditions for optimality and existence
of optimal controls for these problems have been investigated for example in [8-19,22-43].
In (Refs. [15,18,19,29,32]), the optimal control problems for systems described by parabolic and hyperbolic operators with infinite order and consist of one equation have been discussed. Also we extended the discussion in [9-12] to n x n coupled systems of elliptic, parabolic and hyperbolic types involving different types of operators. To obtain optimality conditions the arguments of (Ref.[38]) have been applied.
Making use of the Dubovitskii-Milyutin theorem from [20]), following (Refs. [22-35]) Kotarski have obtained necessary and sufficient conditions of optimality for similar systems governed by second order operator with an infinite number of variables and with Dirichlet and Neumann boundary conditions. The interest in the study of this class of operators is stimulated by problems in quantum field theory.
In [1] a distributed Pareto optimal control problem for the parabolic operator with an infinite number of variables and with Neumann boundary conditions is considered. In [2] a time optimal control problem for parabolic equations with an infinite number of variables is considered. In [29] a distributed control problem for a hyperbolic system with mixed control state constraints involving operator of infinite order is considered. In [31] a distributed control problem for Neumann parabolic problem with time delay is considered. Also in [32], a distributed control problem for a hyperbolic system involving operator of infinite order with Dirichlet conditions is considered.
In this paper, the application of the generalized Dubovitskii- Milyutin Theorem from [41,42] will be demonstrated on an optimization Neumann problem for system described by parabolic operator with infinite order. A time optimal control problem for infinite order parabolic equations is considered. A time optimal control problem is replaced by an equivalent one with a performance index in the form of integral form. Constraints on controls are assumed.
This paper is organized as follows. In section 2, we introduce some functional spaces with an infinite order. In section 3, we formulate the optimal control problem and we introduce the main results of this paper. In section 4, we introduce a real example for this problem.
2 Some Functional Spaces (Refs.[3-5]).
The object of this section is to give the definition of some function spaces of infinite order, and the chains of the constructed spaces which will be used later. We define the Sobolev space W™ {aa, 2} (Rn) (which we shall denoted by W™ {aa, 2}) of infinite order of periodic functions 0(x) defined on all boundary r of Rn,n> 1, as follows,
W™{aa, 2} = I 0(x) e C~(Rn) : ^ aa\\Da0\\2 <
{ |a|=0
where aa > 0 is a numerical sequence and \\.\\2 is the canonical norm in the space L2(Rn)( all functions are assumed to be real valued), and
dH
Da =---,
(dxi)ai ....(3xn)an
where a = (a1, ...,an) is a multi-index for differentiation, \a\ = ^^ ai.
n
a multi-index tor differentiation, \a\ = ^ ai.
i=i
The space W-<x{aa, 2} is defined as the formal conjugate space to the space
Wœ{aa, 2}, namely:
W-™{aa, 2} = {^(x) : ^(x) = ^ aaDa^a(x)},
\a\=0
to
where ^a G L2(Rn) and y^ aa||^a||2 < to..
|a|=0
The duality pairing of the spaces Wto {aa, 2} and W-to {aa, 2} is postulated by the formula
to „
=y2 aa ^a(x)Da0dx,
|a|=0
where
0 G WtoK, 2} , ^ G W-TO{aa, 2} .
From above, Wto {aa, 2} is everywhere dense in L2(Rn) with topological inclusions and W-to {aa, 2} denotes the topological dual space with respect to L2(Rn), so we have the following chain:
Wto {aa, 2} ç L2(Rn) ç W-to {aa, 2} .
We now introduce L2(0, T; L2(Rn)) which we shall denoted by L2(Q), where Q = Rnx]0,T[, denotes the space of measurable functions t ^ $(t) such that
fT i (q) = ( \Wt)\\2dt)2 < (,
j 0
fT
endowed with the scalar product (f,g) = / (f(t),g(t))L2(Rn)dt, L2(Q) is a
02
Hilbert space. In the same manner we define the spaces L2(0,T; W( {aa, 2}), and L2(0, T; W{aa, 2}), as its formal conjugate resp.
Finally we have the following chains:
L2(0, T; W({aa, 2}) C L2(Q) C L2(0,T; W-(K, 2}),
Finally, let us introduce the space
W(0, T) := jy; y G L2(0, T; W( {aa, 2}), ^ G L2(0, T; W{aa, 2}) j ,
in which a solution of a parabolic equation with an infinite order will be con-tained.The spaces considered in this paper are assumed to be real.
3 Time-Optimal Control Problem For Parabolic Equations
Let us take into account the following optimization problem:
dt + A(t)y = u, x G Rn, t G (0,T), (1) d t
y(x, 0) = yp(x), x G Rn, (2)
dyd^=0, x G r, t G (0,T), (3)
y(x, T) G tf, x G Rn, (4)
u G Uad, (5)
T ^ min, (6)
where yp is a given element in L2(Rn), A(t) is a bounded infinite order self-adjoint elliptic partial differential operator mapping Wonto W which takes the form [18-19]
to
(A$)(x) = ^(-1)H aaD2a$(x,t),
\a\=0
and dV^ is the co-normal derivatives with respect to A, for j^j =0,1, 2,.., j^j < a — 1.
For each t e]0,T[, we define the following bilinear form on WTO(Rn): n(t; 0,-0) = (A(t)^,^)L2(Rn), ^ e
Then
n(t; ^^ ) = (A(t)0^) L2(Rn)
= (A{t)fi{x),^{x)) L2(Rn)
(^(-1)|akD2a #r,t),^(x)) L2(Rn) (7)
| a |=0
r, TO
/ V (-l)1 a 1 Dafi(x)Daiß(x)dx
J Rn
'Rn I I n |a|=0
The bilinear form (3.7) is coercive on W( {aa, 2} that is, there exists n G R, such that:
n(t; 0,0) = n\№\\w~(m.n), n> (8)
It is well known that the ellipticity of A(t) is sufficient for the coercitivness of n(t; ) on W((Rn) [38]. Then
(TO N
^(-1)|a|aaD2a 0(x,t),0(x,t)
|a|=0
(TO
|a|=0
= V^^^w^tRn) ■
e WTO(Rn) the functiont ^ n(t; 4,0) is continuously differentiable in ]0,T[; and
n(t; 0,-0)= n(t; 0,0) (9)
Note. The operator + A(t) is an infinite order parabolic operator which maps L2(0, T; W™ {aa, 2}) onto L2(0,T; W{aa, 2}).
Remark 3.1 The equations (3.1)-(3.3) have the unique generalized solution such as
y e W(0,T) := jy\ y e L2(0,T; W™(Rn)), e L2(0,T; W-TO(Rn))|
continuously dependent on the right-hand side of (3.1) and the initial condition (3.2). Moreover, y e W(0,T) is a continuous function [0,T] ^ L2(Rn) (compare with Theorems 1.1 and 1.2 Chapt. 3 [38])
Let us denote by U := L2(0,T; L2(Rn)) = L2(Q), the space of controls and by Y := L2(0,T; W™(Rn)) the space of states.
We assume that Uad is a closed, convex subset of U and K is a closed, convex subset of L2(Rn) with non-empty interior.
3.1 An equivalent optimization problem
The optimization problem (3.1)-(3.6) can be replaced by another equivalent one with a fixed time T and a performance index in a form of integral (compare with [40]). To show that we need two auxiliary theorems.
Theorem 3.2 Let T0 > 0 be the optimal time for the problem (3.1)-(3.6). If int K = then
y(x, T0) e dK (boundary of K) (10)
for any y satisfying (3.1)-(3.4).
Proof Any solution of (3.1)-(3.3) is continuous with respect to t. If (3.10) is not true, then there exists an admissible state y such as y(x,T0) e intK. Thus a T < T0 exists so that y(x,T) e K. This contradicts the optimality of T0 and hence (3.10) must be fulfilled.
Theorem 3.3 Let T0 > 0 be the optimal time for the problem (3.1)-(3.6), let u0 and y0 be an optimal control and the corresponding state, respectively. Then with the assumptions given above there exists a non-trivial element g e L2(Rn) so that the pair (y0,u0) is optimal for the following control problem with the fixed time T
subject to the constraints (3.1)-(3.3).
Proof The linearity of the equations (3.1)-(3.3) implies that the endpoints y(x,T°) of all admissible states y form a convex set YTo. From Theorem (3.1) we have YTo R int K = 0 and y°(0,T°) G dK. Since intK = 0 thus there exists a closed hyperplane separating YTo and K containing y°(x,T°), i.e. there is a
This completes the proof.
Remark 3.4 If the set K has a special form i.e.
K = {y} (x,T); \\y — z\L(Rn) <
where e > 0 and z e L2(Rn) are given, then g is known explicitly and is expressed by g(x) = 2(y°(x,T) — z(x)).
Remark 3.5 The method fails if intK = e.g. in the case when K consists of a single point.
3.2 Optimality conditions
Now based on Theorem (3.2) we can be ready to formulate the necessary condition of optimality for problem (3.1)-(3.6).
Theorem 3.6 Assuming that T0 > 0 is the optimal time for the problem (3.1)-(3.6), u0 and y0 are the optimal control and the corresponding state, respectively. Then with the assumptions given above at the beginning of section 3, there exist
(11)
nonzero g G [L2(Rn)\* = L2(Rn)
an element g G L2(Rn) and the adjoint state p G W(0,T) so that the following system of partial differential equations and inequalities must be satisfied:
State equations:
dy° ~dt
+ A(t)y° = u0,
dwy (x, t)
= 0,
y0(x,0) = yp(x),
y°(x,T0) G K,
Adjoint equations:
- ft + A(t)P = 0,
d " p(x,t)
= 0,
p(x,T 0)= g(x), Maximum conditions:
x G Rn, t G (0,T0)
x G r, t G (0,T0),
x G Rn, x G Rn ■
x G Rn, t G (0,T0),
x G r, t G (0,T0), xe Rn.
(12)
(13)
(14)
(15)
(16)
(17)
(18)
'Q
' Rn
p(u — u°)dxdt > 0 yu G Uad,
g(x)(y — y°)dx > 0 yy G K.
(19)
(20)
Proof According to Theorem (3.2) our problem is equivalent to the one with the fixed time T0 and the performance index in the integral form (3.11). For such a new problem we formulate the necessary conditions of optimality by applying the generalized Dubovitskii-Milyutin Theorem (Theorem 1.8.1 in [26]). Let us denote by Q\,Q2, Q3 the sets in the space E := Y x U as follows
Qi := <
(y,u) G E;
I + A(t)y
y(x, 0) d " y(x,t) dvA
u
yp(x)
= 0,
Q2 := j(y,u) e E; y e Y,u e Uadj,
Q3 := j(y,u) e E; y(x,T0) e K,u e Ua)j.
Thus the optimization problem may be formulated in such a form
I(y,u) ^ min subject to (y,u) e Q1 R Q2 R Q3.
We approximate the sets Qi, and Q2 by the regular tangent cones (RTC), Q3 by the regular admissible cone (RAC) and the performance functional by the regular cone of decrease (RFC).
The tangent cone to the set Q1 at (y0,u0) has the form
RTC (Qi, (y0,u0)) = { (y,U) G E ; P'(y", u0)(y, u) = 0
= <
dt + A(t)y = U
(y, U) G E; y(x, 0)=0
d"y(x, t)
3v%
= 0,
where Pf(y ,u )(y,u) is the Frechet differential of the operator
dy
P(y, u) := (dt + A(t)y — u, y(x, 0) — yp(x)) mapping from the space
W := L2(0, T; Wœ(Rn)) x L2(Q)
into the space
Z := L2(0,T; W-œ(Rn)) x L2(Rn).
Applying theorem on the existence of the solution to the equation (3.1)-(3.3) Remark (3.1) it is easy to prove that P/(y°,u°) is the mapping from the space W onto Z as required in the Lusternik Theorem (Theorem 9.1 in [20]).
The tangent cone RTC(Q2, (y°,u°)) to the set Q2 at (y°,u°) has the form Y x RTC(Uad,u°), where RTC(Uad,u°) is the tangent cone to the set Uad at the point u°. It is known that the tangent cones are closed [36].
Then we can show that:-
RTC(Qi n Q2, (y , u )) = RTC(Qi, (y°,u°)) n RTCQ (y°,u°)).
Further taking into account Theorem 3.3 in [42] it is easy to see that the adjoint cones [RTC(Qi, (y°,u°))]* and [RTC(Q2, (y°,u°))]* are of the same sense.
The admissible cone RAC(Q3, (y°, u°)) to the set Q3 at (y°, u°) has the form RAC(K, y°(x, T°)) x U, where RAC(K, y°(x, T°s)) is the admissible cone to the set K at the point y°(x,T°).
According to Theorem 7.5 [20] the regular cone of decrease for the performance functional is given by
RFC (I, (y0,u0)) = <J (y,U) G E; / g(x)y(x,T°)dx < 0 }.
' Rn
If RFC (I, (y0,u0)) = 0, then the adjoint cone consists of the elements of the form (Theorem 10.2 [20])
f4(y,U) = —X0 g(x)y(x,TQ)dx, where A0 > 0.
n
J R
The functionals belonging to [RTC(Qi, (y°,u0))]* have the form (Theorem 10.1 [20])
fi(y,u)=0 V(y,ö) e RTC(Qi, (y0,u0)).
The functionals
f2(y, u) e [RTC(Q2, (y0, u0))]* and fa(y, u) e [RAC(Q3, (y0, u0))]* can be expressed as follows
f2(y,u) = f2i(y) + f2(u),
f3(y,u) = fi(y) + ft(u)
where f1 (y) = 0 Vy and f32(y) = 0 Vu e U (Theorem 10.1 in [20]), f22(u) is the support functional to the set Uad at the point u0 and f^(y) is the support functional to the set K at the point y°(x,T°) (Theorem 10.5 in [20]).
Since all assumptions of the Dubovitskii-Milyutin Theorem are satisfied and we know suitable adjoint cones then we are ready to write down the Euler-Lagrange Equation in the following form
f!(u) + f1(y) = A0 / g(x)y(x,T°)dx, V(y,u) e RTC (Qi, (y0,u0)). (21)
J Rn
Introducting the adjoint variable p by the equation (3.16)-(3.18) and taking into account that y is the solution of P'(y°,u°)(y,u) = 0 for any fixed u, we obtain
0 = J ^dt + A(t)pjydxdt = J ^-p(x,T°)y(x,T0)^j dx
Hence
+ p(x, 0)y(x, 0)dx + p( — + A(t)y \dxdt JRn JQ \dt J
= — p(x,T 0)y(x,T0 )dx + pUdxdt.
J Rn JQ
/ g(x)y(x, T°)dx = / pUdxdt. (22)
Rn Q
From (3.21) and (3.22) we get
/2(U) + /31(y) = 1 Ao / pUdxdt + 1 AW g(x)y(x,T0)dx (23)
2 J Q 2 J Rn
A number A0 cannot be equal to 0 because in such a case all functionals in the Euler-Lagrange Equation would be zero which is impossible according to the Dubovitskii-Milyutin Theorem. Using the definition of the support functional and dividing both members of the obtained inequalities by A0 from (3.23) we obtain the maximum conditions (3.19)-(3.20).
If RFC(I, (y0,u0)) = 0 then the optimality conditions (3.12)-(3.20) are fulfilled with equality in the maximum conditions (3.19)-(3.20). This last remark completes the proof.
4 Real example
We shall us the following notation:
Q =QT = ^x]0,T[, Q an open subset of Rn; E =£T = rx]0, T [,
r =boundary of Q, E = lateral boundary of Q, V C H c V
Let us consider the system whose state is given by
d
d^y(t; v)+ A(t)y(t; v) = / + Bv (24)
y(0,v)= y0 (25)
) di
^^ ^ d ddy
where A(t)y = — ^^ ——(a^(x,t)——), aj be given functions in ^x]0,T[= Q
with
f aj e L~(Q),
n
2
Y, aij (x, tMj > +.....+ ^) a > 0, b G R,
i,j=1
almost everywhere in Q,
B G L(U,L2(0,T,V')) (26)
Let Uad be a given closed, convex subset of U and let yi be a given element in H.
We assume that
i there exists a v e Uad such that
(27)
| y(r; v) = y1 for an appropriate t e [0, T] and t < T The optimal time is defined by
t0 = inf t, t such that (4.4) holds (28)
The Problem which we shall study are :
existence of an optimal control, that is, existence of u e Uad such that
y(7o; u) = yi; (29)
Lemma 4.1 We assume that
V0,0 e Wi(Rn) the function t ^ n(t; 0,0) is continuously differentiable in ]0,T[; and
n(t; 0,-0)= n(t; 0,0) (30)
and there exists a X such that
n(t; 0,0) + X|0|2 > a\\0\\2, a > 0, V0 e V,t e]0,T[, (31)
(4-4) and (4.3) hold and that Uad is bounded. Then there exists an optimal control, that is u e Uad such that (4-6) is satisfied.
Proof See Theorem 17.1 in [38] Chapter III, section 17.2. Example Take U = L2(E). Let the state y(v) be given by
d
-y(v) + A(t)y(v) = /,
dy ( s -(v) = v,
dvA
y(x,0,v) = y0(x).
Theorem 17.1 [38] may be applied to this example. Hence the theorem covers the case of boundary control.
Remark 4.2 Theorem 17.1 [38] may be modified without any difficulty to cover the case of systems with Dirichlet boundary conditions (cf [38] section 9) and where the control is exercised through the boundary.
Comments
The main result of the paper contains necessary and sufficient conditions of optimality (of Pontryagin's type) for infinite order parabolic system that give characterization of optimal control. But it is easily seen that obtaining analytical formulas for optimal control is very difficult. This results from the fact that state equations (3.12)-(3.15), adjoint equations (3.16)-(3.18) and maximum conditions (3.19)-(3.20) are mutually connected that cause that the usage of derived conditions is difficult. Therefore we must resign from the exact determining of the optimal control and therefore we are forced to use approximations methods. Those problems need further investigations and form tasks for future research.
Also it is evident that by modifying:
- the boundary conditions,
- the nature of the control (distributed, boundary),
- the nature of the observation,
- the initial differential system,
an infinity of variations on the above problem are possible to study with the help of Dubovitskii-Milyutin formalism.
Acknowledgment The author is grateful for the referees of Differential Equation And Control Process Electronic Journal for their fruitful comments and invaluable suggestions.
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