M. H. F A R A G (Minia)
THE GRADIENT PROJECTION METHOD FOR SOLVING AN OPTIMAL CONTROL PROBLEM
Abstract. A gradient method for solving an optimal control problem described by a parabolic equation is considered. The gradient projection method is applied to solve the problem. The convergence of the projection algorithm is investigated.
1. Introduction. The theory of optimal control systems with distrib- uted parameters is one of the leading sections of optimization theory. It has wide applications in various practical fields. The theory of optimal control problems has been studied by many workers [1, 2, 6, 7]. They have shown [4, 9, 10] that these problems arise in many physical applications such as heat conductivity, filtration and diffusion.
2. Statement of the problem and definitions. Let it be required to minimize the function
(1) f (v) =
l
\
0
|u(x, T ; v) − g(x)|
2dx + β
T
\
0
|v
1(t)|
2dt provided that u(x, t; v) is a solution of the boundary value problem
u
t= a
2u
xx+ B(x, t)u + v
2(x, t), (x, t) ∈ Ω = [0 < x < l, 0 < t ≤ T ], (2)
u(x, 0) = φ(x), 0 ≤ x ≤ l, (3)
u
x(0, t) = 0, u
x(l, t) = ν[v
1(t) − u(l, t)], 0 < t ≤ T, (4)
where a
2, l, ν, T , β are positive numbers, v
1(t) the temperature of the external medium, v
2(x, t) the density of heat sources, and the control v is in
1991 Mathematics Subject Classification: 49J20, 49M07, 65L10, 65K10.
Key words and phrases: optimal control, gradient methods, boundary value problems, distributed parameter systems.
[141]
V = n v
v = (v
1(t), v
2(x, t)); v
1(t) ∈ L
2[0, T ], v
1min≤ v
1(t) ≤ v
1max; v
2(x, t) ∈ L
2(Ω),
l
\
0 T
\
0
|v
2(x, t)|
2dx dt ≤ R
2o , where v
1min< v
1max; R > 0 is a given number; g(x), φ(x) ∈ L
2[0, l], B(x, t)
∈ L
2(Ω) are given functions and H = L
2[0, T ] × L
2(Ω).
Definition 1. The problem of finding a function u = u(x, t; v) satisfy- ing conditions (2)–(4) for a given v ∈ V is called the reduced problem.
Definition 2. The solution of the reduced problem (2)–(4) corre- sponding to v ∈ V is a function u(x, t) ∈ H
1,0(Ω(4)) satisfying the integral identity
(5)
l
\
0 T
\
0
[−uη
t+ a
2u
xη
x+ B(x, t)uη − v
2(x, t)η] dx dt
=
l
\
0
φ(x)η(x, 0) dx + a
2ν
T
\
0
[v
1(t) − u(l, t)]η(l, t) dt for all η = η(x, t) ∈ H
1(Ω) with η(x, T ) = 0.
Equations (1)–(4) are the mathematical formulation of the optimal con- trol problem for a linear parabolic equation with controls in boundary con- ditions and the right side of equation (2). Optimal control problems for linear and nonlinear parbolic equations have been widely considered in the literature (see for instance [4, 8, 18]), and were studied by Madatov [11] and Mokrane [12], where the existence, uniqueness and regularity of the solution were proved. In addition, Farag [3] and Phillipson and Mitter [13] have derived numerical results for the heat equation with strong nonlinearity.
3. The gradient of the function. The principal result in this section is Theorem 3.1. Its proof will be prepared by two lemmas:
Lemma 3.1. Let δu(x, t) be the generalized solution of the boundary value problem
δu
t− a
2δu
xx− B(x, t)δu − δv
2(x, t) = 0, (x, t) ∈ Ω, (6)
δu(x, 0) = 0, 0 ≤ x ≤ l, (7)
δu
x(0, t) = 0, δu
x(l, t) = ν[δv
1(t) − δu(l, t)], 0 < t ≤ T.
(8) Then
l
\
0
|δu(x, T )|
2dx ≤ C h
T\0
|δv
1(t)|
2dt +
l
\
0 T
\
0
|δv
2(x, t)|
2dx dt i (9)
= Ckδvk
2H,
where C > 0 is a constant which is independent of the choice of δv ∈ V.
P r o o f. We multiply (6) by δu and integrate it on the rectangle Ω. By using the conditions (7) and (8), we obtain the reduced equation:
(10) 1 2
l
\
0
|δu(x, T )|
2dx + a
2ν
T
\
0
|δu(l, t)|
2dt + a
2l
\
0 T
\
0
|δu
x|
2dx dt
= a
2ν
T
\
0
δu(l, t)δv
1(t) dt +
l
\
0 T
\
0
δuδv
2dx dt.
Applying the inequality ab ≤
2εa
2+
2ε1b
2, ε > 0, we obtain (11) 1
2
l
\
0
|δu(x, T )|
2dx + a
2ν
T
\
0
|δu(l, t)|
2dt + a
2l
\
0 T
\
0
|δu
x|
2dx dt
≤ 1 2 a
2ε
1ν
T
\
0
|δu(l, t)|
2dt + 1 2ε
1a
2ν
T
\
0
|δv
l(t)|
2dt
+ ε
22
l
\
0 T
\
0
|δu(x, t)|
2dx dt + 1 2ε
2l
\
0 T
\
0
|δv
2(x, t)|
2dx dt.
Since
|δu(x, t)|
2=
\lx
δu
x(θ, t) dθ − δu(l, t)
2≤ 2
\lx
δu
x(θ, t) dθ
2+ 2|δu(l, t)|
2≤ 2l
l
\
0
|δu
x(x, t)|
2dx + 2|δu(l, t)|
2we have
(12)
l
\
0 T
\
0
|δu(x, t)|
2dx dt ≤ 2l
2l
\
0 T
\
0
|δu
x|
2dx dt + 2l
T
\
0
|δu(l, t)|
2dt.
From (11), (12) and by reducing these terms we obtain (13) 1
2
l
\
0
|δu(x, T )|
2dx +
a
2ν − a
2νε
12 − lε
2 T\0
|δu(l, t)|
2dt
+ (a
2− l
2ε
2)
l
\
0 T
\
0
|δu
x|
2dx dt
≤ a
2ν 2ε
1T
\
0
|δv
l(t)|
2dt + 1 2ε
2l
\
0 T
\
0
|δv
2(x, t)|
2dx dt.
Letting ε
2= a
2ε
1and 0 < ε
1< min[1/l
2; 2ν/(ν + 2l)], from (13) we
obtain (9) with C = max[a
2ν/ε
1; 1/(a
2ε
1)]. The lemma is proved.
Lemma 3.2. Let λ(x, t; v) = λ(x, t) be the generalized solution of the conjugate boundary value problem
λ
t= −a
2λ
xx− B(x, t)λ, (x, t) ∈ Ω, (14)
λ(x, T ) = 2[u(x, T ; v) − g(x)], 0 ≤ x ≤ l, (15)
λ
x(0, t) = 0, λ
x(l, t) = −νλ(l, t), 0 < t < T.
(16) Then (17) 2
l
\
0
[u(x, T ; v) − g(x)]δu(x, T ) dx
=
T
\
0
a
2νλ(l, t; v)δv
1(t) dt +
l
\
0 T
\
0
λ(x, t; v)δv
2(x, t) dx dt.
P r o o f. Applying the conditions (6)–(8) and (14)–(16), we obtain (18) 2
l
\
0
[u(x, T, v) − g(x)]δu(x, T ) dx
=
l
\
0
λ(x, T )δu(x, T ) dx
=
l
\
0 T
\
0
[λ
tδu + λδu
t] dx dt
=
l
\
0 T
\
0
[−a
2λ
xxδu + a
2λδu
xx+ λδv
2] dx dt
=
T
\
0
a
2νλ(l, t; v)δv
1(t) dt +
l
\
0 T
\
0
λ(x, t; v)δv
2(x, t) dx dt.
The equality (17) is thus obtained. The lemma is proved.
Definition 3. The solution of the conjugate boundary value problem (14)–(16) corresponding to v ∈ V is a function λ(x, t) ∈ H
1,0(Ω) satisfying the integral identity
(19)
l
\
0 T
\
0
[−λξ
t+ a
2λ
xxξ + B(x, t)λξ] dx dt
= −2
l
\
0
[u(x, T ; v) − g(x)]ξ(x, T ) dx
for all ξ = ξ(x, t) ∈ H
1(Ω) with ξ(x, 0) = 0.
Theorem 3.1. The function (1) is differentiable in H and its gradient at v ∈ V is given by
(20) f
v(v) = ∂f
∂v = − ∂ℜ
∂v ≡
− ∂ℜ
∂v
1, − ∂ℜ
∂v
2, where ℜ is defined by
ℜ(x, t, λ, v
1, v
2) ≡ −[a
2νv
1λ(l, t; v
1) + βv
21+ v
2λ(x, t; v
2)].
P r o o f. Consider the increment of the function (1):
δf (v) = f (v + δv) − f (v) (21)
= 2
l
\
0
[u(x, T, v) − g(x)]δu(x, T ) dx + 2β
T
\
0
v
1(t)δv
1(t) dt
+
l
\
0
|δu(x, T )|
2dx + β
T
\
0
|δv
1(t)|
2dt
where v ∈ V , v + δv ∈ V , δu(x, t) ≡ u(x, t; v + δv) − u(x, t; v), u ≡ u(x, t; v).
By substituting equality (17) and estimate (9) in (21), it follows that the function (1) is differentiable in H and its gradient is given by the expression (20). The theorem is proved.
4. The gradient projection method. One of the first authors who used projection methods for solving constrained problems was J. B. Rosen [16, 17]. A lot of projection algorithms were described by Polak [14] and Pshenichny˘ı and Danilin [15]. Having the gradient function (1), we can use the gradient projection method for solving the problem (1)–(4). According to this method we construct a sequence {v
k= (v
k1(t), v
2k(x, t))} by setting
v
1k+1=
v
1k− γ
kf
v(v
1k) if v
1min≤ Z
1(v
1k) ≤ v
1max, v
1minif Z
1(v
k1) < v
1min,
v
1maxif Z
1(v
k1) > v
1max, (22)
v
2k+1=
v
2k− γ
kf
v(v
2k) if Z
2(v
2k) ≤ R
2, R[v
2k− γ
kf
v(v
2k)]
pZ
2(v
k2) if Z
2(v
2k) > R
2, (23)
where Z
1(v
1k) = v
k1− γ
kf
v(v
k1) and Z
2(v
2k) =
Tl 0
TT
0
|v
2k− γ
kf
v(v
2k)|
2dx dt.
The values γ
k≥ 0 in (21)–(22) may be selected in one of the following ways:
(i) γ
kis defined by
(24) γ
k= min
γ≥0
f (γ) = min
γ≥0
(f (v
k) − γf
v(v
k)).
(ii) If the gradient f
v(v) satisfies the condition (25) kf
v(v) − f
v(w)k
H≤ Lkv − wk
Hfor any v, w ∈ V , L = const > 0, then γ
kmay be found from the conditions
(26) 0 < c
1≤ γ
k≤ 2
L + 2c
2. Here, c
1, c
2> 0 are parameters selected by computer.
(iii) The parameter γ
k∈ [0, 1] can be chosen from the monotonicity condition f (v
k+1) < f (v
k).
(iv) γ
kcan be chosen from the condition
(27) f (v
k) − f (v
k− γ
kf
v(v
k)) ≥ εγ
kkf
v(v
k)k
2, ε > 0.
Theorem 4.1. Let V be a closed convex subset of H, and f ∈ C
1,1(V ) with f
∗= inf
Vf (v) > −∞. Let {v
k} be the sequence of controls generated by the projection algorithm formulated in (22)–(27) for an arbitrary ini- tial approximation {v
0} ∈ V . Then the sequence {f (v
k)} decreases and lim
k→∞kv
k− v
k+1k = 0. Moreover , if f is convex in H and the set M (v
0) = {v
0∈ V : f (v) ≤ f (v
0)} is bounded, then the sequence {v
k} minimizes the function f (v) in V and converges to v
∗weakly in H, and it also satisfies the estimate
(28) 0 ≤ f (v
k) − f
∗≤ c
3k , k = 1, 2, . . . ; c
3= const ≥ 0.
If f is also strongly convex in V , then {v
k} converges to the unique mini- mum control v
∗such that
(29) kv
k− v
∗k
2≤ c
4k , k = 1, 2, . . . ; c
4= const ≥ 0.
The proof directly follows from that of Theorem 5.2.1 of [19].
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M. H. Farag
Department of Mathematics Faculty of Science
Minia University Minia, Egypt
Received on 18.8.1995;
revised version on 16.2.1996
