- Email: [email protected]
Optimal control of the convective velocity coefficient in a parabolic problem
Nonlinear Analysis 63 (2005) e1383 – e1390 www.elsevier.com/locate/na
Optimal control of the convective velocity coefficient in a parabolic problem Hem Raj Joshi Mathematics and Computer Science Department, Xavier University, 3800 Victory Parkway, Cincinnati, OH 45207-4441, USA
Abstract We consider optimal control of the coefficient of the first spatial derivative in a parabolic equation. For our convective velocity coefficient h(x, t), we treat the case of one spatial dimension and the control depends on space and time. Control of such a coefficient requires more regularity on the control set than controlling the coefficient of a zeroth-order term. The existence of an optimal control is proved and the optimality system is derived. We obtain the characterization of an optimal control through an elliptic partial differential equation. Thus the optimality system will consist of two parabolic equations coupled with an elliptic variational inequality. 䉷 2005 Elsevier Ltd. All rights reserved. MSC: 35K55; 49K20; 92D25 Keywords: Fluid problem; Optimal control; Convective velocity coefficient
1. Introduction We consider optimal control of the coefficient of the first spatial derivative in a parabolic equation. We treat the case of one spatial dimension and the control depends on space and time. Control of such a coefficient requires more regularity on the control set than controlling the coefficient of a zeroth-order term, and it leads to a partial differential equation (PDE) characterization of an optimal control instead of the usual explicit characterization.
E-mail address: [email protected]. 0362-546X/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2005.02.025
e1384
H.R. Joshi / Nonlinear Analysis 63 (2005) e1383 – e1390
We consider the parabolic equation (in non-divergent form) ut (x, t) = auxx − h(x, t)ux + f u(x, 0) = u0 (x)
in Q = (0, L) × (0, T ), 0 < x < L, t = 0,
(1.1)
with the boundary conditions u(0, t) = 0 ux (L, t) = 0
for all t ∈ (0, T ), for all t ∈ (0, T ).
The convective velocity coefficient h(x, t) is taken to be the control. The objective functional is given by
T
J (h) = 0
L
u(x, t) dx dt +
0
2
0
T
L 0
[h2t (x, t) + h2x (x, t)] dx dt,
where and are positive constants known as weighting functions. Our goal is to minimize the objective functional: J (h∗ ) = min J (h) h∈U
with control set U, where U = {h ∈ H01 (Q)||h(x, t)| M a.e.}. The example is designed to illustrate this technique with more regular controls and to generalize two recent flow applications. Objective functionals with more general differentiable dependence on u can also be treated. Handagama and Lenhart [4] considered an optimal control of the time-dependent flow rate into a gas-phase bioreactor, where the degradation of a contaminant is through the metabolism of bacteria in the bioreactor. Lenhart [6] studied an optimal control of a parabolic differential equation, modeling one-dimensional fluid through a soil-packed tube in which a contaminant is initially distributed. She considered the convective velocity coefficient as a function of time only. See Chawla and Lenhart [1] for a similar application involving bioreactor flow. In Section 2, the problem is formulated and the existence of an optimal control is proved. In Section 3, the optimality system is derived by differentiating the objective functional with respect to the control (in a directional derivative) and differentiating the function mapping the control to the state. We obtain the characterization of an optimal control through an elliptic variational inequality. Thus, the optimality system will consist of two parabolic equations coupled with an elliptic variational inequality.
2. Existence of optimal control The solution of the state equation is a weak solution in W = L2 (0, T ; V ), 1 (0, L). (Subscript {0} means the functions vanish at x = 0.) where V = H{0}
H.R. Joshi / Nonlinear Analysis 63 (2005) e1383 – e1390
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Definition 2.1. A function u ∈ W with ut ∈ L2 (0, T ; V ∗ ) is a solution of ut (x, t) = auxx − h(x, t)ux + f u(0, t) = 0 ux (L, t) = 0 u(x, 0) = u0 (x) if
T
L
u (t), v + 0
0
in Q = (0, L) × (0, T ), on (0, T ), x = 0, on (0, T ), x = L, on (0, L), t = 0,
(2.1)
(aux vx + hux v − f v) dx dt = 0
(2.2)
for all v ∈ V and a.e. time 0 t T , and u(x, 0) = u0 (x), where the brackets , in the integrand denote the duality between V and V ∗ . The following assumptions are made: (a) a, , are positive constants, (b) u0 ∈ H 1 (0, L), with compact support, (c) f ∈ L∞ (Q). For h ∈ U , the existence of a unique solution u = u(h) in W satisfying (2.1) in the sense (2.2) follows from a standard result in [3]. Theorem 2.1. There exists an optimal control h∗ ∈ U minimizing the objective functional J (h). Proof. Choose a minimizing sequence {hn }, n = 1, 2, . . . , such that lim J (hn ) = inf J (h).
n→∞
h∈U
Let un = u(hn ) be the corresponding state solutions to (2.1). To obtain the necessary a priori estimates on our minimizing sequence and associated states, we need a change of variable w = e−t u, where is a positive constant to be chosen. Thus Eq. (1.1) becomes wt + w = aw xx − h(x, t)wx + e−t f . We use the weak definition on the above equation and integrate in time to obtain T T L wtn , wn dt + (aw nx wxn + hn wxn w n + (w n )2 − e−t w n f ) dx dt = 0 0
and then 1 2
0
0
T L T L (w n )2 (x, T ) dx + a (wxn )2 dx dt + (w n )2 dx dt 0 0 0 0 0 T L T L 1 L 2 = u0 dx − hn wxn w n dx dt + e−t w n f dx dt. 2 0 0 0 0 0
L
Note: hn H 1 (Q) C by the form of the objective functional.
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H.R. Joshi / Nonlinear Analysis 63 (2005) e1383 – e1390
Now we apply Cauchy’s inequality in the right-hand side to obtain T L T L (w n )2 (x, T ) dx + a (wxn )2 dx dt + (w n )2 dx dt 0 0 0 0 0 1 L 2 a T L n 2 M2 T L n 2 u0 dx + (wx ) dx dt + (w ) dx dt 2 0 2 0 0 2a 0 0 T L 1 T L n 2 + (w ) dx dt + f 2 dx dt. 4 0 0 0 0
1 2
L
Choose such that > 1 + M 2 /a. Thus we obtain 1 2
T L M2 T L n 2 (w n )2 (x, T ) dx + a (wxn )2 dx dt + (w ) dx dt 2a 0 0 0 0 0 T L 1 L 2 u0 dx + f 2 dx dt. 2 0 0 0
L
(2.3)
By (2.3) we can conclude that w n W are uniformly bounded independent of n. Thus un W are uniformly bounded independent of n. Also using (1.1) and the above bounds, we have uniform bounds on unt L2 (0,T ;V ∗ ) . Using the above bounds, we can extract a subsequence satisfying the following convergence properties. un unt hn hnx hnt
u∗ weakly in W , u∗t weakly in L2 (0, T ; V ∗ ), → h∗ strongly in H 1 (Q), h∗x weakly in L2 (Q), h∗t weakly in L2 (Q).
(2.4)
.The strong convergence of hn in L2 makes sense as H 1 (Q)L2 (Q) [8]. Now, we need to show u∗ = u(h∗ ),
i.e. u∗ is the state solution associated with h∗ .
Consider the weak form of the PDE (integrated in time) satisfied by un (for v ∈ V ): 0
T
unt , v dt+a
0
T
L 0
unx vx dx dt+
T 0
L 0
hn unx v dx dt−
T 0
L 0
f v dx dt = 0.
H.R. Joshi / Nonlinear Analysis 63 (2005) e1383 – e1390
The convergence of the third term uses the fact that T L n n h ux v − hux v dx dt 0 0 T T L |(hn − h)v||unx | dx dt + 0
0
0
T
L
1/2
n
T
L 0
e1387
hv(unx − ux ) dx dt
L
|unx |2 dx dt
1/2
|(h − h)v| dx dt 0 0 T0 L hv(unx − ux ) dx dt + 0 0 T L 1/2 T L 1/2 C1 |hn − h|2 dx dt |unx |2 dx dt 0 0 0T 0L n + hv(ux − ux ) dx dt 2
0
0
→0
0
as n → ∞,
where the C1 depends on the L∞ bound on v. The convergence of the first term depends on the L2 strong convergence of the {hn } sequence and the L2 uniform bound on the {unx } sequence. Since hv is in L2 (Q), the second term converges by weak L2 convergence of the {unx } sequence. Passing to the limit in the PDE satisfied by un , we obtain u∗ = u(h∗ ). Next, we verify that h∗ is an optimal control. Using the lower semicontinuity of the objective functional with respect to these weak convergences, we obtain T L T L n 2 J (h∗ ) inf un (x, t) dx + inf [(ht ) + (hnx )2 ] dx dt n n 2 0 0 0 0 = lim J (hn ) = inf J (h). n→∞
Hence
h∗
h
is an optimal control.
Remark. The convergence of the term hn unx with hn = hn (x, t) is precisely the reason for having a H 1 control set. If the controls were only functions of space, then the L∞ control sets and techniques of [1,6] would work. 3. Characterization of an optimal control By differentiating the objective functional J (h) with respect to h, we derive the optimality system. We will obtain the characterization of an optimal control through an elliptic differential equation. Thus the optimality system will consist of two parabolic equations coupled with a variational inequality [5]. Lemma 3.1. The solution map h ∈ U −→ u = u(h) ∈ W
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H.R. Joshi / Nonlinear Analysis 63 (2005) e1383 – e1390
is differentiable in the following sense: u(h + l) − u(h)
weakly in W
as → 0, where h + l ∈ U, l ∈ L∞ (Q). Moreover the sensitivity function = (h; l) satisfies t (x, t) = axx − hx − lux (x, 0) = 0 (0, t) = 0 x (L, t) = 0
in Q = (0, L) × (0, T ), when t = 0, when x = 0, when x = L,
(3.1)
where u = u(h). Proof. In order to get a priori estimates on the difference quotients, we do the following change of variables, w = e−t u,
w = e−t u ,
where u = u(h) and u = u(h + l). By considering the PDE solved by w − w, we obtain (multiplying by test function (w − w)) 1 2
T L T L (w −w)2 (x, T ) dx+a ((w −w)x )2 dx dt+ (w −w)2 dx dt 0 0 0 0 0 T L T L =− h(w − w)x (w − w) dx dt − lwx (w − w) dx dt 0 0 0 0 T L a T L 1 2 ((w − w)x ) dx dt + (h(w − w))2 dx dt 2 0 0 2a 0 0 T L 2 T L + (lw x )2 dx dt + (w − w)2 dx dt 4 0 0 0 0 a T L M2 T L ((w − w)x )2 dx dt + (w − w)2 dx dt 2 0 0 2a 0 0 T L 2 T L + (lw x )2 dx dt + (w − w)2 dx dt + K1 2 , (3.2) 4 0 0 0 0
L
denoting the L∞ bound on h by M. Using the facts that wx L2 is uniformly bounded from (2.3), we obtain T L a T L M2 2 −1 ((w − w)x ) dx dt + − (w − w)2 dx dt K1 2 , 2 0 0 2a 0 0 where K1 depends on l∞ and wx L2 . For large , we have w − w K2 W
H.R. Joshi / Nonlinear Analysis 63 (2005) e1383 – e1390
or
e1389
u − u K3 . W
Thus weak convergence to is achieved. We also have u −u t t weakly in L2 (0, T ; V ∗ ) and u − u −→ strongly in L2 (Q) as in Theorem 2.1. Hence, we conclude that satisfies (3.1).
We now derive a variational inequality characterizing an optimal control by differentiating J (h) with respect to h at an optimal control. Theorem 3.1. Given an optimal control h in U, there exists a solution p in W to the adjoint problem −pt = ap xx + (hp)x + 1 p(x, T ) = 0 p(0, t) = 0 apx (L, t) + hp(L, t) = 0
in Q, on (0, L), t = T , on (0, T ), x = 0, on (0, T ), x = L.
Furthermore h(x, t) satisfies the following variational inequality: min max − htt + hxx + pux , h − M , h + M = 0,
(3.3)
(3.4)
where u = u(h). Remark. See [2,5] for the definition of weak solution for such a variational inequality. Proof. Suppose h(x, t) is an optimal control. Let l ∈ H01 (Q) such that h + l ∈ U for > 0. The derivative of J (h) with respect to h in the direction l satisfies J (h + l) − J (h) T L u −u (x, t) dx dt = lim + →0 0 0 T L 2 2 2 2 + {(ht + lt ) − ht + (hx + lx ) − hx }(x, t) dx dt 2 0 0 T L T L = (x, t) dx dt + [ht lt + hx lx ](x, t) dx dt.
0 lim
→0+
0
0
0
0
(3.5)
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H.R. Joshi / Nonlinear Analysis 63 (2005) e1383 – e1390
Since there are no pointwise inequality constraints on our control set, we have this directional derivative being zero. We used Lemma 3.1 for the weak convergence of (u − u)/. Let p be the solution of the adjoint problem (3.3) [7]; then substituting the weak form of (3.3) in (3.5) gives T T L (−pt + ap x x − (hp)x )(x, t) dx dt + hp(L, t) dt 0 0 0 0 T L + [ht lt + hx lx ](x, t) dx dt. 0
0
Integrating by parts yields L T T L T 0 [hp]0 dt + hp(L, t) dt −[p]0 dx − 0 0 0 T L T L (pt +hpx +apx x ) dx dt + [ht lt +hx lx ](x, t) dx dt + 0 0 0 0 L T = [−p(x, T ) + p(x, 0)] dx − [hp(L, t) − hp(0, t)] dt 0 0 T L T L + (pt +ap x x +hpx ) dx dt+ [ht lt +hx lx ](x, t) dx dt 0 0 0 0 T L = (−plux + [ht lt + hx lx ])(x, t) dx dt (using (3.1) and (3.3)). 0
0
We substituted the inhomogenous term −lux from the sensitivity (3.1) to obtain the last equality. The last equality gives the variational inequality (3.4) that an optimal control h must satisfy. Remark. Our optimality system consists of two parabolic PDEs and an elliptic variational inequality, (1.1), (3.3), and (3.4). References [1] S. Chawla, S. Lenhart, Application of optimal control theory to bioremediation, J. Comput. Appl. Math. 114 (2000) 81–102. [2] M. Chipot, Variational Inequalities and Flow in Porous Media, Springer, New York, 1984. [3] L.C. Evans, Partial Differential Equations, Graduate Studies In Mathematics, Vol. 19, American Mathematical Society, Providence, RI, 1998. [4] N. Handagama, S. Lenhart, Optimal control of a PDE/ODE system modeling a gas-phase bioreactor, Math. Models Med. Health Sci. (1997) 197–212. [5] D. Kinderlehrer, G. Stampacchia, An Introduction to Variational Inequalities and their Applications, Academic Press, New York, 1980. [6] S. Lenhart, Optimal control of a convective–diffusive fluid problem, Math. Models Method Appl. Sci. 5 (2) (1995) 225–237. [7] J.L. Lions, E. Magenes, Non-Homogeneous Boundary Value Problems and Applications I & II, Springer, New York, 1972. [8] J. Yong, X. Li, Optimal Control Theory for Infinite Dimensional Systems, Birkhäuser, Basel, 1995.
Optimal control of the convective velocity coefficient in a parabolic problem Hem Raj Joshi Mathematics and Computer Science Department, Xavier University, 3800 Victory Parkway, Cincinnati, OH 45207-4441, USA
Abstract We consider optimal control of the coefficient of the first spatial derivative in a parabolic equation. For our convective velocity coefficient h(x, t), we treat the case of one spatial dimension and the control depends on space and time. Control of such a coefficient requires more regularity on the control set than controlling the coefficient of a zeroth-order term. The existence of an optimal control is proved and the optimality system is derived. We obtain the characterization of an optimal control through an elliptic partial differential equation. Thus the optimality system will consist of two parabolic equations coupled with an elliptic variational inequality. 䉷 2005 Elsevier Ltd. All rights reserved. MSC: 35K55; 49K20; 92D25 Keywords: Fluid problem; Optimal control; Convective velocity coefficient
1. Introduction We consider optimal control of the coefficient of the first spatial derivative in a parabolic equation. We treat the case of one spatial dimension and the control depends on space and time. Control of such a coefficient requires more regularity on the control set than controlling the coefficient of a zeroth-order term, and it leads to a partial differential equation (PDE) characterization of an optimal control instead of the usual explicit characterization.
E-mail address: [email protected]. 0362-546X/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2005.02.025
e1384
H.R. Joshi / Nonlinear Analysis 63 (2005) e1383 – e1390
We consider the parabolic equation (in non-divergent form) ut (x, t) = auxx − h(x, t)ux + f u(x, 0) = u0 (x)
in Q = (0, L) × (0, T ), 0 < x < L, t = 0,
(1.1)
with the boundary conditions u(0, t) = 0 ux (L, t) = 0
for all t ∈ (0, T ), for all t ∈ (0, T ).
The convective velocity coefficient h(x, t) is taken to be the control. The objective functional is given by
T
J (h) = 0
L
u(x, t) dx dt +
0
2
0
T
L 0
[h2t (x, t) + h2x (x, t)] dx dt,
where and are positive constants known as weighting functions. Our goal is to minimize the objective functional: J (h∗ ) = min J (h) h∈U
with control set U, where U = {h ∈ H01 (Q)||h(x, t)| M a.e.}. The example is designed to illustrate this technique with more regular controls and to generalize two recent flow applications. Objective functionals with more general differentiable dependence on u can also be treated. Handagama and Lenhart [4] considered an optimal control of the time-dependent flow rate into a gas-phase bioreactor, where the degradation of a contaminant is through the metabolism of bacteria in the bioreactor. Lenhart [6] studied an optimal control of a parabolic differential equation, modeling one-dimensional fluid through a soil-packed tube in which a contaminant is initially distributed. She considered the convective velocity coefficient as a function of time only. See Chawla and Lenhart [1] for a similar application involving bioreactor flow. In Section 2, the problem is formulated and the existence of an optimal control is proved. In Section 3, the optimality system is derived by differentiating the objective functional with respect to the control (in a directional derivative) and differentiating the function mapping the control to the state. We obtain the characterization of an optimal control through an elliptic variational inequality. Thus, the optimality system will consist of two parabolic equations coupled with an elliptic variational inequality.
2. Existence of optimal control The solution of the state equation is a weak solution in W = L2 (0, T ; V ), 1 (0, L). (Subscript {0} means the functions vanish at x = 0.) where V = H{0}
H.R. Joshi / Nonlinear Analysis 63 (2005) e1383 – e1390
e1385
Definition 2.1. A function u ∈ W with ut ∈ L2 (0, T ; V ∗ ) is a solution of ut (x, t) = auxx − h(x, t)ux + f u(0, t) = 0 ux (L, t) = 0 u(x, 0) = u0 (x) if
T
L
u (t), v + 0
0
in Q = (0, L) × (0, T ), on (0, T ), x = 0, on (0, T ), x = L, on (0, L), t = 0,
(2.1)
(aux vx + hux v − f v) dx dt = 0
(2.2)
for all v ∈ V and a.e. time 0 t T , and u(x, 0) = u0 (x), where the brackets , in the integrand denote the duality between V and V ∗ . The following assumptions are made: (a) a, , are positive constants, (b) u0 ∈ H 1 (0, L), with compact support, (c) f ∈ L∞ (Q). For h ∈ U , the existence of a unique solution u = u(h) in W satisfying (2.1) in the sense (2.2) follows from a standard result in [3]. Theorem 2.1. There exists an optimal control h∗ ∈ U minimizing the objective functional J (h). Proof. Choose a minimizing sequence {hn }, n = 1, 2, . . . , such that lim J (hn ) = inf J (h).
n→∞
h∈U
Let un = u(hn ) be the corresponding state solutions to (2.1). To obtain the necessary a priori estimates on our minimizing sequence and associated states, we need a change of variable w = e−t u, where is a positive constant to be chosen. Thus Eq. (1.1) becomes wt + w = aw xx − h(x, t)wx + e−t f . We use the weak definition on the above equation and integrate in time to obtain T T L wtn , wn dt + (aw nx wxn + hn wxn w n + (w n )2 − e−t w n f ) dx dt = 0 0
and then 1 2
0
0
T L T L (w n )2 (x, T ) dx + a (wxn )2 dx dt + (w n )2 dx dt 0 0 0 0 0 T L T L 1 L 2 = u0 dx − hn wxn w n dx dt + e−t w n f dx dt. 2 0 0 0 0 0
L
Note: hn H 1 (Q) C by the form of the objective functional.
e1386
H.R. Joshi / Nonlinear Analysis 63 (2005) e1383 – e1390
Now we apply Cauchy’s inequality in the right-hand side to obtain T L T L (w n )2 (x, T ) dx + a (wxn )2 dx dt + (w n )2 dx dt 0 0 0 0 0 1 L 2 a T L n 2 M2 T L n 2 u0 dx + (wx ) dx dt + (w ) dx dt 2 0 2 0 0 2a 0 0 T L 1 T L n 2 + (w ) dx dt + f 2 dx dt. 4 0 0 0 0
1 2
L
Choose such that > 1 + M 2 /a. Thus we obtain 1 2
T L M2 T L n 2 (w n )2 (x, T ) dx + a (wxn )2 dx dt + (w ) dx dt 2a 0 0 0 0 0 T L 1 L 2 u0 dx + f 2 dx dt. 2 0 0 0
L
(2.3)
By (2.3) we can conclude that w n W are uniformly bounded independent of n. Thus un W are uniformly bounded independent of n. Also using (1.1) and the above bounds, we have uniform bounds on unt L2 (0,T ;V ∗ ) . Using the above bounds, we can extract a subsequence satisfying the following convergence properties. un unt hn hnx hnt
u∗ weakly in W , u∗t weakly in L2 (0, T ; V ∗ ), → h∗ strongly in H 1 (Q), h∗x weakly in L2 (Q), h∗t weakly in L2 (Q).
(2.4)
.The strong convergence of hn in L2 makes sense as H 1 (Q)L2 (Q) [8]. Now, we need to show u∗ = u(h∗ ),
i.e. u∗ is the state solution associated with h∗ .
Consider the weak form of the PDE (integrated in time) satisfied by un (for v ∈ V ): 0
T
unt , v dt+a
0
T
L 0
unx vx dx dt+
T 0
L 0
hn unx v dx dt−
T 0
L 0
f v dx dt = 0.
H.R. Joshi / Nonlinear Analysis 63 (2005) e1383 – e1390
The convergence of the third term uses the fact that T L n n h ux v − hux v dx dt 0 0 T T L |(hn − h)v||unx | dx dt + 0
0
0
T
L
1/2
n
T
L 0
e1387
hv(unx − ux ) dx dt
L
|unx |2 dx dt
1/2
|(h − h)v| dx dt 0 0 T0 L hv(unx − ux ) dx dt + 0 0 T L 1/2 T L 1/2 C1 |hn − h|2 dx dt |unx |2 dx dt 0 0 0T 0L n + hv(ux − ux ) dx dt 2
0
0
→0
0
as n → ∞,
where the C1 depends on the L∞ bound on v. The convergence of the first term depends on the L2 strong convergence of the {hn } sequence and the L2 uniform bound on the {unx } sequence. Since hv is in L2 (Q), the second term converges by weak L2 convergence of the {unx } sequence. Passing to the limit in the PDE satisfied by un , we obtain u∗ = u(h∗ ). Next, we verify that h∗ is an optimal control. Using the lower semicontinuity of the objective functional with respect to these weak convergences, we obtain T L T L n 2 J (h∗ ) inf un (x, t) dx + inf [(ht ) + (hnx )2 ] dx dt n n 2 0 0 0 0 = lim J (hn ) = inf J (h). n→∞
Hence
h∗
h
is an optimal control.
Remark. The convergence of the term hn unx with hn = hn (x, t) is precisely the reason for having a H 1 control set. If the controls were only functions of space, then the L∞ control sets and techniques of [1,6] would work. 3. Characterization of an optimal control By differentiating the objective functional J (h) with respect to h, we derive the optimality system. We will obtain the characterization of an optimal control through an elliptic differential equation. Thus the optimality system will consist of two parabolic equations coupled with a variational inequality [5]. Lemma 3.1. The solution map h ∈ U −→ u = u(h) ∈ W
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H.R. Joshi / Nonlinear Analysis 63 (2005) e1383 – e1390
is differentiable in the following sense: u(h + l) − u(h)
weakly in W
as → 0, where h + l ∈ U, l ∈ L∞ (Q). Moreover the sensitivity function = (h; l) satisfies t (x, t) = axx − hx − lux (x, 0) = 0 (0, t) = 0 x (L, t) = 0
in Q = (0, L) × (0, T ), when t = 0, when x = 0, when x = L,
(3.1)
where u = u(h). Proof. In order to get a priori estimates on the difference quotients, we do the following change of variables, w = e−t u,
w = e−t u ,
where u = u(h) and u = u(h + l). By considering the PDE solved by w − w, we obtain (multiplying by test function (w − w)) 1 2
T L T L (w −w)2 (x, T ) dx+a ((w −w)x )2 dx dt+ (w −w)2 dx dt 0 0 0 0 0 T L T L =− h(w − w)x (w − w) dx dt − lwx (w − w) dx dt 0 0 0 0 T L a T L 1 2 ((w − w)x ) dx dt + (h(w − w))2 dx dt 2 0 0 2a 0 0 T L 2 T L + (lw x )2 dx dt + (w − w)2 dx dt 4 0 0 0 0 a T L M2 T L ((w − w)x )2 dx dt + (w − w)2 dx dt 2 0 0 2a 0 0 T L 2 T L + (lw x )2 dx dt + (w − w)2 dx dt + K1 2 , (3.2) 4 0 0 0 0
L
denoting the L∞ bound on h by M. Using the facts that wx L2 is uniformly bounded from (2.3), we obtain T L a T L M2 2 −1 ((w − w)x ) dx dt + − (w − w)2 dx dt K1 2 , 2 0 0 2a 0 0 where K1 depends on l∞ and wx L2 . For large , we have w − w K2 W
H.R. Joshi / Nonlinear Analysis 63 (2005) e1383 – e1390
or
e1389
u − u K3 . W
Thus weak convergence to is achieved. We also have u −u t t weakly in L2 (0, T ; V ∗ ) and u − u −→ strongly in L2 (Q) as in Theorem 2.1. Hence, we conclude that satisfies (3.1).
We now derive a variational inequality characterizing an optimal control by differentiating J (h) with respect to h at an optimal control. Theorem 3.1. Given an optimal control h in U, there exists a solution p in W to the adjoint problem −pt = ap xx + (hp)x + 1 p(x, T ) = 0 p(0, t) = 0 apx (L, t) + hp(L, t) = 0
in Q, on (0, L), t = T , on (0, T ), x = 0, on (0, T ), x = L.
Furthermore h(x, t) satisfies the following variational inequality: min max − htt + hxx + pux , h − M , h + M = 0,
(3.3)
(3.4)
where u = u(h). Remark. See [2,5] for the definition of weak solution for such a variational inequality. Proof. Suppose h(x, t) is an optimal control. Let l ∈ H01 (Q) such that h + l ∈ U for > 0. The derivative of J (h) with respect to h in the direction l satisfies J (h + l) − J (h) T L u −u (x, t) dx dt = lim + →0 0 0 T L 2 2 2 2 + {(ht + lt ) − ht + (hx + lx ) − hx }(x, t) dx dt 2 0 0 T L T L = (x, t) dx dt + [ht lt + hx lx ](x, t) dx dt.
0 lim
→0+
0
0
0
0
(3.5)
e1390
H.R. Joshi / Nonlinear Analysis 63 (2005) e1383 – e1390
Since there are no pointwise inequality constraints on our control set, we have this directional derivative being zero. We used Lemma 3.1 for the weak convergence of (u − u)/. Let p be the solution of the adjoint problem (3.3) [7]; then substituting the weak form of (3.3) in (3.5) gives T T L (−pt + ap x x − (hp)x )(x, t) dx dt + hp(L, t) dt 0 0 0 0 T L + [ht lt + hx lx ](x, t) dx dt. 0
0
Integrating by parts yields L T T L T 0 [hp]0 dt + hp(L, t) dt −[p]0 dx − 0 0 0 T L T L (pt +hpx +apx x ) dx dt + [ht lt +hx lx ](x, t) dx dt + 0 0 0 0 L T = [−p(x, T ) + p(x, 0)] dx − [hp(L, t) − hp(0, t)] dt 0 0 T L T L + (pt +ap x x +hpx ) dx dt+ [ht lt +hx lx ](x, t) dx dt 0 0 0 0 T L = (−plux + [ht lt + hx lx ])(x, t) dx dt (using (3.1) and (3.3)). 0
0
We substituted the inhomogenous term −lux from the sensitivity (3.1) to obtain the last equality. The last equality gives the variational inequality (3.4) that an optimal control h must satisfy. Remark. Our optimality system consists of two parabolic PDEs and an elliptic variational inequality, (1.1), (3.3), and (3.4). References [1] S. Chawla, S. Lenhart, Application of optimal control theory to bioremediation, J. Comput. Appl. Math. 114 (2000) 81–102. [2] M. Chipot, Variational Inequalities and Flow in Porous Media, Springer, New York, 1984. [3] L.C. Evans, Partial Differential Equations, Graduate Studies In Mathematics, Vol. 19, American Mathematical Society, Providence, RI, 1998. [4] N. Handagama, S. Lenhart, Optimal control of a PDE/ODE system modeling a gas-phase bioreactor, Math. Models Med. Health Sci. (1997) 197–212. [5] D. Kinderlehrer, G. Stampacchia, An Introduction to Variational Inequalities and their Applications, Academic Press, New York, 1980. [6] S. Lenhart, Optimal control of a convective–diffusive fluid problem, Math. Models Method Appl. Sci. 5 (2) (1995) 225–237. [7] J.L. Lions, E. Magenes, Non-Homogeneous Boundary Value Problems and Applications I & II, Springer, New York, 1972. [8] J. Yong, X. Li, Optimal Control Theory for Infinite Dimensional Systems, Birkhäuser, Basel, 1995.