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CONT6-QQ.SIF
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***************************
* SET UP THE INITIAL DATA *
***************************
NAME CONT6-QQ
* Problem :
* *********
* A quadratic-quadratic control problem as suggested by
* A. Leung and S. Stojanovic,
* ``Optimal control for elliptic Volterra--Lotka equations'',
* J. Math. Analysis and Applications, vol. 173, pp. 603-619, 1993.
* S. Stojanovic,
* ``Optimal damping control and nonlinear elliptic systems'',
* SIAM J. Control Optimization, vol. 29, pp. 594-608, 1991.
* and
* H. Maurer and H.D. Mittelmann,
* ``Optimization techniques for solving elliptic control problems with
* control and state constraints. Part 2: Distributed control``,
* to appear in Comp. Optim. Applic 2000.
* See also Hans Mittelmann's WWW article
* http://plato.la.asu.edu/papers/paper88/paper.html
* where the problem is Example 6.
* SIF input: Nick Gould, September 2000
* classification QQR2-AN-V-V
* Number of nodes in x direction
IE n 100 $-PARAMETER
*IE n 200 $-PARAMETER
*IE n 400 $-PARAMETER
*IE n 600 $-PARAMETER
*IE n 800 $-PARAMETER
* Other useful parameters
RE ub 6.09
RF pi/4 ARCTAN 1.0
RM 2pi pi/4 8.0
RE r 1.4
RE d 1.6
IE 0 0
IE 1 1
RE one 1.0
IA n1 n -1
RI rn n
R/ h one rn
R* h2 h h
RM -h2 h2 -1.0
R* 2pih2 2pi h2
VARIABLES
DO j 0 n
DO i 0 n
X u(i,j)
ND
DO j 1 n1
DO i 1 n1
X f(i,j)
ND
GROUPS
* objective function (see later)
N ff
* constraints
* linear part of p{i in 1..n1, j in 1..n1}:
* 4*u[i,j] - u[i-1,j] - u[i,j-1] - u[i+1,j] - u[i,j+1]
* - u[i,j] * (a[i,j] - f[i,j] - b*u[i,j])*h2 == 0;
DO j 1 n1
IA j+1 j 1
IA j-1 j -1
DO i 1 n1
IA i+1 i 1
IA i-1 i -1
XE p(i,j) u(i,j) 4.0
XE p(i,j) u(i+1,j) -1.0 u(i-1,j) -1.0
XE p(i,j) u(i,j-1) -1.0 u(i,j+1) -1.0
ND
* bc1 {i in 1..n1}: u[i,0] - u[i,1] + h*u[i,1] = 0
DO i 1 n1
XE bc1(i) u(i,0) 1.0 u(i,1) -1.0
ZE bc1(i) u(i,1) h
ND
* bc2 {j in 1..n1}: u[0,j] - u[1,j] + h*u[1,j] = 0
DO j 1 n1
XE bc2(j) u(0,i) 1.0 u(1,i) -1.0
ZE bc2(j) u(1,j) h
ND
* bc3 {i in 1..n1}: u[i,n] - u[i,n1] = 0;
DO i 1 n1
XE bc3(i) u(i,n) 1.0 u(i,n1) -1.0
ND
* bc4 {j in 1..n1}: u[n,j] - u[n1,j] = 0;
DO j 1 n1
XE bc4(j) u(n,j) 1.0 u(n1,j) -1.0
ND
BOUNDS
DO j 0 n
DO i 0 n
ZU CONT6-QQ u(i,j) ub
ND
DO j 1 n1
DO i 1 n1
ZL CONT6-QQ f(i,j) r
ZU CONT6-QQ f(i,j) d
ND
START POINT
DO j 0 n
DO i 0 n
X CONT6-QQ u(i,j) 6.0
ND
DO j 1 n1
DO i 1 n1
X CONT6-QQ f(i,j) 1.5
ND
ELEMENT TYPE
EV OBJ F U
EV NLC F U
EP NLC A
ELEMENT USES
DO j 1 n1
RI rj j
DO i 1 n1
RI ri i
* f[i,j]*(sm*f[i,j] - sk*u[i,j])
XT o(i,j) OBJ
ZV o(i,j) F f(i,j)
ZV o(i,j) U u(i,j)
* u[i,j] * (a[i,j] - f[i,j] - b*u[i,j])
* where a[i,j] := 7 + 4 * sin(2*pi*h2*i*j);
R* ij ri rj
R* 2pih2ij 2pih2 ij
R( sin SIN 2pih2ij
RM 4sin sin 4.0
RA a 4sin 7.0
XT c(i,j) NLC
ZV c(i,j) F f(i,j)
ZV c(i,j) U u(i,j)
ZP c(i,j) A a
ND
GROUP USES
DO j 1 n1
DO i 1 n1
* ff: h2*sum{i in 1..n1, j in 1..n1} f[i,j]*(sm*f[i,j] - sk*u[i,j])
ZE ff o(i,j) h2
* nonlinear part of p{i in 1..n1, j in 1..n1}:
* 4*u[i,j] - u[i-1,j] - u[i,j-1] - u[i+1,j] - u[i,j+1]
* - u[i,j] * (a[i,j] - f[i,j] - b*u[i,j])*h2 == 0
ZE p(i,j) c(i,j) -h2
ND
OBJECT BOUND
* Solution
*XL SOLUTION -4.32546D+00 $ n = 100
ENDATA
***********************
* SET UP THE FUNCTION *
* AND RANGE ROUTINES *
***********************
ELEMENTS CONT6-QQ
TEMPORARIES
R SK
R SM
R B
INDIVIDUALS
* f * (sm * f - sk * u )
T OBJ
A SK 0.8
A SM 1.0
F F * ( SM * F - SK * U )
G F 2.0 * SM * F - SK * U
G U - SK * F
H F F 2.0 * SM
H F U - SK
* u * (a - f - b*u)
T NLC
A B 1.0
F U * ( A - F - B * U )
G U A - F - 2.0 * B * U
G F - U
H U U - 2.0 * B
H F U - 1.0
ENDATA