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HVYCRASH.SIF
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***************************
* SET UP THE INITIAL DATA *
***************************
NAME HVYCRASH
* Problem :
* *********
* A problem freely inspired by the heavy spacecraft landing problem by
* Ivashkevich, as reported by Tyatushkin, Zholudev and Erinchek. The
* modifications of the original problem involves dropping the final state
* constraint on the second variable and the setting of EPS to zero
* in the second constraint.
* (No feasible point was found for the original formulation by any of the
* packages at hand.)
* This is a badly scaled degenerate nonlinear nonconvex optimal control
* problem, with lots of negative curvature.
* Source: inspired by
* A.I Tyatushkin, A.I. Zholudev and N. M. Erinchek,
* "The gradient method for solving optimal control problems with phase
* constraints",
* in "System Modelling and Optimization", P. Kall, ed., pp. 456--464,
* Springer Verlag, Lecture Notes in Control and Information Sciences 180, 1992.
* This reference itself refers to:
* A.K. Ivashkevich,
* "Multistage US spacecraft Space Shuttle",
* Survey of foreign papers (Part 11), Orbital Craft, Moscow, 1976.
* SIF input: Ph. Toint, February 1994.
* classification LOR2-AN-V-V
* Number of discretized points in (0,TT].
* The number of variables is 4 * ( N + 1 )
* The number of constraints is 3 * N
*IE N 50 $-PARAMETER n=204, m=150 original value
*IE N 100 $-PARAMETER n=404, m=300
*IE N 500 $-PARAMETER n=2004, m=1500
IE N 1000 $-PARAMETER n=4004, m=3000
* Problem parameters
RE TT 0.2185
RE EPS 0.1
RE ALPHA 1.62079
* Mesh
IA N-1 N -1
RI RN N
R/ H TT RN
RM -H H -1.0
RM -EPS EPS -1.0
* Constants
IE 0 0
IE 1 1
IE 2 2
IE 3 3
VARIABLES
* State variables
DO T 0 N
X X(1,T)
X X(2,T)
X X(3,T)
OD T
* Control variables
DO T 0 N
X U(T)
OD T
GROUPS
* Objective function
XN OBJ X(1,N) 1.0
* State constraints
DO T 1 N
IA T-1 T -1
XE C(1,T) X(1,T-1) 1.0 X(1,T) -1.0
ZE C(2,T)
ZE C(3,T) X(3,T-1) EPS
ZE C(3,T) X(3,T) -EPS
ND
BOUNDS
FR HVYCRASH 'DEFAULT'
XX HVYCRASH X(1,0) 0.0
XX HVYCRASH X(2,0) 2.19905
XX HVYCRASH X(3,0) 0.0
XX HVYCRASH X(3,N) 0.0
DO T 0 N
XL HVYCRASH U(T) 0.08
XU HVYCRASH U(T) 0.417
XL HVYCRASH X(3,T) 0.0
XU HVYCRASH X(3,T) 6.2831854
OD T
START POINT
DO T 1 N-1
XV HVYCRASH X(2,T) 1.5
OD T
XV HVYCRASH X(1,0) 0.0
XV HVYCRASH X(2,0) 2.19905
XV HVYCRASH X(3,0) 0.0
XV HVYCRASH X(2,N) 1.09905
XV HVYCRASH X(3,N) 0.0
ELEMENT TYPE
EV C2 XX3 XX2
EV C2 UU
EP C2 A
EV C3 XX3 XX2
EV C3 UU
EP C3 A
EV C4 XX3 XX2
EV C4 UU
EP C4 A
EV INV V
EV UX2 UU XX2
ELEMENT USES
DO T 1 N
* State constraints elements
XT E1(T) C2
ZV E1(T) XX3 X(3,T)
ZV E1(T) XX2 X(2,T)
ZV E1(T) UU U(T)
ZP E1(T) A ALPHA
XT E2(T) INV
ZV E2(T) V X(2,T)
XT E3(T) C3
ZV E3(T) XX3 X(3,T)
ZV E3(T) XX2 X(2,T)
ZV E3(T) UU U(T)
ZP E3(T) A ALPHA
XT E4(T) UX2
ZV E4(T) XX2 X(2,T)
ZV E4(T) UU U(T)
XT E5(T) C4
ZV E5(T) XX3 X(3,T)
ZV E5(T) XX2 X(2,T)
ZV E5(T) UU U(T)
ZP E5(T) A ALPHA
OD T
GROUP USES
* State constraints
DO T 1 N
ZE C(1,T) E1(T) H
* ZE C(2,T) E2(T) -H
* ZE C(2,T) E3(T) -H
XE C(2,T) E2(T) -1.0 E3(T) -1.0
ZE C(3,T) E4(T) H
ZE C(3,T) E5(T) -H
OD T
OBJECT BOUND
* Solution
*LO SOLTN -0.21850
*LO SOLTN(100) 3.26705331D-8
*LO SOLTN(500) 2.36171208D-8
*LO SOLTN(1000) 8.48265630D-8
*LO SOLTN(3000)
ENDATA
*********************
* SET UP THE GROUPS *
* ROUTINE *
*********************
ELEMENTS HVYCRASH
TEMPORARIES
R C
R DC
R D2C
R X2SQ
R CS
R SS
R D
M COS
M SIN
INDIVIDUALS
T INV
F 1.0 / V
G V - 1.0 / ( V * V )
H V V 2.0 / ( V ** 3 )
T UX2
A C 0.01 + 0.3 * UU * UU
A DC 0.6 * UU
A D2C 0.6
A X2SQ XX2 * XX2
F UU / ( C * X2SQ )
G UU ( 1.0 - UU * DC / C ) / ( C * X2SQ )
G XX2 - 2.0 * UU / ( C * X2SQ * XX2 )
H UU UU ( 2.0 * UU * DC * DC / C - D2C * UU
H+ - 2.0 * DC ) / ( C * C * X2SQ )
H UU XX2 - 2.0 * ( 1.0 - UU * DC / C ) /
H+ ( C * X2SQ * XX2 )
H XX2 XX2 6.0 * UU / ( C * X2SQ * X2SQ )
T C2
A C 0.01 + 0.3 * UU * UU
A DC 0.6 * UU
A D2C 0.6
A X2SQ XX2 * XX2
A CS COS( XX3 )
A SS SIN( XX3 )
A D A * C * X2SQ
F CS / D
G UU - CS * DC / ( D * C )
G XX2 - 2.0 * CS / ( D * XX2 )
G XX3 - SS / D
H UU UU CS * ( 2.0 * DC * DC / C - D2C ) /
H+ ( D * C )
H UU XX2 2.0 * CS * DC / ( D * C * XX2 )
H UU XX3 SS * DC / ( D * C )
H XX2 XX2 6.0 * CS / ( D * X2SQ )
H XX2 XX3 2.0 * SS / ( D * XX2 )
H XX3 XX3 - CS / D
T C3
A C 0.01 + 0.3 * UU * UU
A DC 0.6 * UU
A D2C 0.6
A X2SQ XX2 * XX2
A CS COS( XX3 )
A SS SIN( XX3 )
A D A * C * X2SQ * XX2
F CS / D
G UU - CS * DC / ( D * C )
G XX2 - 3.0 * CS / ( D * XX2 )
G XX3 - SS / D
H UU UU CS * ( 2.0 * DC * DC / C - D2C ) /
H+ ( D * C )
H UU XX2 3.0 * CS * DC / ( D * C * XX2 )
H UU XX3 SS * DC / ( D * C )
H XX2 XX2 12.0 * CS / ( D * X2SQ )
H XX2 XX3 3.0 * SS / ( D * XX2 )
H XX3 XX3 - CS / D
T C4
A C 0.01 + 0.3 * UU * UU
A DC 0.6 * UU
A D2C 0.6
A X2SQ XX2 * XX2
A CS COS( XX3 )
A SS SIN( XX3 )
A D A * C * X2SQ * X2SQ
F CS / D
G UU - CS * DC / ( D * C )
G XX2 - 4.0 * CS / ( D * XX2 )
G XX3 - SS / D
H UU UU CS * ( 2.0 * DC * DC / C - D2C ) /
H+ ( D * C )
H UU XX2 4.0 * CS * DC / ( D * C * XX2 )
H UU XX3 SS * DC / ( D * C )
H XX2 XX2 20.0 * CS / ( D * X2SQ )
H XX2 XX3 4.0 * SS / ( D * XX2 )
H XX3 XX3 - CS / D
ENDATA