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Appendix F
A Sample Test Problem




The following example has two controls and one state,
and gradients are calculated.

subs=cell(1,9);
subs={™t3x™,™t3j™,™t3f™,™t3c™,>t3k™,™t3l™,™t3g™,™t3a™,™t3p™};
par=[1, 2, 20, 0, 1, 0.25]; % Parameters
% nx, nu, nn, npa, grad, c

nnb=par(3); nx=par(1); nu=par(2);
xinit=[0.5];
% Initial value for the state

u0=zeros(nn,nu);
% Starting values for computing the control

ul=zeros(nn,nu);
% Lower bound (vector or matrix) for the control

uu=ones(nn,nu);
% Upper bound (vector or matrix) for the control

figure



Control=constr(™fqq™,u0,[],ul,uu,™gqq™,par,subs,xinit)
% Calls constr package

[Objective,Constraint,State,Integral,Costate,Gradient] =
cqq(Control,par,subs,xinit) % Right hand side of differential
equation

function yy=t3x(t,it,z,yin,hs,um,xm,lm,ps)
192 OPTIMAL CONTROL MODELS IN FINANCE

yy(1)=z(1)*um(floor(it),1)+um(floor(it),2);
% Dependent variables on the right hand side
of a differential equation are coded as
z(1), z(2), etc.

function ff=t3j(t,it,z,yin,hs,um,xm,lm,ps)
% Integrand of objective function

c=ps(6)
ff(1)=(um(floor(it),1)-1)*li3(xm,hs,t);
ff(1)=ff(1)+c*um(floor(it),2);

function ff=t3f(xf,um,xm,ps)
% Endpoint term

ff(1)=0; % Control constraint

function gg=t3c(ii,hs,um,xm,lm,ps)
gg=um(ii,1) + um(ii,1) - 1;

function dg=t3k(ii,hs,um,xm,lm,ps)
% Gradient of constraint

dg=[eye(nn);eye(nn)];

function yy=t3g(t,hs,um,xm,lm,nn)
% Gradient of objective

temp= 0.5*(lm(t,1)+lm(t+1,1));
t2=t/nn;
yy=[(1+temp)*li3(xm,hs,t2), 0.25+temp];

function yy=t3a(nn,xf,um,xm,ps)
%Boundary condition for adjoint equation (at t=1)

yy=0;

function yy=t3l(t,it,z,yin,hs,um,xm,lm,ps)
% Right hand side of adjoint equation

yy=-(1+z(1))*um(floor(it),1)+1;
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Index




control policy, 10, 29, 46, 54, 61, 67, 88, 101,
admissible controllers, 95
103, 109
aggregate dynamic financial system, 40
control restraint, 95
analytical solution, xv, 1, 91, 104“108, 143
control strategy, 16
approximate solution methods, 10
control system, 11, 17
approximation methods, 11
control variables, 93, 139
approximation problem, 12
convergence properties, 12
associated problems, 6, 7
convex function, 96
augmented Lagrangian algorithm, 11, 15, 98
cost analysis, 10
cost function, 2, 11“13, 19, 27, 28, 30, 35, 36,
bang-bang control, 1, 4, 7, 16, 17, 19, 26, 41,
41, 45, 91, 142
47, 97, 101
cost of changing control, 8, 19, 20, 27, 30, 36,
bang-bang optimal control problems, 26, 27,
41, 86, 89, 142
143
CPET, 16, 19

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