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106 Singular Control 6
Viscocity Solutions of Hamilton-Jacobi Equations and Optimal Control Problems. Alberto Bressan, S.I.S.S.A, Trieste, Italy.
A singular control example.
106.1 Problem Description
Find u over t in [0; 10 ] to maximize:
subject to:
Reference: [8]
106.2 Problem setup
toms t
t_F = 10;
p = tomPhase('p', t, 0, t_F, 80);
setPhase(p);
tomStates x1 x2 x3
tomControls u
x = [x1; x2; x3];
% Initial guess
x0 = {icollocate({x1 == 0, x2 == 0, x3 == 0})
collocate(u==0)};
% Box constraints
cbox = {-1 <= collocate(u) <= 1};
% Boundary constraints
cbnd = initial(x == [0;0;0]);
% ODEs and path constraints
ceq = collocate({dot(x1) == u; dot(x2) == -x(1)
dot(x3) == x(2)-x(1).^2});
% Objective
objective = -final(x(3));
106.3 Solve the problem
options = struct;
options.name = 'Singular Control 6';
solution = ezsolve(objective, {cbox, cbnd, ceq}, x0, options);
% Extract optimal states and controls from solution
t = collocate(subs(t,solution));
u = collocate(subs(u,solution));
x1 = collocate(subs(x1,solution));
x2 = collocate(subs(x2,solution));
x3 = collocate(subs(x3,solution));
Problem type appears to be: lpcon
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - Tomlab Optimization Inc. Development license 999001. Valid to 2011-02-05
=====================================================================================
Problem: --- 1: Singular Control 6 f_k -55.555568442322624000
sum(|constr|) 0.000000011396340832
f(x_k) + sum(|constr|) -55.555568430926286000
f(x_0) 0.000000000000000000
Solver: snopt. EXIT=0. INFORM=1.
SNOPT 7.2-5 NLP code
Optimality conditions satisfied
FuncEv 1 ConstrEv 58 ConJacEv 58 Iter 49 MinorIter 678
CPU time: 1.750000 sec. Elapsed time: 1.781000 sec.
106.4 Plot result
subplot(3,1,1)
plot(t,x1,'*-',t,x2/10,'*-',t,x3/50,'*-');
legend('x1','x2/10','x3/50');
title('Singular Control 6 state variables');
subplot(3,1,2)
plot(t,u,'+-');
legend('u');
title('Singular Control 6 control');
subplot(3,1,3)
plot(t,-1*(t<t_F/3)+1/2*(t>=t_F/3),'*-');
legend('Known u');
title('Singular Control 6 known solution');
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