Eigenvector approximate dichotomic basis method for solving hyper-sensitive optimal control problems 2000, Anil V. Rao and Kenneth D. Mease
3.1. Motivating example, a hyper-sensitive HBVP
Find u(t) over t in [0; t_f ] to minimize
J = | ∫ |
| (x2 + u2) dt |
subject to:
| = −x3+u |
x0 = 1 |
xtf = 1.5 |
tf = 10 |
Reference: [27]
toms t p = tomPhase('p', t, 0, 10, 50); setPhase(p); tomStates x tomControls u % Initial guess x0 = {icollocate(x == 0) collocate(u == 0)}; % bounds and ODEs bceq = {collocate(dot(x) == -x.^3+u) initial(x) == 1; final(x) == 1.5}; % Objective objective = integrate(x.^2+u.^2);
options = struct; options.name = 'Hyper Sensitive'; solution = ezsolve(objective, bceq, x0, options); t = subs(collocate(t),solution); x = subs(collocate(x),solution); u = subs(collocate(u),solution);
Problem type appears to be: qpcon Starting numeric solver ===== * * * =================================================================== * * * TOMLAB - Tomlab Optimization Inc. Development license 999001. Valid to 2011-02-05 ===================================================================================== Problem: --- 1: Hyper Sensitive f_k 6.723925391388356800 sum(|constr|) 0.000000002440650080 f(x_k) + sum(|constr|) 6.723925393829007100 f(x_0) 0.000000000000000000 Solver: snopt. EXIT=0. INFORM=1. SNOPT 7.2-5 NLP code Optimality conditions satisfied FuncEv 1 ConstrEv 26 ConJacEv 26 Iter 21 MinorIter 70 CPU time: 0.093750 sec. Elapsed time: 0.093000 sec.
subplot(2,1,1) plot(t,x,'*-'); legend('x'); title('Hyper Sensitive state variables'); subplot(2,1,2) plot(t,u,'+-'); legend('u'); title('Hyper Sensitive control');