ITERATIVE DYNAMIC PROGRAMMING, REIN LUUS
10.2.1 Example 1
CHAPMAN & HALL/CRC Monographs and Surveys in Pure and Applied Mathematics
Find u over t in [0; 2 ] to minimize
J = x2(tF) |
subject to:
| = u |
| = 0.5*x12 |
The initial condition are:
x(0) = [1 0] |
−1 <= u <= 1 |
Reference: [25]
toms t p = tomPhase('p', t, 0, 2, 50); setPhase(p); tomStates x1 x2 tomControls u % Initial guess x0 = {icollocate({x1 == 1; x2 == 0}) collocate(u == 0)}; % Box constraints cbox = {-1 <= collocate(u) <= 1}; % Boundary constraints cbnd = initial({x1 == 1; x2 == 0}); % ODEs and path constraints ceq = collocate({dot(x1) == u dot(x2) == 0.5*x1.^2}); % Objective objective = final(x2);
options = struct; options.name = 'Singular Control 1'; solution = ezsolve(objective, {cbox, cbnd, ceq}, x0, options); t = subs(collocate(t),solution); u = subs(collocate(u),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 1 f_k 0.166665695130345510 sum(|constr|) 0.000000330654862346 f(x_k) + sum(|constr|) 0.166666025785207870 f(x_0) 0.000000000000000000 Solver: snopt. EXIT=0. INFORM=1. SNOPT 7.2-5 NLP code Optimality conditions satisfied FuncEv 1 ConstrEv 41 ConJacEv 41 Iter 39 MinorIter 164 CPU time: 0.218750 sec. Elapsed time: 0.219000 sec.
figure(1) plot(t,u,'+-'); legend('u'); title('Singular Control 1 control');