Function Space Complementarity Methods for Optimal Control Problems, Dissertation, Martin Weiser
Find u over t in [-0.5; 0.5 ] to minimize:
J = | ∫ |
| t*u dt |
subject to:
|u| <= 1 |
Reference: [34]
toms t p = tomPhase('p', t, -0.5, 1, 20); setPhase(p); tomStates x tomControls u % Initial guess x0 = {collocate(u == 1-2*(t+0.5)) icollocate(x == 1-2*(t+0.5))}; % Box constraints cbox = {-1 <= icollocate(x) <= 1 -1 <= collocate(u) <= 1}; % ODEs and path constraints ceq = collocate(dot(x) == 0); % Objective objective = integrate(t.*u);
options = struct; options.name = 'Simple Bang Bang Problem'; solution = ezsolve(objective, {cbox, ceq}, x0, options); t = subs(collocate(t),solution); u = subs(collocate(u),solution);
Problem type appears to be: lp Starting numeric solver ===== * * * =================================================================== * * * TOMLAB - Tomlab Optimization Inc. Development license 999001. Valid to 2011-02-05 ===================================================================================== Problem: --- 1: Simple Bang Bang Problem f_k -0.250490325030179710 sum(|constr|) 0.000000000000810402 f(x_k) + sum(|constr|) -0.250490325029369300 f(x_0) 0.000000000000000000 Solver: CPLEX. EXIT=0. INFORM=1. CPLEX Dual Simplex LP solver Optimal solution found
figure(1); plot(t,u,'*-'); legend('u'); ylim([-1.1,1.1]);