103  Singular Control 3

ITERATIVE DYNAMIC PROGRAMMING, REIN LUUS

10.2.3 Example 3

CHAPMAN & HALL/CRC Monographs and Surveys in Pure and Applied Mathematics

103.1  Problem Formulation

Find u over t in [0; 5 ] to minimize

 J = x3(tF)

subject to:

 dx1 dt
= x2
 dx2 dt
= u
 dx3 dt
= x12 + x22

The initial condition are:

 x(0) = [0  1  0]
 −1 <= u <= 1

Reference: 

103.2  Problem setup

toms t
p = tomPhase('p', t, 0, 5, 60);
setPhase(p);

tomStates x1 x2 x3
tomControls u

% Initial guess
x0 = {icollocate({x1 == 0; x2 == 1; x3 == 0})
collocate(u == 0)};

% Box constraints
cbox = {-1 <= collocate(u) <= 1};

% Boundary constraints
cbnd = initial({x1 == 0; x2 == 1; x3 == 0});

% ODEs and path constraints
ceq = collocate({dot(x1) == x2
dot(x2) == u; dot(x3) == x1.^2 + x2.^2});

% Objective
objective = final(x3);

103.3  Solve the problem

options = struct;
options.name = 'Singular Control 3';
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 3             f_k       0.753994561590098700
sum(|constr|)      0.000000015978054113
f(x_k) + sum(|constr|)      0.753994577568152800
f(x_0)      0.000000000000000000

Solver: snopt.  EXIT=0.  INFORM=1.
SNOPT 7.2-5 NLP code
Optimality conditions satisfied

FuncEv    1 ConstrEv   43 ConJacEv   43 Iter   34 MinorIter  366
CPU time: 0.671875 sec. Elapsed time: 0.688000 sec.

103.4  Plot result

figure(1)
plot(t,u,'+-');
legend('u');
title('Singular Control 3 control'); 