# PROPT Singular Control 4

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This page is part of the PROPT Manual. See PROPT Manual. |

ITERATIVE DYNAMIC PROGRAMMING, REIN LUUS

10.2.3 Example 4

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

## Contents |

## Problem Formulation

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

*J* = *x*_{4}(*t*_{F})

subject to:

The initial condition are:

− 1 < = *u* < = 1

`% Copyright (c) 2007-2008 by Tomlab Optimization Inc.`

## Problem setup

toms t p = tomPhase('p', t, 0, 5, 100); setPhase(p) tomStates x1 x2 x3 x4 tomControls u % Initial guess x0 = {icollocate({x1 == 1; x2 == 0 x3 == 0; x4 == 0}) collocate(u == 0)}; % Box constraints cbox = {-1 <= collocate(u) <= 1}; % Boundary constraints cbnd = initial({x1 == 1; x2 == 0 x3 == 0; x4 == 0}); % ODEs and path constraints ceq = collocate({dot(x1) == x2; dot(x2) == x3 dot(x3) == u; dot(x4) == x1.^2}); % Objective objective = final(x4);

## Solve the problem

options = struct; options.name = 'Singular Control 4'; solution = ezsolve(objective, {cbox, cbnd, ceq}, x0, options); t = subs(collocate(t),solution); u = subs(collocate(u),solution);

Problem type appears to be: lpcon Time for symbolic processing: 0.089315 seconds Starting numeric solver ===== * * * =================================================================== * * * TOMLAB - TOMLAB Development license 999007. Valid to 2011-12-31 ===================================================================================== Problem: --- 1: Singular Control 4 f_k 1.252389645383044100 sum(|constr|) 0.000000063932037643 f(x_k) + sum(|constr|) 1.252389709315081800 f(x_0) 0.000000000000000000 Solver: snopt. EXIT=0. INFORM=1. SNOPT 7.2-5 NLP code Optimality conditions satisfied FuncEv 1 ConstrEv 92 ConJacEv 92 Iter 89 MinorIter 652 CPU time: 9.313260 sec. Elapsed time: 2.553000 sec.

## Plot result

figure(1) plot(t,u,'+-'); legend('u'); title('Singular Control 4 control');