## 42  Global Dynamic System

Deterministic Global Optimization of Nonlinear Dynamic Systems, Youdong Lin and Mark A. Stadtherr, Department of Chemical and Biomolecular Engineering, University of Notre Dame

### 42.1  Problem Description

Find u over t in [0; 1 ] to minimize:

 J = −x2(tf)

subject to:

 dx dt
= −x2+u
 x(t0) = 9
 −5 <= u <= 5

Reference: [36]

### 42.2  Problem setup

```toms t
p = tomPhase('p', t, 0, 1, 20);
setPhase(p);

tomStates x
tomControls u

% Box constraints, bounds and odo
c = {-10 <= icollocate(x) <= 10
-5  <= collocate(u)  <= 5
initial(x == 9)
collocate(dot(x) == -x.^2+u)};
```

### 42.3  Solve the problem

```options = struct;
options.name = 'Global Dynamic System';
Prob = sym2prob('con',-final(x)^2, c, [], options);
Prob.xInit = 20;
Result = tomRun('multiMin', Prob, 1);
solution = getSolution(Result);
t = subs(collocate(t),solution);
x = subs(collocate(x),solution);
u = subs(collocate(u),solution);
```
```===== * * * =================================================================== * * *
TOMLAB - Tomlab Optimization Inc. Development license  999001. Valid to 2011-02-05
=====================================================================================
Problem: ---  1: Global Dynamic System - Trial 13  f_k      -8.232621699103969300
sum(|constr|)      0.000000001208069200
f(x_k) + sum(|constr|)     -8.232621697895901000

Solver: multiMin with local solver snopt.  EXIT=0.  INFORM=0.
Find local optima using multistart local search
Did 20 local tries. Found 1 global, 2 minima. TotFuncEv 980. TotConstrEv 941

FuncEv  980 GradEv  940 ConstrEv  941 ConJacEv   33 Iter  489
CPU time: 1.453125 sec. Elapsed time: 1.500000 sec.
```

### 42.4  Plot result

```figure(1);
subplot(2,1,1);
plot(t,x,'*-');
legend('x');

subplot(2,1,2);
plot(t,u,'*-');
legend('u');
```