How to introduce a constraint and stop condition in ode45 ？

There are two questions. The first question is how to introduce a constraint in ode45 for a variable.

For example, the constraint of y(2) is 0<=y(2)<=2 in the following code，and how to introduce it?

The second question is how to introduce a stop condition in ode45？

For example, how to stop the integration of ode45 at two conditions when y(1)<-1 and y(2)<-2 in the following code?

Please give me help. Thanks.

y10=2;
y20=0;
Y0=[y10,y20];
[t, Y] = ode45(@fun,[0:0.1:20], Y0);

plot(t,Y(:,1),'-o',t,Y(:,2),'-o')
xlabel('Time t');
ylabel('Solution y');
legend('y_1','y_2')

function Dy=fun(t,y)
Dy=zeros(2,1);
Dy(1)=y(2);
Dy(2)=(1-y(1)^2)*y(2)-y(1)
end

[Jan]

Note: [] is Matlab's operator for concatenations. [0:0.1:20] concates the vector 0:0.1:20 with nothing, so tzhe square brackets are a waste of time only.

[Cola]

You have my gratitude.

Answer 1

[Torsten]

First question:

You can't prescribe a constraint for a solution variable of an ODE. The ODE variable results automatically from the ODE equation, so you cannot influence it: either it respects your constraint or not.

Second question:

Use the "event" facility of the ODE integrators:

https://de.mathworks.com/help/matlab/math/ode-event-location.html

[Cola]

Thank you so much. The second question is solved by the ''event''. The first question is just what you said, but I still want to try it.

Answer 2

[Jan]

Both is done by an event function. There you can check the values of the current trajectory and stop the integration. If the values should be limited only, you restart the integration from the current location setting the corresponding derivative to 0.

[Cola]

Thank you for your prompt response. The second question is solved, and I am trying to introduce a constraint by an event function for the first question .