find the minimum objective for a given ode as boundary value problem

Hello Community,

could you give me hint how to tackle this problem.

I have an ode (c has a positive value):

dxdt2 = a+b*t - c*dxdt^2;

with this boundary conditions:

t(0)   = t0;
x(0)   = x0;
dxdt(0) = v0;
x(end) = xEnd;
dxdt(end) = vEnd;
dxdt2(end) = 0;

and with these unkowns:

t(end)
a is positive;
b is either positive or negative

also the objective funktion an energy metric have to be minimzed:

E = Integral of c*dxdt^2 from t0 to tEnd

So at first i have to define the ode-function and the objective-function.

But do i need the bvp solver and the fmincon-solver?

Thank you in advance.

[Torsten]

You prescribe 5 conditions for your ODE:

x(0)   = x0;
dxdt(0) = v0;
x(end) = xEnd;
dxdt(end) = vEnd;
dxdt2(end) = 0;

and you have 5 degrees of freedom:

a,b,c and two boundary conditions

This should uniquely fix your solution. E follows automatically - no minimization possible.

[Marko]

Thank you Torsten.

But for the case that i prescribe only 4 conditions for the ODE, how could i couple the bvp-solver with fmincon?

[Torsten]

I see now that tend is also a free parameter. So you have 5 conditions and 6 degrees of freedom. Ok.

By dxdt2 you mean d^2x/dt^2 ? And by dxdt^2 you mean (dx/dt)^2 ?

[Marko]

yes, your interpretation is correct.

Answer 1

[Alan Weiss]

For examples of optimizing a function given as the solution to an ODE, see Fit ODE, Problem-Based and Fit an Ordinary Differential Equation (ODE). These examples minimize a sum of squared difference between a given curve and the ODE solution, but the techniques apply to your problem as well. For more information, see Optimizing a Simulation or Ordinary Differential Equation.

Good luck,

Alan Weiss

MATLAB mathematical toolbox documentation

Answer 2

[Torsten]

Vector of unknowns for fmincon:
(y1,y2,y3,y4)=(a,b,c,tend)

Objective for fmincon:
f(a,b,c,tend) = integral_{t=0}^{tend} c*(dx/dt)^2
Can be obtained by solving with ODE45
z' = c*x2^2 , z(0) = 0
x1' = x2    , x1(0) = x0
x2' = a+b*t-c*x2^2, x2(0) = v0
in [0 tend]
z(end) is the value to be returned


Constraints (to be supplied in nlcon):
ceq(1) = x2(end) - vend == 0
ceq(2) = a+b*tend-c*x2(end)^2 == 0
x2(end) can be obtained by solving with ODE45
x1' = x2    , x1(0) = x0
x2' = a+b*t-c*x2^2, x2(0) = v0
in [0 tend]

Upper and lower bounds:
a >= 0
tend >= 0 ?