# Non linear Parametric BVP with inequalities

Consider a non linear ode in dimension $$10$$: $$\dot x = f(t,x,\lambda)$$ where $$\lambda$$ is a vector of $$p$$ parameters.

Consider a boundary value problem of the form :

$$\dot x(t) = f(t,x(t),\lambda)$$ for $$t\in [0,T]$$ such that $$g(x(0), x(T),\lambda)\le 0$$ and $$h(x(0),x(T),\lambda)=0$$ for some vector valued function $$g$$ and $$h$$.

Is there exists some numerical scheme to solve this kind of problem.

First I think I can add the parameters $$\lambda$$ to the state $$x$$ that is consider the state $$\tilde x=(x,\lambda)$$ such that $$\dot \lambda = 0$$ so that I don't consider parametric BVP but BVP.

Second, to handle the inequalities, I am thinking to add also some constants $$c$$ such that $$g(\tilde x(0),\tilde x(T)) + c =0$$ and such that $$\dot c=0$$.

Then I can apply a Newton type algorithm to find a zero of my shooting function $$S(\tilde x,c)$$.

I can't implement it in python, fortran, matlab, c++ ...

How serious this idea sound ? Do you have other method, maybe a relaxation of my problem leading to an optimization problem ?