thanks in advance for any help. My question relates to a homework problem. I have been given a system of generalized differential equations with the goal of creating an algorithm that will solve problems with the form of the given system. I know what I need to do to implement the algorithm, I just don't fully understand the problem formulation.
I have been given $$\frac{df}{dt} = (a*f*(fmax - f)) - (b*f*u(t)) $$ $$\frac{dm}{dt} = (c*m) + (b*f*u(t)) $$ and have been told that a,b, and c are scalars such that a and g are positive. I know that f is a population of some sort, fmax is the maximum size of that population, and f(0) = f0. t is an element of T where T is an interval of time (for example 1,2,3,...,n). The unknown is u(t) (which is obviously a function of time). My goal is maximize m(T).
This is where I get confused. After the above system is given to me and the variables are given above in my assignment explanation, I am told that our goal is to minimize u over the function g(X(T)) subject to $\frac{dX}{dt} = F(X(t),u(t))$, with X(0) = X0 and the constraint u(t) $\in U$, $\forall t$. U in this case is the set of possible values of u(t) (the feasible set). I have been given nothing else and after talking to others I am under the impression that this makes perfect sense. The rest of the problem description is spent talking about how an algorithm may be implemented.
How does $\frac{dX}{dt}$ relate to the system of equations that I have been given, and where did the functions g and F come from (not to mention X(t))? I understand that adding g and F are meant to generalize the problem further, but I don't know how it relates to the system of differential equations that I have been given. Also, I was under the impression that I needed to maximize m(T) so why would I want to minimize u given g(X(T))?
If i understand you correctly, you want to do
$\min_u g(x)$ Such that $ dx/dt = f(x,u,t)$ is that correct?
In that case check out an optimization technique called the adjoint method, which is nothing more than an application of the implicit function theorem to the solution of the differential equation.