# Parameter manipulation while solving ODE

Hi,

I am trying to solve an optimal control problem on the following form of Lorenz system:

``````function lorenz(du, u, p, t)
theta = p
du = 10.0 * (u - u) + theta
du = u * (28.0 - (u)) - u + theta
du = u * u - (8/3) * (u) + theta
end

tspan = (0.0, T)             # time span for nonlinear problem
prob = ODEProblem(lorenz,u0,tspan,theta)                 #ODE problem definition
sol = solve(prob, RK4(),saveat = Δt, dt=Δt)          #solving the NL problem

``````

where (theta) is the control parameter. In the previous code, theta is assumed to be constant throughout the time horizon T. However, in an optimal control problem, the control action (theta)
should be updated at each time step. Is there a way to do that without having to manually do the RK4 algorithm?

1 Like

The differential equations library allows you to define parameters of any type, so functions should also be fine afaik.
Assuming your control parameter only depends on quantities that are known at each step of the integration (i.e time and the state), the following should do the trick:

``````function controlActionTheta(du,u,t)
Theta = (maximum(u),t,0) #or whatever you want to define there is
return Theta
end

function lorenz(du, u, p, t)
theta = p(du,u,t)
du = 10.0 * (u - u) + theta
du = u * (28.0 - (u)) - u + theta
du = u * u - (8/3) * (u) + theta
end

prob = ODEProblem(lorenz,u0,tspan,controlActionTheta)
``````

In case you want to do something different, you might also want to checkout the callback library of DifferentialEquations.jl:
https://diffeq.sciml.ai/stable/features/callback_library/

3 Likes

Indeed, as @Salmon points out, the parameters can also be functions. This is used frequently, see also the Example 3 Ordinary Differential Equations · DifferentialEquations.jl.

3 Likes

Thanks a lot for your prompt response!

1 Like