'fmincon' is equipped to handle such constraints by accepting a function that provides both equality and inequality constraints. In your case, you want an inequality constraint that keeps the restriction (integral of y(4)) within certain bounds.
Let's break it down:
- Objective Function: This remains the same. It would be a function that returns:
- Nonlinear Constraints: You'd create another function, say 'myConstraints', that when given the same inputs as your objective function, returns:
This is the upper bound. 
This is the lower bound.
Where c should be less than or equal to 0 for the optimization constraints to be satisfied.
Let's draft the constraints function:
function [c, ceq] = myConstraints(parameters)
% Solve the DAE using the given parameters
% Assuming the DAE's are solved using a function named solveDAE
[t, y] = solveDAE(parameters);
% Inequality constraints
c(1) = trapz(t, y(:, 4)) - 2;
c(2) = 1 - trapz(t, y(:, 4));
% No equality constraints
ceq = [];
end
Call to fmincon:
x0 = initial_parameters; % starting point
A = []; b = []; % No linear inequality constraints
Aeq = []; beq = []; % No linear equality constraints
lb = lower_bounds; % Lower bounds on parameters
ub = upper_bounds; % Upper bounds on parameters
nonlcon = @myConstraints; % Nonlinear constraints function
[x_optimal, fval] = fmincon(@objectiveFunction, x0, A, b, Aeq, beq, lb, ub, nonlcon);
Where 'objectiveFunction' is another function similar to 'myConstraints' but only returns the value of 'obj_function'.
With this, fmincon will optimize your parameters such that the integral of y(17) is minimized while ensuring that the integral of y(4) remains between 1 and 2.
