Recently, I have meet an interesting question? How to let the objective function must be non-negative? Here we have several parameters, for example,
$$\dot x(t)=u_1+u_2$$
$$k_1(x_0,u_1,u_2,T-t_0)=\int_{t_0}^{T}[(b_1-\frac{1}{2}*u_1)*u_1-d_1x(t)]dt-D_1x(T)$$
$$k_2(x_0,u_2,u_3,T-t_0)=\int_{t_0}^{T}[(b_2-\frac{1}{2}*u_2)*u_2-d_2x(t)]dt-D_2x(T)$$
Here $$u_1\in [0,b_1], u_2\in [0,b_2]$$
Question: If the function (1) and (2) must be positive, for arbitrary control variable $$u_1,u_2$$, how to formulate additional condition of parameters?
Remark: normally, we need maximize or minimize the objective function, but here i want consider another way, any suggestions? Thanks a lot.