I'm working on a constrained optimization problem in which an unknown forcing function, $u(\eta)$, is in the integrand of a convolution integral. To find an optimal shape for $u(\eta)$, the integral must be discretized over the time interval $[0,T]$ so that it can be expressed as a function of $u_1$, $u_2,$...$u_n$ (my design variables, to be programmed using Matlab's fmincon). Essentially, I'm trying to break up this convolution integral:
$\int_{0}^{t}u(\eta)\sin{\omega(t-\eta)}d\eta$
using the following integral property:
$\int_a^cf(x)dx=\int_a^bf(x)dx+\int_b^cf(x)dx$
If I want to discretize using, for example, two grid points, I have the following:
$\int_0^Tu(\eta)\sin{\omega(T-\eta)}d\eta=\int_0^{\eta_1}u(\eta)\sin{\omega(\eta_1-\eta)}d\eta+\int_{\eta_1}^{\eta_2}u(\eta)\sin{\omega(\eta_2-\eta)}d\eta+\int_{\eta_2}^{T}u(\eta)\sin{\omega(T-\eta)}d\eta$
The idea is that, because the individual regions of integration are so small, $u(\eta)$ can be approximated as a constant, pulled outside the integral, and the remaining integrals can be evaluated analytically in Matlab via Simpson's Rule, Trapezoidal Rule, etc. The constants $u_1$, $u_2,$...$u_n$ can then be found using fmincon with appropriate objective and constraint functions (not mentioned in this post).
Can anyone confirm that my formulation is correct? Because the integrand is a function of the integral bounds, I'm not sure I'm doing this correctly. Any input is much appreciated. Thank you!