constrained optimal control excluding the optimal state

Question

I don't know if this is a very basic question:

Let's say there is the typical optimal control problem with the cost function $$J = \int_{0}^{T} \mathcal{L}(x(t),u(t),t)\mathrm{d}t + \Psi(x(0),t=0)$$ with constraint $\dot{x}(t) = f(x(t),u(t),t)$ and optimal (feedback) control $u(t)$. Suppose there is a (globally) optimal solution $(x^*(t),u^*(t), t^*)$.

Is there a way to search for the 'next best solution' excluding the optimal solution, i.e. putting in an additional constraint where for example the optimal feedback control $u^*(x(t),t))$ is not allowed?

Answer 1

If the problem is convex then a local optimum is the global optimum. By constraining the control set the next-best solution will either be:

• "bang-bang"
• inexistent, if the problem is over constrained

If, however, the problem is non-convex then there might be multiple local optima and depending on the additional constraints the next-best solution can be:

• totaly different than the previous one (but still satisfying all necessary conditions).
• "bang-bang"
• inexistent, if the problem is over constrained

When you find an optimal solution for a problem that is non-convex, there is no way of saying if it is global or not, it migh already be an alternative solution!

I guess that there are global search methods, but I don't know anything about about those.