Are there any known results on the uniqueness of solution of an optimal control problem?

Question

In particular, I am looking for a result for the uniqueness of an optimal control problem in which the dynamical system is nonlinear ODE, with pure state constraint. The optimal control problem is to minimise the cost functional $$ J(x(t),u(t)) = \int_0^1 F(x(t),u(t)) dt, $$ subject to the differential equations $$ \dot x=f(x(t),u(t)), $$ the constraint $$ h(x(t),t) \geq 0, $$ with initial and final condititons $$ x(0)=x_0,x(1)=x_1.$$ I have found some results but those were mostly concerned with PDEs.

Answer 1

Optimal control solutions are not unique in general. In fact, because of its infinite-dimensional nature, it is even difficult to classify the uniqueness of the solution. In my limited knowledge, I have not seen any literature claiming the uniqueness of the solution. Even optimality can only be guaranteed in a local setting for a particular initial condition and not in a global setting. If optimality cannot be guaranteed in a global setting, uniqueness is much more difficult to prove.