Some nonlinear optimal control problems with closed-form solutions

Optimal controllers guarantee many desirable properties including stability and robustness of the closed-loop system. Unfortunately, the design of optimal controllers is generally very difficult because it requires solving an associated Hamilton-Jacobi-Bellman equation. In this paper we develop a new approach that allows the formulation of some nonlinear optimal control problems whose solution can be stated explicitly as a state-feedback controller. The approach is based on using Young's inequality to derive explicit conditions by which the solution of the associated Hamilton-Jacobi-Bellman equation is simplified. This allows us to formulate large families of nonlinear optimal control problems with closed-form solutions. We demonstrate this by developing optimal controllers for a Lotka-Volterra system.