Symmetries and analytical solutions of the Hamilton‐Jacobi‐Bellman equation for a class of optimal control problems

The main contribution of this paper is to identify explicit classes of locally controllable second‐order systems and optimization functionals for which optimal control problems can be solved analytically, assuming that a differentiable optimal cost‐to‐go function exists for such control problems. An additional contribution of the paper is to obtain a Lyapunov function for the same classes of systems. The paper addresses the Lie point symmetries of the Hamilton–Jacobi–Bellman (HJB) equation for optimal control of second‐order nonlinear control systems that are affine in a single input and for which the cost is quadratic in the input. It is shown that if there exists a dilation symmetry of the HJB equation for optimal control problems in this class, this symmetry can be used to obtain a solution. It is concluded that when the cost on the state preserves the dilation symmetry, solving the optimal control problem is reduced to finding the solution to a first‐order ordinary differential equation. For some cases where the cost on the state breaks the dilation symmetry, the paper also presents an alternative method to find analytical solutions of the HJB equation corresponding to additive control inputs. The relevance of the proposed methodologies is illustrated in several examples for which analytical solutions are found, including the Van der Pol oscillator and mass–spring systems. Furthermore, it is proved that in the well‐known case of a linear quadratic regulator, the quadratic cost is precisely the cost that preserves the dilation symmetry of the equation. Copyright 2015 John Wiley & Sons, Ltd.