Symmetry analysis of the Bellman equation

The optimal correction of a material point’s trajectory under small perturbations is an important problem in control theory. The study of such processes can be reduced to solving a boundary value problem for a special nonlinear second-order partial differential equation known as the Bellman equation. This paper deals with a partial differential equation with quadratic nonlinearity, which is a special case of the Bellman equation. The equation contains three independent variables—one temporal and two spatial, and two arbitrary time-dependent functions as multipliers. Multidimensionality is a distinctive feature of this equation, which significantly complicates its analytical study. Therefore, we use Lie group analysis which is an effective technique to analytically study nonlinear partial differential equations. It allows the investigation of not only the symmetry properties of equations, but also to find their particular solutions. The problem of group classification for this Bellman equation is solved with respect to two arbitrary time-dependent functions. It is established that in the case of the arbitrariness of these functions, the equation admits a four-parameter group of point transformations. This group expands to five-parameter and six-parameter groups for the parametric representation and linear dependency of the functions, respectively. Several invariant solutions are also constructed.