  A sparse regression approach for the computation of high-dimensional optimal
feedback laws arising in deterministic nonlinear control is proposed. The
approach exploits the control-theoretical link between Hamilton-Jacobi-Bellman
PDEs characterizing the value function of the optimal control problems, and
first-order optimality conditions via Pontryagin's Maximum Principle. The
latter is used as a representation formula to recover the value function and
its gradient at arbitrary points in the space-time domain through the solution
of a two-point boundary value problem. After generating a dataset consisting of
different state-value pairs, a hyperbolic cross polynomial model for the value
function is fitted using a LASSO regression. An extended set of low and
high-dimensional numerical tests in nonlinear optimal control reveal that
enriching the dataset with gradient information reduces the number of training
samples, and that the sparse polynomial regression consistently yields a
feedback law of lower complexity.
