Pontryagin-Based Solver with Smoothed Hamiltonian, Adaptive $\Delta t$, and PA-Bundle Refinement

Abstract

We present a Pontryagin-based numerical solver for deterministic optimal control problems in Bolza form. The solver regularizes the generally nonsmooth Hamiltonian using a log-sum-exp smoothing of a concave piecewise-affine bundle surrogate, yielding a smoothed Hamiltonian $H_{\delta }$ that is $C^{\infty }$ and concave in the costate and $C^{1,1}$ in the state. The resulting two-point boundary value problem is discretized by a symplectic Euler scheme and solved by damped Newton iteration on the full-space nonlinear system in the discrete state and costate variables. A unified adaptive outer loop jointly controls the time-step distribution $\Delta t$, the number of planes in the bundle surrogate, and the smoothing parameter $\delta$. The time-adaptation strategy follows the error-density framework of Karlsson, Larsson, Sandberg, and Tempone (2015). Related approaches include Pontryagin shooting, Hamilton--Jacobi--Bellman methods, occupation-measure relaxations, max-plus approximations, and direct collocation. Numerical examples, including nonsmooth, singular, hypersensitive, and quadratic-programming-oracle benchmarks, demonstrate the roles of time refinement, bundle enrichment, and smoothing continuation. A linear--quadratic regulator with a closed-form Riccati solution is used as a calibration check.