Lossless Convexification for Linear Systems with Piecewise Linear Controls

Lossless Convexification (LCvx) is a convexification technique that transforms a class of nonconvex optimal control problems$\unicode{x2013}$where the nonconvexity arises from a lower bound on the control norm$\unicode{x2013}$into equivalent convex problems, with the goal being to apply fast polynomial-time solvers. However, to solve these infinite-dimensional problems in practice, they must first be converted into finite-dimensional problems, and it remains an open challenge to ensure the theoretical guarantees of LCvx are maintained across this discretization step. Prior work has proven guarantees for piecewise constant controls, but these methods do not extend to piecewise linear controls, which are more relevant to real world applications. In this work, we present an algorithm that extends LCvx guarantees to piecewise linear controls. Under mild assumptions, our algorithm provably finds a solution violating the nonconvex constraints along at most $2n_x + 2$ trajectory "edges" using $O(\log(\Delta\rho/\varepsilon))$ solver calls (where $n_x$ is the state space dimension and $\Delta\rho = \rho_{\max} - \rho_{\min}$ is the difference in our control norm bounds). A key feature is the perturbation of the control norm lower bound and the addition of rate constraints on the controls, ensuring LCvx holds along the trajectory edges. Finally, we provide numerical results demonstrating the effectiveness of our algorithm.