Hamilton-Jacobi equations are a key tool in optimal control theory and central to the study of mean-field games. Although first-order equations lack parabolic smoothing, viscosity solutions are nevertheless Hölder continuous provided that the Hamiltonian grows superlinearly in the momentum variable (e.g. Cardaliaguet, ESAIM:COCV 2009). However, this coercivity assumption fails whenever the dynamics do not allow control in all directions. Partially coercive Hamilton-Jacobi equations thus arise naturally in the optimal control of systems with constrained or degenerate dynamics, including a broad class of linear control systems.

Even when direct control is unavailable in some directions, the dynamics may still be controllable if the control interacts with the drift in a suitable way. For linear systems this is characterised by the Kalman rank condition. I will present recent work on a class of first-order Hamilton-Jacobi equations motivated by such controllable linear systems. We show that, under the Kalman rank condition, viscosity solutions are Hölder continuous for Hamiltonians with superlinear growth in the controlled directions only. The resulting Hölder regularity is anisotropic, reflecting the geometry induced by the underlying control structure. This is joint work with Alpár Mészáros (Durham).