In this paper, a robust adaptive control problem for a class of uncertain linear systems on a finite-time interval is formulated as a minimax game problem. For this class of systems, with uncertain control coefficients augmenting the state space, the system under consideration has a bilinear structure. Although the minimax solution allows for a finite-dimensional estimator, the resulting compensator structure remains much more complex than the linear-quadratic problem. For this class of partially observed systems, the minimax problem is reduced to an equivalent full information control problem where the state-space dynamics are composed of an estimator equation and its associated Riccati equation. The equivalence between the minimax adaptive controller and a saddle-point certainty equivalence adaptive controller is shown via Hamilton-Jacobi-Bellman theory where the optimal return function is differentiable. Points of discontinuity in the partial derivatives of the optimal return function are shown to form a manifold of Darboux points, from which multiple global optimal trajectories emanate. Therefore, the Dynamic Programming approach can be extended to be valid over the entire phase space, and the uniqueness of the value of the optimal return function is guaranteed. We finally show that with additional assumptions the finite-time problem can be extended to infinite time.