Convexity and monotonicity in nonlinear optimal control under uncertainty

We consider the problem of finite-horizon optimal control design under uncertainty for imperfectly observed discrete-time systems with convex costs and constraints. It is known that this problem can be cast as an infinite-dimensional convex program when the dynamics and measurements are linear, uncertainty is additive, and the risks associated with constraint violations and excessive costs are measured in expectation or in the worst case. In this paper, we extend this result to systems with convex or concave dynamics, nonlinear measurements, more general uncertainty structures and other coherent risk measures. In this setting, the optimal control problem can be cast as an infinite-dimensional convex program if (1) the costs, constraints and dynamics satisfy certain monotonicity properties, and (2) the measured outputs can be reversibly `purified' of the influence of the control inputs through Q- or Youla-parameterization. The practical value of this result is that the finite-dimensional subproblems arising in a variety of suboptimal control methods, notably including model predictive control and the Q-design procedure, are also convex for this class of nonlinear systems. Subproblems can therefore be solved to global optimality using convenient modeling software and efficient, reliable solvers. We illustrate these ideas in a numerical example.