Minimax optimal control

Published versio

Infinite horizon control and minimax observer design for linear DAEs

In this paper we construct an infinite horizon minimax state observer for a linear stationary differential-algebraic equation (DAE) with uncertain but bounded input and noisy output. We do not assume regularity or existence of a (unique) solution for any initial state of the DAE. Our approach is based on a generalization of Kalman's duality principle. The latter allows us to transform minimax state estimation problem into a dual control problem for the adjoint DAE: the state estimate in the original problem becomes the control input for the dual problem and the cost function of the latter is, in fact, the worst-case estimation error. Using geometric control theory, we construct an optimal control in the feed-back form and represent it as an output of a stable LTI system. The latter gives the minimax state estimator. In addition, we obtain a solution of infinite-horizon linear quadratic optimal control problem for DAEs.Comment: This is an extended version of the paper which is to appear in the proceedings of the 52nd IEEE Conference on Decision and Control, Florence, Italy, December 10-13, 201

Chebyshev minimax control theory

General, closed-form, analytical solutions are determined for certain classes of C-minimax control problems, several alternative mathematical theories are derived, and a controller design theory is developed to give optimal control in the presence of unmeasureable external disturbances

Towards Minimax Online Learning with Unknown Time Horizon

We consider online learning when the time horizon is unknown. We apply a minimax analysis, beginning with the fixed horizon case, and then moving on to two unknown-horizon settings, one that assumes the horizon is chosen randomly according to some known distribution, and the other which allows the adversary full control over the horizon. For the random horizon setting with restricted losses, we derive a fully optimal minimax algorithm. And for the adversarial horizon setting, we prove a nontrivial lower bound which shows that the adversary obtains strictly more power than when the horizon is fixed and known. Based on the minimax solution of the random horizon setting, we then propose a new adaptive algorithm which "pretends" that the horizon is drawn from a distribution from a special family, but no matter how the actual horizon is chosen, the worst-case regret is of the optimal rate. Furthermore, our algorithm can be combined and applied in many ways, for instance, to online convex optimization, follow the perturbed leader, exponential weights algorithm and first order bounds. Experiments show that our algorithm outperforms many other existing algorithms in an online linear optimization setting

Path-dependent Hamilton-Jacobi equations in infinite dimensions

We propose notions of minimax and viscosity solutions for a class of fully nonlinear path-dependent PDEs with nonlinear, monotone, and coercive operators on Hilbert space. Our main result is well-posedness (existence, uniqueness, and stability) for minimax solutions. A particular novelty is a suitable combination of minimax and viscosity solution techniques in the proof of the comparison principle. One of the main difficulties, the lack of compactness in infinite-dimensional Hilbert spaces, is circumvented by working with suitable compact subsets of our path space. As an application, our theory makes it possible to employ the dynamic programming approach to study optimal control problems for a fairly general class of (delay) evolution equations in the variational framework. Furthermore, differential games associated to such evolution equations can be investigated following the Krasovskii-Subbotin approach similarly as in finite dimensions.Comment: Final version, 53 pages, to appear in Journal of Functional Analysi