Objective function design for robust optimality of linear control under state-constraints and uncertainty

Abstract

Abstract. We consider a model for the control of a linear network flow system with unknown but bounded demand and polytopic bounds on controlled flows. We are interested in the problem of finding a suitable objective function that makes robust optimal the policy represented by the so-called linear saturated feedback control. We regard the problem as a suitable differential game with switching cost and study it in the framework of the viscosity solutions theory for Bellman and Isaacs equations.Mathematics Subject Classification. 49L25, 49N90, 90C35.

Mean-Field Games and Dynamic Demand Management in Power Grids

“…The function m[·] is continuous in time, andm is symmetric with respect to its zero mean (note that f (x, 0) = −f (x, 1)). Recalling that f (0, u * (0, t)) = f (0, 1/2) = 0, we can infer that such a symmetric and mean zero-valued distribution is a solution of (6). To see this observe that…”

“…In this paper we adopt the same perspective in that we show that network frequency stabilization can be achieved by giving incentives to the agents to adjust their strategies in order to converge to a a mean field equilibrium. To do this, in the spirit of prescriptive game theory [6], a central planner or game designer has to design the individual objective function so to penalize those agents that are in ON state in peak hours, as well as those who are in OFF state in off-peak hours.…”

“…Several examples of optimal control problems of this kind are studied, for instance, in [4], [5] and [6]. In particular, it is proved that the value function is characterized as the unique viscosity solutions of a system of Hamilton-Jacobi equations (one per every branch), suitably coupled via the boundary conditions.…”

Game Theoretic Decentralized Feedback Controls in Markov Jump Processes

“…In the same spirit as in inverse control problems, [10], we provide an explicit expression of the running cost function in order to obtain our desired optimal feedback control as solution of the optimal control problem.…”

Infinite horizon optimal control problems with multiple thermostatic hybrid dynamics

“…Such discontinuities may lead to nonexistence of trajectories and (in a dynamic programming framework) to discontinuous Hamilton-Jacobi equations. The insertion of thermostatic approximations of such discontinuities may certainly help in treating such problems (see for instance Liberzon [18] and, for applications to a problem with uncertainty, Bagagiolo-Bauso [7]). …”