From averaged to simultaneous controllability

Abstract

International audienceWe consider a linear finite dimensional control system depending on unknown parameters. We aim to design controls, independent of the parameters, to control the system in some optimal sense. We discuss the notions of averaged control, according to which one aims to control only the average of the states with respect to the unknown parameters, and the notion of simultaneous control in which the goal is to control the system for all values of these parameters. We show how these notions are connected through a penalization process. Roughly, averaged control is a relaxed version of the simultaneous control property, in which the differences of the states with respect to the various parameters are left free, while simultaneous control can be achieved by reinforcing the averaged control property by penalizing these differences. We show however that these two notions require of different rank conditions on the matrices determining the dynamics and the control. When the stronger conditions for simultaneous control are fulfilled, one can obtain the later as a limit, through this penalization process, out of the averaged control property

Averaged controllability of parameter dependent conservative semigroups

“…This issue is analysed in [29], where this idea is discussed in detail in the finite-dimensional control context. , F ζ is solution of:…”

“…Obviously, the later requires also the control of all possible parameter dependent states, and not only the control of their average. In the concluding section, we will show the link between these two notions via penalization, an issue that is treated in more detail in [29]. This procedure, quickly explained in the concluding remarks, is similar to the one implemented by J.-L. Lions in [28] in order to link optimal control and approximate controllability for the heat equation.…”

Convergence results for an averaged LQR problem with applications to reinforcement learning

“…We stress that a GP is a probability distribution on the set X of bounded, continuous functions, and thus, it constitutes a perfect candidate to play the role of π in Problem B. In the policy improvement step, the optimal policy is then computed minimizing the averaged cost over all possible realizations of the GP, in a similar way to how we defined the cost functional in (16).…”

“…A general, rigorous framework capturing PILCO as well as other Bayesian model-based RL approaches (see, e.g., [3,4,[10][11][12]29]) has been developed in [18,19]. In particular, it is important to mention that the framework developed in [18] is closely related to the averaging control framework and Riemann-Stieltjes optimal control [2,16,21,24,30].…”

Output controllability in a long-time horizon

“…It is obvious that if the system (4) is simultaneously controllable, then it is controllable in average. More detailed relations between these notions have been studied in [6], in which the authors analyse deviations of each system component from the averaged value. To this effect they identify the optimal averaged control as the one minimising a quadratic functional which, together with the control norm, contains a penalisation term measuring deviation of each system component from the average.…”