Estimates for the controls of the wave equation with a potential

Abstract

This article studies the L2-norm of the boundary controls for the one dimensional linear wave equation with a space variable potential a = a(x). It is known these controls depend on a and their norms may increase exponentially with ||a||L∞. Our aim is to make a deeper study of this dependence in correlation with the properties of the initial data. The main result of the paper shows that the minimal L2−norm controls are uniformly bounded with respect to the potential a, if the initial data have only sufficiently high eigenmodes.

Constrained BSDEs Driven by a Non-Quasi-Left-Continuous Random Measure and Optimal Control of PDMPs on Bounded Domains

“…Arguing as in Proposition 2.2 in [3], one can easily see that under P x α the process X in (2.1) is Markovian with respect to F. In particular, for every n ≥ 1, the conditional survival function of the inter-jump time…”

“…It remains to prove (5.10). This can be done proceeding as in the proof of Proposition 5.6 in [3], that is as in the context of PDMPs with no jumps from the boundary, since the presence of predictable jumps does not induce here any additional technical difficulty.…”

“…In the non-diffusive framework, the correct formulation of the randomization method requires some efforts and different techniques from the diffusive case, since the controlled process is naturally described only in terms of its local characteristics and not as a solution to some stochastic differential equation. A first step in the generalization of the randomization method to the non-diffusive framework was done in [4], where a probabilistic representation for the value function associated to an optimal control problem for pure jump Markov processes was provided; afterwards, the randomization techniques have been implemented in [3] to solve PDMPs optimal control problems on unbounded state spaces (notice that in both [4] and [3] the jump measure of the controlled state process is quasi-left continuous). In the present paper we are interested to extend those results to the case of optimal control problems for PDMPs on bounded state spaces, where additional forced jumps appear whenever the process hits the boundary.…”