Optimal control of semi-Markov processes with a backward stochastic differential equations approach

Abstract

We consider a general class of operator-valued irrational positive-real functions with an emphasis on their frequency-domain properties and the relation with stabilization by output feedback. Such functions arise naturally as the transfer functions of numerous infinite-dimensional control systems, including examples specified by PDEs. Our results include characterizations of positive realness in terms of imaginary axis conditions, as well as characterizations in terms of stabilizing output feedback, where both static and dynamic output feedback are considered. In particular, it is shown that stabilizability by all static output feedback operators belonging to a sector can be characterized in terms of a natural positive-real condition and, furthermore, we derive a characterization of positive realness in terms of a mixture of imaginary axis and stabilization conditions. Finally, we introduce concepts of strict and strong positive realness, prove results which relate these notions and analyse the relationship between the strong positive realness property and stabilization by feedback. The theory is illustrated by examples, some arising from controlled and observed partial differential equations.

On the Solution of Locally Lipschitz BSDE Associated to Jump Markov Process

“…To fulfill this, we need to impose the following sufficient condition, f (s, X s , 0, 0) ≥ 0. Indeed, to benefit from the comparison Theorem for globally Lipschitz BSDE, we first construct a sequence of globally Lipschitz generators (f n ) n∈N , by replacing f in BSDE (4.1) by f n defined in Lemma (3), to obtain the following family of approximating BSDEs Z n r (y)q(dr dy), s ∈ [t, T ] , (4.2) In view of Theorem (1), BSDE (4.2) has a unique solution (Y n , Z n ), for each integer n. Since h (X T ) ≥ 0, and f n ≥ 0, the comparison Theorem (cf.Theorem 3.9 in [6]), leads to Y n ≥ 0. Using similar arguments in the proof of Theorem (2), one can easily show that (Y n , Z n ) is a Cauchy sequence then it converges to (Y, Z) .…”

Optimal control for stochastic Volterra equations with multiplicative Lévy noise

“…Nonlinear BSDEs were first introduced by Pardoux and Peng [44] and are currently used in the field of the stochastic control theory : see, e.g., [29,55]. Recently BSDEs driven by random measures have been introduced to solve optimal control problem for marked point processes [16,17,20,18,3].…”

The identification problem for BSDEs driven by possibly non quasi-left-continuous random measures

“…BSDEs with a discontinuous driving term of the form (1.1) have been studied as well; in almost all cases, the random measure µ is quasi-left-continuous, i.e. µ({S} × R) = 0 on {S < ∞} for every predictable time S, see, e.g., [34], [12], [33], [5], [4]. Existence and uniqueness for BSDEs driven by a random measure which is not necessarily quasi-left-continuous are very recent, and were discussed in [1] in the purely discontinuous case, and in [27] in the jump-diffusion case.…”