Stochastic evolution equations of jump type with random coefficients: existence, uniqueness and optimal control

Abstract

We study a class of stochastic evolution equations of jump type with random coefficients and its optimal control problem. There are three major ingredients. The first is to prove the existence and uniqueness of the solutions by continuous dependence theorem of solutions combining with the parameter extension method. The second is to establish the stochastic maximum principle and verification theorem for our optimal control problem by the classic convex variation method and dual technique. The third is to represent an example of a Cauchy problem for a controlled stochastic partial differential equation with jumps which our theoretical results can solve.

Almost sure stabilization of hybrid systems by feedback control based on discrete-time observations of mode and state

“…with the initial state x(0) = x 0 ∈ R n and mode r(0) = r 0 ∈ S, where f : R n × S → R n . As before, we assume that f meets conditions (9) and (11), namely there are constants K 1 > 0 and…”

“…In recent years, stochastic systems have been considered by many researchers since many practical systems can be modeled using these kinds of systems. Many significant results for stochastic systems have been reported (see [1][2][3][4][5][6][7][8][9][10][11][12][13]). Markovian jump systems are a special class of hybrid stochastic systems, which can be found in some engineering systems including power systems, manufacturing systems, ecosystems, and so forth.…”

Stochastic Evolution Equation Driven by Teugels Martingale and Its Optimal Control

“…In 2005, Øksendal, Prosk and Zhang [20] studied the optimal control problem of quasilinear semielliptic SPDEs driven by Poisson random measure and gave sufficient maximum principle results, not necessary ones. In 2017, Tang and Meng [30]studied the optimal control problem of more general stochastic evolution equations driven by Poisson random measure with random coefficients and gave necessary and sufficient maximum principle results.In this paper, for a controlled stochastic evolution equation (1.1), we suppose that the control domain is convex. We adopt the convex variation method and the first adjoint duality analysis to show a necessary maximum principle where the continuous dependence theorem (see Theorem 3.2) plays a key role in proving the variation inequality for the cost functional (see Lemma 6.2).…”

Optimal Control with State Constraints for Stochastic Evolution Equation with Jumps in Hilbert Space

“…In 2005, Øksendal, Proske, Zhang [12] studied the optimal control problem of quasilinear semielliptic SPDEs driven by Poisson random measure and gave sufficient maximum principle results, not necessary ones. In 2017, Tang and Meng [13] studied the optimal control problem for a controlled stochastic evolution equation (1.1) with the cost functional (1.2), where the control domain is assumed to be convex. [13] adopt the convex variation method and the first adjoint duality analysis to show a necessary maximum principle.…”

“…In 2017, Tang and Meng [13] studied the optimal control problem for a controlled stochastic evolution equation (1.1) with the cost functional (1.2), where the control domain is assumed to be convex. [13] adopt the convex variation method and the first adjoint duality analysis to show a necessary maximum principle. And Under the convexity assumption of the Hamiltonian and the terminal cost, a sufficient maximum principle for this optimal problem which is the so-called verification theorem is obtained…”