A class of stochastic hyperbolic evolution equations via weighted pseudo almost periodic coefficients and optimal controls

Abstract

Summary This paper introduces the concept of the piecewise weighted pseudo periodicity for stochastic processes. In addition, it establishes a new composition theorem for weighted pseudo almost periodic functions under the non‐Lipschitz conditions. Using this composition theorem, evolution family together with a fixed‐point theorem for condensing maps, we investigate the existence of p‐mean piecewise weighted pseudo almost periodic mild solutions and optimal mild solutions for a class of impulsive stochastic hyperbolic evolution equations in Hilbert spaces. Then, the existence conditions of optimal pairs of systems governed by the nonlinear impulsive stochastic evolution equations are presented. Finally, an example is provided to illustrate the obtained theory.

Square-mean piecewise almost automorphic mild solutions to a class of impulsive stochastic evolution equations

“…In [28], the authors introduced the concept of square-mean piecewise almost periodic functions for impulsive stochastic processes, and studied the existence and stability of square-mean piecewise almost periodic solutions for linear and nonlinear impulsive stochastic differential equations. Furthermore, Yan et al [31][32][33][34] discussed the square-mean piecewise pseudo almost periodic solutions and the square-mean piecewise weighted pseudo almost periodic solutions for impulsive stochastic evolution equations. However, there are few studies on the square-mean piecewise almost automorphic solutions of impulsive stochastic evolution equations.…”