This paper investigates the guaranteed cost control problem and guaranteesthe cost Nash equilibrium for discrete-time Markovian jump Linearsystem (MJLS) with incomplete transition rates. By employing matrixinequalities and the free-connection weighting matrix approach, severalsufficient conditions are proposed to ensure the applicability of theguaranteed cost control strategy. Moreover, we provide sufficient conditionsto highlight further the correctness and lower conservativeness ofthe utilized theorems. Ensuring these conditions for the existence of aguaranteed cost Nash equilibrium affords to transform them into an optimizationproblem that simultaneously satisfies a set of BMI and matrixinequalities. Furthermore, to make the results more general, this paperdesigns guaranteed cost Nash equilibrium strategies for multiple players.Finally, two detailed numerical examples validate the paper’s theorems.

Aiming at the objective uncertainty, subjective uncertainty, and extreme events may be in a dynamic system simultaneously. This paper focuses on the differential game problem of a linear quadratic jump uncertain stochastic system. The system is described by both a jump uncertain differential equation and a stochastic differential equation. The principle of optimality and equation of optimality are established. Then, a differential game model based on linear quadratic jump uncertain stochastic system is constructed. Furthermore, Nash equilibrium are discussed by using the obtained equation. Finally, its application in the dynamic investment decision of enterprises is given, and numerical simulations are performed. This approach offers a new method for quantitative analysis in future studies.