Existence and stability analysis for nonlinear optimal control problems with $1$-mean equicontinuous controls
Hongyong Deng Wei Wei
Journal of Industrial & Management Optimization 2015, 11(4): 1409-1422 doi: 10.3934/jimo.2015.11.1409
In this paper, the existence and stability of solutions of nonlinear optimal control problems with $1$-mean equicontinuous controls are discussed. In particular, a new existence theorem is obtained without convexity assumption. We investigate the stability of the optimal control problem with respect to the right-hand side functions, which is important in computational methods for optimal control problems when the function is approximated by a new function. Due to lack of uniqueness of solutions for an optimal control problem, the stability results for a class of optimal control problems with the measurable admissible control set is given based on the theory of set-valued mappings and the definition of essential solutions for optimal control problems. We show that the optimal control problems, whose solutions are all essential, form a dense residual set, and so every optimal control problem can be closely approximated arbitrarily by an essential optimal control problem.
keywords: Optimal control stability essential solution. existence set-valued mapping
A fully non-linear PDE problem from pricing CDS with counterparty risk
Bei Hu Lishang Jiang Jin Liang Wei Wei
Discrete & Continuous Dynamical Systems - B 2012, 17(6): 2001-2016 doi: 10.3934/dcdsb.2012.17.2001
In this study, we establish a financial credit derivative pricing model for a contract which is subject to counterparty risks. The model leads to a fully nonlinear partial differential equation problem. We study this PDE problem and obtained a solution as the limit of a sequence of semi-linear PDE problems which also arise from financial models. Moreover, the problems and methods build a bridge between two main risk frameworks: structure and intensity models. We obtain the uniqueness, regularities and some properties of the solution of this problem.
keywords: structure model fully non-linear PDE. intensity model Credit derivative pricing counterparty risk
A unified theory of maximum principle for continuous and discrete time optimal control problems
Zaidong Zhan Shuping Chen Wei Wei
Mathematical Control & Related Fields 2012, 2(2): 195-215 doi: 10.3934/mcrf.2012.2.195
Traditionally, the time domains that are widely used in mathematical descriptions are limited to real numbers for the case of continuous-time optimal control problems or to integers for the case of discrete-time optimal control problems. In this paper, based on a family of "needle variations", we derive maximum principle for optimal control problem on time scales. The results not only unify the theory of continuous and discrete optimal control problems but also conclude problems involving time domains in partly continuous and partly discrete ingredients. A simple optimal control problem on time scales is discussed in detail. Meanwhile, the results also unify the theory of some hybrid systems, for example, impulsive systems.
keywords: maximum principle Ekeland's variational principle optimal control Time scales impulsive control.
Honglei Xu Cheng-Chew Lim Wei Wei
Discrete & Continuous Dynamical Systems - B 2011, 16(4): i-i doi: 10.3934/dcdsb.2011.16.4i
This special issue of Discrete and Continuous Dynamical Systems Series B is dedicated to Professor Kok Lay Teo and Professor Jie Sun for their fundamental contributions to optimization and optimal control and their computational methods and applications. The 2010 International Conference on Optimization and Control (ICOCO2010) was held at Guizhou Park Hotel in Guiyang, China on July 18-23, 2010, in honour of Professors Teo and Sun on their 65th birthdays.

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