Research On The Optimal Control Problems To Variational Inequality

Variational inequality is an important part of nonlinear functional analysis.It is mainly used to solve practical problems such as multi-rigid body dynamic friction contact mechanics and hybrid engineering systems under inequality constraints or discontinuous jump boundary conditions.We will study the optimal control problems govern by several types of variational inequalities in reflexive B anach spaces,mainly including the optimal control problem govern by H-semi-variational inequalities,the optimal control problem govern by multi-value quasi-semivariational inequalities,the optimal control problem govern by development type variational inequalities and optimal control problem govern by hybrid quasi-equilibrium problems.(1)We study the optimal control problem govern by the H-semi-variational inequality.Firstly,we use the KKM lemma to prove the existence of the solution of the H-semi-variational inequality.Secondly,by establishing its dual problem,we prove the existence of the optimal solution by using the zero dual gap property.Finally,an corollary about the result is given in the end.This chapter extends the optimal control problem of variational inequality to the optimal control problem of H-semi-variational inequality.(2)We discuss the optimal control problem govern by a class of multivalued quasi-semivariational inequalities.We first prove the existence of solutions to multi-valued quasi-variational inequalities.then,we uses the zero duality gap property to obtain the optimal solution of the optimal control problem.In the end,some assumptions are put forward and a convergence result of the optimal solution of the optimal control system is obtained.(3)We study the optimal control problem govern by the developmental variational inequality.We prove the existence of the solution of the developmental variational inequality in the first,and secondly,we obtain the optimal solution of the optimal control problem by using the necessary conditions for the zero duality gap property.Then,some assumptions are put forward,and the result of the convergence of the optimal solution is obtained.Finally,on this basis,the regular optimization result of least squares is given.(4)We discuss the optimal control problem govern by a class of mixed quasi-equilibrium problems.In the first,we prove the existence of the solution of the equilibrium problem.then,combine the minimization sequence of the optimal problem with the property of the bounded sequence in the reflexive Banach space to obtain the necessary conditions for the zero duality gap property.Secondly,based on the existence of the optimal solution of the original problem,some assumptions are put forward to obtain the convergence result of the optimal solution of the optimal control system.Finally,the least squares regular optimization result of the equilibrium problem is obtained,and an example is given to explain the feasibility of the results.