Hongwei Lou

Optimality Conditions for Semilinear Elliptic Equations with Leading Term Containing Controls

An optimal control problem for semilinear elliptic partial differential equations is considered. The equation is in divergence form with the leading term containing the control. Necessary conditions for optimal controls are established by the method of homogenized spike variation. The key to such a method is to modify the usual spike variational technique by taking into account the homogenization techniques for elliptic equations, together with an idea from the theory of relaxed controls. Problems with state constraints are also discussed by further adding some well-known penalty arguments involving the application of Ekelands variational principle and finite codimensionality of certain sets.

Maximum Principle of Optimal Control for Degenerate Quasi-Linear Elliptic Equations

Optimal control problems governed by degenerate quasi-linear partial differential equations of elliptic type are considered. The optimal control systems considered may lack Cesari-type conditions, and therefore the corresponding approximate optimal control problem may have no solution. To yield the maximum principle of optimal pairs, relaxed controls are used to overcome the difficulties occurring when considering approximate problems. The relaxed controls used are defined by finite additive measures so that the case of the control set being noncompact can be treated.