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Higher eigenvalues of composite materials for anisotropic conductors are considered. To get the existence result for minimizing problems, relaxed problems are introduced by the homogenization method. Then, necessary conditions for minimizers are yielded. Based on the necessary conditions, it is shown that in some cases, optimal conductivities of relaxed minimizing problems can be replaced equivalently by a weighted harmonic mean of conductivities.

Blowup/Quenching time optimal control problems for controlled autonomous ordinary differential equations are considered. The main results are maximum principles for these time optimal control problems, including the transversality conditions.

An optimal control problem governed by semilinear elliptic partial differential equation is considered. The equation is in divergence form with the leading term containing controls. A relaxed problem is constructed by homogenization. By studying the G-closure problem, a local representation of admissible set of relaxed control is given. Finally, the maximum principle of relaxed problem is established via homogenization spike variation.

An optimal control problem for semilinear hyperbolic partial differential equations is considered. The control variable appears in coefficients. Necessary conditions for optimal controls are established by method of two-scale convergence and homogenized spike variation. Results for problems with state constraints are also stated.

An optimal control problem governed by semilinear elliptic partial differential equation is considered. The equation is in divergence form with the leading term containing controls. By studying the $G$-closure of the leading term, an existence result is established under a Cesari-type condition.

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 Ekeland's variational principle and finite codimensionality of certain sets.

The author proves that the right-hand term of a p-Laplace equation is zero on the singular set of a local solution to the equation. Such a result is used to study the existence of an optimal control problem.

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.

An optimal control problem governed by quasi-linear variational inequality is considered. The cost functional to be minimized contains the solution of a quasi-linear variational inequality. To get the existence, regularity, and necessary condition for the optimal pair, a new related control problem is introduced. By proving the existence of an optimal pair to such a new problem, the existence and regularity of the optimal pair to the original problem are obtained. It turns out that the regularity obtained is sharp in general. Some necessary conditions of the optimal pair are also obtained.

An optimal control problem of the obstacle for an elliptic variational inequality is considered, in which the obstacle is regarded as the control. To get the regularity of the optimal pair, a new related control problem is introduced. By proving the existence of an optimal pair to such a new control problem, the regularity of the optimal pair to the original problem is obtained. It turns out that the regularity obtained is sharp in general. Some other interesting properties of the optimal pair are also established.

An optimal control problem for semilinear parabolic partial differential equations is considered. The control variable appears in the leading term of the equation. Necessary conditions for optimal controls are established by the method of homogenizing spike variation. Results for problems with state constraints are also stated., 22 apges, 1 figures

An optimal control problem for seconde order semilinear parabolic partial differential equation is considered. The equation is in divergence form with the leading term containing the control. By the method of homogenizing spike variation, necessary conditions for optimal controls are established. The key to such a method is to modify the usual spike variational technique by taking into account the homogenization techniques for parabolic equations, together with some idea from the theory of relaxed controls. The method we used here can also treat problems with state constraints by adding some well‐known penalty arguments involving the application of Ekeland’s variational principle and finite co‐dimensionality of certain sets.

Sharp constants for an inequality of Poincare type are studied. The problem is solved by using optimal control theory.