Cesari-type conditions for semilinear elliptic equation with leading term containing controls
Bo Li Hongwei Lou
Mathematical Control & Related Fields 2011, 1(1): 41-59 doi: 10.3934/mcrf.2011.1.41
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.
keywords: existence condition homogenization Cesari-type condition. elliptic equation Optimal control
Optimal blowup/quenching time for controlled autonomous ordinary differential equations
Hongwei Lou Weihan Wang
Mathematical Control & Related Fields 2015, 5(3): 517-527 doi: 10.3934/mcrf.2015.5.517
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.
keywords: optimal quenching time Optimal blowup time transversality condition. maximum principle
Second-order necessary conditions for optimal control of semilinear elliptic equations with leading term containing controls
Hongwei Lou Jiongmin Yong
Mathematical Control & Related Fields 2018, 8(1): 57-88 doi: 10.3934/mcrf.2018003

An optimal control problem for a semilinear elliptic equation of divergenceform is considered. Both the leading term and the semilinear term of the state equationcontain the control. The well-known Pontryagin type maximum principle for the optimal controls is the first-order necessary condition. When such a first-order necessary condition is singular in some sense, certain type of the second-order necessary condition will come in naturally. The aim of this paper is to explore such kind of conditions for our optimal control problem.

keywords: Optimal control semilinear elliptic equation control in leading term second-order necessary conditions
Second-order necessary/sufficient conditions for optimal control problems in the absence of linear structure
Hongwei Lou
Discrete & Continuous Dynamical Systems - B 2010, 14(4): 1445-1464 doi: 10.3934/dcdsb.2010.14.1445
Second-order necessary conditions for optimal control problems are considered, where the "second-order" is in the sense of that Pontryagin's maximum principle is viewed as a first-order necessary optimality condition. A sufficient condition for a local minimizer is also given.
keywords: ordinary differential equations. optimal control second-order necessary conditions sufficient conditions
Minimization of the elliptic higher eigenvalues for multiphase anisotropic conductors
Hongwei Lou Xueyuan Yin
Mathematical Control & Related Fields 2018, 8(3&4): 855-877 doi: 10.3934/mcrf.2018038

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.

keywords: Optimal control homogenization higher eigenvalue anisotropic conductor harmonic mean
Preface: A tribute to professor Jiongmin Yong on his 60th birthday
Hongwei Lou Qi Lü Gengsheng Wang Xu Zhang
Mathematical Control & Related Fields 2018, 8(3&4): ⅰ-ⅰ doi: 10.3934/mcrf.201803i
Time optimal control problems for some non-smooth systems
Hongwei Lou Junjie Wen Yashan Xu
Mathematical Control & Related Fields 2014, 4(3): 289-314 doi: 10.3934/mcrf.2014.4.289
Time optimal control problems for some non-smooth systems in general form are considered. The non-smoothness is caused by singularity. It is proved that Pontryagin's maximum principle holds for at least one optimal relaxed control. Thus, Pontryagin's maximum principle holds when the optimal classical control is a unique optimal relaxed control. By constructing an auxiliary controlled system which admits the original optimal classical control as its unique optimal relaxed control, one get a chance to get Pontryagin's maximum principle for the original optimal classical control. Existence results are also considered.
keywords: optimal quenching time maximum principle monotonicity. existence Optimal blowup time

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