Abnormal and Singular Solutions in the Target Guarding Problem with Dynamics

  • Matthew W. HarrisEmail author


The topic of this paper is a two-player zero-sum differential game known as the target guarding problem. After a brief review of Isaacs’ original problem and solution, a problem with second-order dynamics and acceleration control is considered. It is shown that there are four solution classes satisfying the necessary conditions. The four classes are (i) abnormal and non-singular, (ii) normal and non-singular, (iii) normal and pursuer singular, (iv) normal and evader singular. The normal and totally singular case is ruled out. Closed-form solutions are provided for cases ii–iv. The order of singularity in all cases is infinite. Thus, the problem exhibits many interesting properties: normality, abnormality, non-singularity, infinite-order singularity, and non-uniqueness. A practical example of each class is provided.


Abnormality Singularity Differential games Target guarding 



On the occasion of his retirement as editor of the Journal of Optimization Theory and Applications, I thank Dr. David G. Hull for his service to the journal and The University of Texas. More importantly, I thank him for his personal mentorship and friendship, and I wish him the best.


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Authors and Affiliations

  1. 1.Mechanical and Aerospace EngineeringUtah State UniversityLoganUSA

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