The Necessary Conditions for Optimal Hybrid Systems of Variable Dimensions

2021 ◽  
Vol 60 (6) ◽  
pp. 883-894
Author(s):  
A. S. Bortakovsky
Keyword(s):  
Hybrid Systems ◽  
Necessary Conditions

scholarly journals FRAMEWORKS SYMMETRY AND RIGIDITY

2010 ◽  
Vol 20 (06) ◽  
pp. 723-750 ◽  
Author(s):  
J. C. OWEN ◽  
S. C. POWER
Keyword(s):  
Symmetry Group ◽  
Hybrid Systems ◽  
Short Proof ◽  
Necessary Conditions ◽  
Character Formula ◽  
Rigidity Matrix ◽  
Counting Rules ◽  
Point Line

Symmetry equations are obtained for the rigidity matrix of a bar-joint framework in ℝd. These form the basis for a short proof of the Fowler-Guest symmetry group generalisation of the Calladine-Maxwell counting rules. Similar symmetry equations are obtained for the Jacobian of diverse framework systems, including constrained point-line systems that appear in CAD, body-pin frameworks, hybrid systems of distance constrained objects and infinite bar-joint frameworks. This leads to generalised forms of the Fowler-Guest character formula together with counting rules in terms of counts of symmetry-fixed elements. Necessary conditions for isostaticity are obtained for asymmetric frameworks, both when symmetries are present in subframeworks and when symmetries occur in partition-derived frameworks.

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