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• W2076231114 abstract "Planar linear discrete systems with constant coefficients and delays<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M1><mml:mi>x</mml:mi><mml:mrow><mml:mo stretchy=false>(</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn mathvariant=normal>1</mml:mn></mml:mrow><mml:mo stretchy=false>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>A</mml:mi><mml:mi>x</mml:mi><mml:mrow><mml:mo stretchy=false>(</mml:mo><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mo stretchy=false>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:msubsup><mml:mo stretchy=false>∑</mml:mo><mml:mrow><mml:mi>l</mml:mi><mml:mo>=</mml:mo><mml:mn mathvariant=normal>1</mml:mn></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:mi mathvariant=normal>‍</mml:mi></mml:mrow></mml:mrow><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=false>(</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy=false>)</mml:mo></mml:mrow></mml:math>are considered where<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M2><mml:mi>k</mml:mi><mml:msubsup><mml:mrow><mml:mo>∈</mml:mo><mml:mi>ℤ</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant=normal>0</mml:mn></mml:mrow><mml:mrow><mml:mi mathvariant=normal>∞</mml:mi></mml:mrow></mml:msubsup><mml:mo>:</mml:mo><mml:mo>=</mml:mo><mml:mo stretchy=false>{</mml:mo><mml:mn mathvariant=normal>0,1</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mi mathvariant=normal>∞</mml:mi><mml:mo stretchy=false>}</mml:mo></mml:math>,<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M3><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant=normal>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant=normal>2</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math>are constant integer delays,<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M4><mml:mn mathvariant=normal>0</mml:mn><mml:mo><</mml:mo><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant=normal>1</mml:mn></mml:mrow></mml:msub><mml:mo><</mml:mo><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant=normal>2</mml:mn></mml:mrow></mml:msub><mml:mo><</mml:mo><mml:mo>⋯</mml:mo><mml:mo><</mml:mo><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math>,<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M5><mml:mi>A</mml:mi><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant=normal>1</mml:mn></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:math>are constant<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M6><mml:mn mathvariant=normal>2</mml:mn><mml:mo>×</mml:mo><mml:mn mathvariant=normal>2</mml:mn></mml:math>matrices, and<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M7><mml:mi>x</mml:mi><mml:mo>:</mml:mo><mml:msubsup><mml:mrow><mml:mi>ℤ</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mo>-</mml:mo><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi mathvariant=normal>∞</mml:mi></mml:mrow></mml:msubsup><mml:msup><mml:mrow><mml:mo>→</mml:mo><mml:mi>ℝ</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant=normal>2</mml:mn></mml:mrow></mml:msup></mml:math>. It is assumed that the considered system is weakly delayed. The characteristic equations of such systems are identical with those for the same systems but without delayed terms. In this case, after several steps, the space of solutions with a given starting dimension<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M8><mml:mn mathvariant=normal>2</mml:mn><mml:mo stretchy=false>(</mml:mo><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mn mathvariant=normal>1</mml:mn><mml:mo stretchy=false>)</mml:mo></mml:math>is pasted into a space with a dimension less than the starting one. In a sense, this situation is analogous to one known in the theory of linear differential systems with constant coefficients and special delays when the initially infinite dimensional space of solutions on the initial interval turns (after several steps) into a finite dimensional set of solutions. For every possible case, explicit general solutions are constructed and, finally, results on the dimensionality of the space of solutions are obtained." @default.
• W2076231114 created "2016-06-24" @default.
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• W2076231114 date "2014-01-01" @default.
• W2076231114 modified "2023-10-03" @default.
• W2076231114 title "General Explicit Solution of Planar Weakly Delayed Linear Discrete Systems and Pasting Its Solutions" @default.
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• W2076231114 doi "https://doi.org/10.1155/2014/627295" @default.
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