A systematical method for complete stability problem of a class of delayed neural networks in parameter space

This paper presents a systematical method to address the complete stability problem of delayed neural networks with heterogeneous free parameters. First, we adopt an algebraic method to investigate the complete stability problem with respect to the free delay parameter <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>τ</mml:mi> </mml:math> . Then, the stability analysis is extended to the scenario with additional free system parameters, denoted by a vector <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> . We can investigate the complete root classification for the auxiliary characteristic equation in the entire <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi>τ</mml:mi> <mml:mo>)</mml:mo> </mml:math> -space. As a result, we can analytically calculate the number of stability <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>τ</mml:mi> </mml:math> -intervals and characterize all classifications of stability property over the whole <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi>τ</mml:mi> <mml:mo>)</mml:mo> </mml:math> -space. Finally, we will give a systematical method for determining the stability set in the whole <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi>τ</mml:mi> <mml:mo>)</mml:mo> </mml:math> -space. Some representative examples show the effectiveness of the approach.