/home/bill/Neural_Nets/My Reviews/0_Lyapunov functionals.txt

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Paper review for IJCNN2014 Beijing : 
/home/bill/a_IJCNN14 Beijing/My Reviews/N-14155 p Yang, Liu, Wei - Near-Optimal Online Control of Uncertain Nonlinear Continuous-Time Systems Based on Concurrent Learning.pdf
p2c2h0.85 Equation (11)
   J(t) = J1(t) + J2(t) 
   J1(t) = 1/2*x_t_T*P*x_t
   J2(t) = 1/2*tr(W1_t_T*l1_inv*W1_t) + 1/2*tr(V1_t_T*l2_inv*V1_t)
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Paper review for IJCNN2014 Beijing : 
/home/bill/a_IJCNN14 Beijing/Others reviews/N-14324 Yi, Lu, Chen p - Stability of Hopfield Neural Networks with Event-triggered Feedbacks.pdf 
p3c1h0.0.5 Equation (4) : 
L(x) = 
      Sum { i=1 to m of : 
          + 1/(lambda_i*R_i)) * integral{ds from s=0 to g_i(lambda_i*x_i) of g_i_inv(s)} 
          - theta_i*g_i(lambda_i*x_i) 
          - 1/2*Sum{j=1 to m of : w_ij*g_i(lambda_i*x_i)*g_j(lambda_j*x_j)}
          } 
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Neural Networks paper review
/home/bill/Neural_Nets/My Reviews/NEUNET-D-13-00401 p He etal - Anti-Windup for time-varying delayed CNNs subject to Input Saturation.pdf
p3h0.8 Define a Lyapunov–Krasovskii functional (variable time delays)
   V(x_t) = V1 + V2     where 
   V1 = x_T(t)*P_inv*x(t)
   V2 = integl{ d_theta from (t - tau(t)) to t,  x_T(theta)*P_inv*x(theta) }
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ICICIP 54 m Sirisongkol, Liu - Stability Analysis of RNNs with Time-Varying Delay and Disturbances via Quadratic Convex Technique.txt
09Jun2014 

(8) V(x(t)) = sum(i=1 to 4: V_i(x(t))

(9) Lyapunov-Krasovski (L-K) functional candidate : 
   V1(x(t)) = sigma1_T(t)*P*sigma1(t)
   V2(x(t)) = 2*sum(i=1 to n:lambda_i*integral(ds from 0 to x_i(t): g_i(s) )
   V3(x(t)) = integral(ds from (t-d(t)) to t: 
                 [ + sigma2_T(t,s)*Q1*sigma2(t,s) 
                   + g_T(x(s))*S*g(x,s)
                 ]
   V4(x(t)) = integral(ds from (t-h) to t: 
                 [ +               sigma2_T(t,s)*Q2*sigma2(t,s) 
                   + (h - t + s)^1*sigma3_T(t,s)*Q3*sigma3(t,s) 
                   + (h - t + s)^2* x_dot_T(s)  *R1* x_dot(s)
                   + (h - t + s)^3* x_dot_T(s)  *R1* x_dot(s)
                 ]
.where 
   sigma1_T(t)   = [x_T(t)      integral(ds from (t-h) to t: x_T(s) ) ]
   sigma2_T(t,s) = [x_T(t)      x_T(s)                                ] 
   sigma3_T(t,s) = [x_T(t)      x_dot_T(s)                            ]

p2c2h0.85 Remark 1: [29] Our newly constructed function, compared with the ones in the literature, has three differences:
1) An independent augmented variable integral(ds from (t-h) to t: x_T(s) );
2) The cross terms between entries in sigma1_T(t), sigma2_T(t,s), sigma3_T(t,s) respectively;
3) Quadratic terms multiplied by first, second, and third degrees of a scalar function (h-t+s), where the degree increase (h-t+s) means the number increase of the integral by 1.
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