/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 Lyapunovâ€“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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enddoc