By Hall E.H.
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Additional info for Electric Conductivity and Optical Absorption of Metals
3! ∞ T The improper integral P = 0 eA t QeAt dt is finite, since each integrating term tends to zero. Matrix P, because of the properties of theorem 5, is the integral of a positive definite matrix for each t and therefore is positive definite. Note that matrix P is also symmetrical (if Q is symmetrical). 4) can be verified. By substituting P into the Lyapunov equation we obtain: ∞ AT ∞ T eA t QeAt dt + 0 T eA t QeAt dtA = 0 ∞ T T (AT eA t QeAt + eA t QeAt A)dt = = 0 ∞ = 0 T d AT t At (e Qe )dt = eA t QeAt dt where we used the fact that lim eAt = 0 t→∞ and that eA0 = I.
A2 0 0 0 0 .. 96. As expected at least one of the minors is not positive (H3 < 0 and H4 < 0). 4) has only one solution. 5) Fundamentals of stability 19 where A, B and C are matrices of Rn×n (in the most general case, they are non-symmetrical), and X is a n × n unknown matrix. Even here, the equation can be vectorized: x11 c11 x12 c12 x13 M = − c13 .. .. . . xnn cnn 2 2 where M is a suitable matrix of Rn ×n (if B = AT and C is a symmetrical and M ∈ Rk×k ).
Since P is a positive definite matrix, V (x) is a positive definite function. The derivate against time of this function is: 16 Optimal and Robust Control: Advanced Topics with MATLAB V˙ (x) = x˙ T Px + xT Px˙ Since x˙ = Ax, then: V˙ (x) = xT AT Px + xT PAx = = xT (AT P + PA)x = −xT Qx Since Q is a positive definite matrix, V˙ (x) is a negative definite function. Function V (x) = xT Px satisfies the hypotheses of Lyapunov II criterion for dynamical systems and thus the equilibrium point (and so the system) is asymptotically stable.