# Numerical Inversion of the Laplace Transform: Applications by Bellman/Kalaba/Lockett PDF By Bellman/Kalaba/Lockett

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The scope of the book has also been outlined at the end of this chapter. Exercises 1. Suppose a person P evaluates his emotion using three psychological state variables x, y and z. Let the positive support of P on x, y and z be F1, and the strong positive support of P on x, y and Z be F2, where F1(x, y, z) = x2+ y2+ z2 , and F2(x, y, z) = (x + y + z)2. Show that F2 is unconditionally greater than F1. [Hints: F2(x, y, z) = (x + y + z)2 = x2 + y2 + z2 + 2 (xy + yz + zx) ≥ (x2 + y2 + z2) for x, y and z to be real numbers.

For higher dimensional systems, the conditions of equilibrium are slightly different. It is thus apparent that the condition for equilibrium of a physical system can be derived from the basic laws of statics introduced above. A logical system, which too is free from the notion of time, is a static system, but the laws that govern the equilibrium of the interpretation of the logical statements are different from the laws of mechanics (statics). The examples below provide an insight to the equilibrium of a mechanical and a logical system.

For convenience of the readers, we briefly outline the rule of stability analysis. , xn), i =1 to n. 19) We construct a Lyapunov energy function L(x1, x2, …, xn), and check whether dL/dt <0 for the given dynamics. If it is so, the dynamics is said to be stable in the Lyapunov sense. We now present an illustrative neural system called Hopfield dynamics to study the Lyapunov approach for stability analysis for continuous systems. 20) for i=1 to n. Here, C, βi and wij are system parameters. θi is the input, xj is the system state, and ui is the stored signal .