TABLE OF CONTENTS Semidefinite programming is a natural language with which to pose and process numerous engineering problems associated with stability. Let us look at several examples.
Free motions of many linearly elastic mechanical systems, i.e., their behavior in absence of external loads, are governed by systems of differential equations of the type
where x(t) ? R n is the state vector of the system at time t, M is the (generalized) mass matrix, and A is the stiffness matrix of the system. Basically, (N) is the Newton law for a system with the potential energy 
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As a simple example, consider a system of k points of masses 1, ... , ? k linked by springs with given elasticity coefficients. Here x is the vector of the displacements x i ? R d of the points from their equilibrium positions e i ( d = 1/2/3 is the dimension of the model). The Newton equations become
where v ij are given by
where k ij > 0 are the elasticity coefficients of the springs. The resulting system is of the form (N) with a diagonal matrix M and a positive semidefinite symmetric matrix A. The well-known simplest system of this type is a pendulum (a single point able to slide along a given axis and linked by a spring to a fixed point on the...
