In the sequel we use for simplicity the term quasi-definite linear functional (

positive definite linear functional) for linear functionals that are quasi-definite (

positive definite) on the space of polynomials of sufficiently large degree.

It is clear that [C.sub.k], [C.sup.*.sub.k], [C.sub.d], and [C.sup.*.sub.d] are

positive definite and [C.sub.h], [C.sub.hk], and [C.sub.hd] are nonnegative definite.

Given a

positive definite matrix R and three positive constants [c.sub.1], [c.sub.2], [T.sub.f] ([c.sub.1] < [c.sub.2]), the polynomial fuzzy system (3) with control input u(t) is said to be finite-time stabilization with respect to [mathematical expression not reproducible].

Let X be the Hermitian

positive definite solution of (1).

if [rho](A) < 1, C is a fixed

positive definite matrix, matrix H solves the corresponding Lyapunov matrix equation (5), and

Equation (15) is a matrix Lyapunov equation where matrices P and Q are

positive definite matrices.

If [A.sub.d] is

positive definite and E is negative definite or [A.sub.d] is negative definite and E is

positive definite, then the S-system in (1) is stable.

The Ghost-Fluid Method leads to a symmetric

positive definite linear systems that captures the discontinuity in the normal derivative while smearing out the discontinuity in the tangential direction to the interface.

where S [member of] [R.sup.nxn] is a

positive definite symmetric matrix and T [member of] [R.sup.nxn] is a unity upper triangular matrix.

In classical analysis a complex valued continuous function is said

positive definite (resp.

If there exists symmetric

positive definite matrix Q, such that the following LMI holds:

For two symmetric matrices of the same dimensions X and Y, X > Y means that X-Y is

positive definite. [R.sup.+] is the set of positive real numbers.