that is, there exists a matrix function such that.
where is a positive diagonal matrix and is a nonnegative matrix function such that,.
Furthermore, condition (3.28) assures the existence of a matrix function such that condition (3.18) is fulfilled.
Assume there exists a matrix function such that the matrix function is invertible, and the following exponential function on a time scale is bounded: (2.3).
For a matrix-valued function (Phiin H^{2}_{M_{ntimes r}}), we say that (Deltain H^{2}_{M_{ntimes m}}) is a left inner divisor of Φ if Δ is an inner matrix function such that (Phi=Delta A) for some (A in H^{2}_{M_{mtimes r}}).
Suppose that (q(t)in C^{2} ( [ 0,1 ], operatorname{End} C^{ell} )) is a matrix function such that for each (in [ 0,1 ]), (q(t)) has ℓ distinct nonzero simple eigenvalues (mu_{1} ( t ), ldots, mu_{ell} (t)) in the complex plane such that (mu_{j} (t)in C^{2} [ 0,1 ]) for (j=1, ldots, ell).
Also, decorin is known to be a functional component of the extracellular matrix with biological functions such as regulating collagen fibrillogenesis and controlling cell proliferation by upregulating p21 [ 22, 23].
Then, there exist sets of nonunique controller gain matrix functions defined by such that the closed-loop system (3.1)–(3.3) is globally asymptotically stable independent of the sizes of the delays provided that with defined in (3.12) and is the stability abscissa of the uncontrollable (stable and invariant under state feedback) dominant eigenvalue of.
Then, there exist sets of nonunique controller gain matrix functions defined by such that the closed-loop system (3.1)–(3.3) is globally asymptotically stable independent of the sizes of the delays and has an arbitrary spectrum of distinct eigenvalues of modulus not larger than a prescribed upper bound.
