By Panos J. Antsaklis

In accordance with a streamlined presentation of the authors’ winning paintings Linear structures, this textbook presents an creation to structures concept with an emphasis on keep an eye on. preliminary chapters current helpful mathematical history fabric for a basic realizing of the dynamical habit of structures. every one bankruptcy contains priceless bankruptcy descriptions and directions for the reader, in addition to summaries, notes, references, and routines on the finish. The emphasis all through is on time-invariant structures, either non-stop- and discrete-time.

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**Example text**

87) reduces to φ(t, t0 , x0 ) φh (t) = Φ(t, t0 )x0 . 86) may be viewed as consisting of a component that is due to the initial conditions (t0 , x0 ) and another component that is due to the forcing term g(t). This type of separation is in general possible only in linear systems of diﬀerential equations. 86) and φh the homogeneous solution. 86), summarized by φ, is known for all t. 86) at time t. The components φi of φ, i = 1, . . 86). , A(t) and g(t) may have 32 1 System Models, Diﬀerential Equations, and Initial-Value Problems (at most) a ﬁnite number of discontinuities over any ﬁnite time interval].

This solution is continuable to the right in more than one way. 44) for t ≥ 0. 3) In the next result, ∂D denotes the boundary of a domain D and ∂J denotes the boundary of an interval J. 12. 11) on an open interval J, then φ can be continued to a maximal open interval J ∗ ⊃ J in such a way that (t, φ(t)) tends to ∂D as t → ∂J ∗ when ∂D is not empty and |t| + |φ(t)| → ∞ if ∂D is empty. The extended solution φ∗ on J ∗ is noncontinuable. 7]. When D = J ×Rn for some open interval J and f satisﬁes a Lipschitz condition there (with respect to x), we have the following very useful continuation result.

15, the sequence {φm }, m = 0, 1, 2, . . 77) on compact subsets of R. 79) where t Φ(t, t0 ) = I + t A(s1 )ds1 + t0 t + t0 s1 A(s1 ) t0 t + t0 s1 A(s2 )ds2 ds1 t0 s2 A(s2 ) A(s1 ) t0 s1 A(s1 ) A(s3 )ds3 ds2 ds1 + · · · t0 sm−1 A(s2 ) . . t0 A(sm )dsm dsm−1 · · · ds1 + · · · . 80) is called the Peano–Baker series. 80) we immediately note that Φ(t, t) = I. 77), we obtain that ˙ t0 ) = A(t)Φ(t, t0 ). 77) evolving in time t is known. 77). 77). We can specialize the preceding discussion to linear systems of equations x˙ = Ax.